Special issue of ADAM devoted to The International Conference and PhD-Master Summer School “Groups and Graphs, Designs and Dynamics”, 2019 This special issue collects nine papers from participants of The International Conference and PhD-Master Summer School “Groups and Graphs, Designs and Dynamics” (G2D2) (https://math.sjtu.edu.cn/conference/G2D2/), held at the Three Gorges Mathematical Research Center in Yichang, China, August 12-25, 2019. Our authors come from Belarus, China, India, Myanmar, Russia and the United Kingdom, and their papers discuss various aspects of groups and graphs. We would like to thank the authors and refer- ees for their valuable contributions. We are also grateful to the Three Gorges Mathematical Research Center for hosting this two-weeks international event. The G2-series is about strong and beautiful mathematics involving group actions on combinatorial objects. It has the Cayley graph Cay(Z8, {±1,±2}) as its logo and has been an annual event since 2014; see the webpage of the seventh one G2G2 for its history: https://conferences.famnit.upr.si/event/13/. Besides the nine papers presented in this special issue, lecturers of the four short courses of the G2D2 summer school, coming from Italy, Japan, the United Kingdom, and the United States, have pre- pared four sets of nice lecture notes, and our more than 150 G2D2 participants from around the globe have been communicating an incredibly rich variety of mathematics during and after the event. Following the poet William Blake, let us hope that this special issue will be a grain of sand in which our reader may see the world of G2D2 2019. Guest Editors: Alexander Ivanov, a.ivanov@imperial.ac.uk Elena Konstantinova, e konsta@math.nsc.ru Jack Koolen, koolen@ustc.edu.cn Yaokun Wu, ykwu@sjtu.edu.cn ISSN 2590-9770 The Art of Discrete and Applied Mathematics 4 (2021) #P2.01 https://doi.org/10.26493/2590-9770.1335.2fa (Also available at http://adam-journal.eu) On strictly Deza graphs derived from the Berlekamp-Van Lint-Seidel graph ⇤ Soe Soe Zaw School of Mathematical Sciences and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, P. R. China, and Department of Mathematics, Dagon University, Yangon 11422, Myanmar Received 30 October 2019, accepted 22 December 2019, published online 10 March 2021 Abstract In this paper, we find strictly Deza graphs that can be obtained from the Berlekamp-Van Lint-Seidel graph by dual Seidel switching. Keywords: Dual Seidel switching, Berlekamp-Van Lint-Seidel graph, divisible design graph, Deza graph. Math. Subj. Class.: 05C25, 05E30 1 Introduction Goryainov et al. [8] gave a characterisation of strictly Deza graphs with parameters (n, k, k 1, a) and = 1. They found that such strictly Deza graphs necessarily come from strongly regular graphs having the property µ = 1 and can be obtained via two operations: strong product with an edge and the dual Seidel switching [9]. We are still far away from getting a classification of strongly regular graphs with µ = 1 [1]. It is known that if = 0 and µ = 1, then such a strongly regular graph is either the pentagon, or the Petersen graph, or the Hoffman-Singleton graph, or a hypothetical strongly regular graph with parameters (3250, 57, 0, 1). Berlekamp et al. studied strongly regular graphs with = 1 and µ = 2 [3]. It was shown that such a strongly regular graph has parameters either (9, 4, 1, 2) (the only such a graph is 3 ⇥ 3-lattice), or (99, 14, 1, 2), or (243, 22, 1, 2), or (6273, 112, 1, 2), ⇤The author thanks both anonymous referees for their comments and suggestions, which significantly im- proved the paper. The author thanks Professor Yaokun Wu for his continued support and warm hospitality and Sergey Goryainov for valuable discussions. This work is supported by NSFC(11671258 and 11971305) and STCSM(17690740800). E-mail address: soesoezaw313@gmail.com (Soe Soe Zaw) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 2 Art Discrete Appl. Math. 4 (2021) #P2.01 or (494019, 994, 1, 2). Berlekamp et al. further constructed a graph with parameters (243, 22, 1, 2), which is known as the Berlekamp-Van Lint-Seidel graph, but its uniqueness as well as the existence of graphs for the other three parameter tuples remain undecided. In particular, for the tuple (99, 14, 1, 2), this problem is known as the Conway’s 99-graph problem. The smallest feasible parameter tuples of strongly regular graphs with = 2, µ = 3 and = 3, µ = 4 are (364, 33, 2, 3), (676, 45, 2, 3) and (901, 60, 3, 4), respectively [4], and it is unknown if strongly regular graphs with such parameters exist. In [8], some examples of strictly Deza graphs with parameters (n, k, k1, a) and = 1 were given. In particular, dual Seidel switching was applied to the Petersen graph, the Hoffman-Singleton graph, Paley graphs of square order. In this paper, we investigate if dual Seidel switching can be applied to the Berlekamp-Van Lint-Seidel graph or its complement. 2 Preliminaries We consider undirected graphs without loops or multiple edges. A k-regular graph on v vertices is called strongly regular with parameters (v, k,, µ), 0 < k < v 1, if any two distinct vertices x, y in have common neighbours when x, y are adjacent and µ common neighbours if x, y are non-adjacent. For a vertex x in a graph , the neighbourhood (x) is the set of all neighbours of x in . Lemma 2.1 ([5], Theorem 1.3.1(i)). Let be a strongly regular graph with parameters (v, k,, µ), µ 6= 0, µ 6= k. Then has three distinct eigenvalues k, r, s, where k > r > 0 > s and the eigenvalues r, s satisfy the quadratic equation x2+(µ)x+(µk) = 0. For a graph , denote by the complement of . Lemma 2.2 ([5], Theorem 1.3.1(vi)). Let be a strongly regular graph with parameters (v, k,, µ). Then the complement of is a strongly regular graph with parameters (v, v k 1, v 2k + µ 2, v 2k + ) and eigenvalues v k 1,s 1,r 1. A k-regular graph on v vertices is called a Deza graph with parameters (v, k, b, a), b a, if the number of common neighbours of any two distinct vertices in takes on the two values a or b. A Deza graph is called a strictly Deza graph, if it has diameter 2 and is not strongly regular. The following lemma gives a construction of strictly Deza graphs, which is known as dual Seidel switching. Lemma 2.3 ([7], Theorem 3.1). Let be a strongly regular graph with parameters (v, k,, µ), k 6= µ, 6= µ and adjacency matrix M . Let P be a permutation matrix that represents an involution of that interchanges only non-adjacent vertices. Then PM is the adjacency matrix of a strictly Deza graph with parameters (v, k, b, a), where b = max(, µ) and a = min(, µ). Since in Lemma 2.3 represents an involution, the matrix PM is obtained from the matrix M by a permutation of rows in all pairs of rows with indexes i1 and i2, such that (i1) = i2 and (i2) = i1. Lemma 2.4 follows immediately from Lemma 2.3 and shows what is the neighbourhood of a vertex of the graph . Lemma 2.4. For the neighbourhood (u) of a vertex u of the graph from Lemma 2.3, the following conditions hold: (u) = ⇢ (u), if (u) = u; ((u)), if (u) 6= u. S. S. Zaw: On strictly Deza graphs derived from the Berlekamp-Van Lint-Seidel graph 3 In [8, Theorem 2], it was shown that the strong product with an edge and dual Seidel switching is the only method to obtain strictly Deza graphs with k = b + 1. Recall that the graph strong product of two graphs 1 and 2 has vertex set V (G1)⇥ V (G2) and two distinct vertices (v1, v2) and (u1, u2) are connected iff they are adjacent or equal in each coordinate, i.e., for i 2 1, 2, either vi = ui or {vi, ui} in E(i), where E(i) is the edge set of i [2]. It follows from Lemma 2.2 that, if a strongly regular graph has the property µ = 1, then the complementary graph has the property µ = 1 as well. Thus, according to [8, Theorem 2], we concentrate on order 2 automorphisms of that interchange either only non-adjacent vertices or only adjacent vertices. Let G be a group and S be an inverse-closed identity-free subset in G. The graph on G with two vertices x, y being adjacent whenever xy1 belongs to S is called the Cayley graph of the group G with connection set S and is denoted by Cay(G,S). 3 The Berlekamp-Van Lint-Seidel graph and dual Seidel switching The Berlekamp-Van Lint-Seidel graph, denoted by , is the coset graph of the ternary Golay code [5, Section 11.3]. This graph is known to be strongly regular with parameters (243, 22, 1, 2). In this section, we deal with two more ways to define this graph and give a description of the involutions of and suitable for dual Seidel switching. The main result of this paper is the following theorem. Theorem 3.1. The following statements hold. (1) The Berlekamp-Van Lint-Seidel graph has no order 2 automorphisms that inter- change only adjacent vertices; (2) The Berlekamp-Van Lint-Seidel graph has the unique (up to conjugation) order 2 automorphism that interchanges only non-adjacent vertices. To prove Theorem 3.1, we prove two lemmas, which imply the truth of the theorem statements. 3.1 from the Mathieu group M11 By ATLAS of Group Representations the Mathieu group M11 can be represented [10] by 5⇥ 5 matrices over GF (3) as follows. Put x := 0 BBBB@ 0 2 1 0 0 2 1 1 2 2 0 1 1 2 2 1 0 2 2 1 1 2 2 2 0 1 CCCCA , y := 0 BBBB@ 0 0 2 0 2 1 1 2 2 0 2 2 2 2 2 1 2 1 1 0 2 2 0 2 1 1 CCCCA . Then the group G := hx, yi is isomorphic to M11, where x is an involution. Let V (5, 3) denote the 5-dimensional vector space of over GF (3). We regard the elements of V (5, 3) 4 Art Discrete Appl. Math. 4 (2021) #P2.01 as rows and consider the action of G on V (5, 3) by the right multiplication, which has two orbits of size 22 and 220 on the nonzero vectors. The orbit of size 22 is given by the set S1 := {±(1, 0, 0, 0, 0),±(0, 0, 1, 0, 1),±(0, 1, 0, 1, 0),±(0, 1, 2, 0, 0), ±(0, 0, 1, 2, 1),±(0, 1, 0, 1, 2),±(1, 1, 2, 0, 2),±(1, 0, 0, 1, 2), ±(1, 0, 2, 1, 0),±(1, 1, 0, 0, 2),±(1, 1, 2, 1, 0)}, and, moreover, is isomorphic to the Cayley graph Cay(V (5, 3), S1). Since G stabilises S1 setwise, G is a subgroup in the automorphism group of , which is known (see [6]) to be isomorphic to the group 35 : (2 ⇥ M11). The fact that M11 has precisely one class of conjugate involutions implies that the automorphism group of has precisely three classes of conjugate involutions. Let e be the identity matrix from G. Note that e does not belong to G, but the multiplication by e is an involution of the automorphism group of Cay(V (5, 3), S1), which means that the three pairwise non-conjugate involutions of the automorphism group of Cay(V (5, 3), S1) are given by the right multiplication by e, x and x. Lemma 3.2. The following statements hold. (1) The involution e interchanges adjacent vertices as well as non-adjacent ones; (2) The involution x interchanges adjacent vertices as well as non-adjacent ones. Proof. (1) This involution fixes the zero vector and moves all non-zero vectors by swapping every two elements that are additive inverses of each other. In the graph Cay(V (5, 3), S1), two additive inverses are adjacent whenever both of them belong to S1. It means that the involution e interchanges adjacent vertices as well as non-adjacent ones. (2) On the one hand, the involution x swaps the vertices (0, 1, 0, 1, 0) and (0, 2, 0, 2, 0), which are adjacent in Cay(V (5, 3), S1). On the other hand, x swaps the vertices (1, 0, 0, 0, 0) and (0, 2, 1, 0, 0), which are not adjacent in Cay(V (5, 3), S1). In view of Lemma 3.2, it remains to check the inner involution x. In the next subsection, we explore one more definition of the Berlekamp-Van Lint-Seidel graph and give a very natural description of the involution x. 3.2 Specific parity-check matrix Recall that, for a positive integer n and a prime power q, V (n, q) denotes the n-dimensional vector space over the finite field Fq . The ternary Golay code can be constructed as the 6- dimensional subspace in V (11, 3) consisting of all row-vectors c such that the equality Hc T = 0 holds, where H := 2 66664 1 1 1 2 2 0 1 0 0 0 0 1 1 2 1 0 2 0 1 0 0 0 1 2 1 0 1 2 0 0 1 0 0 1 2 0 1 2 1 0 0 0 1 0 1 0 2 2 1 1 0 0 0 0 1 3 77775 is the specific parity check matrix of this code. Let x1, x2, x3, . . . , x11 denote the vectors from V (5, 3) that correspond to the columns of H . There are 22 vectors of type ±xi S. S. Zaw: On strictly Deza graphs derived from the Berlekamp-Van Lint-Seidel graph 5 and 220 vectors of type ±xi ± xj where i 6= j; i, j = 1, 2, . . . , 11. The Cayley graph Cay(V (5, 3), S2), where S2 := {±x1, . . . ,±x11}, is known to be isomorphic with the Berlekamp-Van Lint-Seidel graph (see [3]). Lemma 3.3. The reversal of vectors is an involution of Cay(V (5, 3), S2) that interchanges only non-adjacent vertices. Proof. Obviously, the reversal of vectors is a permutation of the vertex set of . For a vector 2 V (5, 3), denote by r the reversed vector. Note that (S2)r = S2 holds. Since, for any two vertices 1, 2 in , we have r1 r2 = (1 2)r, the reversal is an automorphism of Cay(V (5, 3), S2). For a vector (a, b, c, d, e) 2 V (5, 3), consider the difference (a, b, c, d, e) (a, b, c, d, e)r = (a e, b d, 0, d b, e a). Note that the first and the fifth coordinates and the second and fourth ones are additive inverses. Since S2 has no such vectors with zero third coordinate, the reversal interchanges only non-adjacent vertices. 4 Concluding remarks The following three strictly Deza graphs can be derived from the Berlekamp-Van Lint- Seidel graph . First, Lemma 2.3 and Theorem 1(2) give a strictly Deza graph with parameters (243, 22, 2, 1). It has spectrum {221, 548, 472, (4)60, (5)62} and its automorphism group of order 2592 is a subgroup in the automorhism group of . Further, in view of [8, Construction 1], the strong product [K2] of the Berlekamp-Van Lint-Seidel graph with an edge is a strictly Deza graph with parameters (486, 45, 44, 4). It has spectrum {451, 9132, (1)243, (9)110}. Finally, the order 2 automorphism from Theorem 1(2) induces an order 2 automorphism of [K2] that interchanges only non-adjacent vertices. Applying the dual Seidel switching to [K2], we obtain one more strictly Deza graph with parameters (486, 45, 44, 4), which has spectrum {451, 9120, 1108, (1)135, (9)122}. In the connection with the results from [8], we point out that both graphs with parame- ters (486, 45, 44, 4) are divisible design graphs. ORCID iDs Soe Soe Zaw https://orcid.org/0000-0002-2621-2138 References [1] K. T. Arasu, D. Jungnickel, S. L. Ma and A. Pott, Strongly regular Cayley graphs with µ = 1, J. Combin. Theory Ser. A 67 (1994), 116–125, doi:10.1016/0097-3165(94)90007-8. [2] L. W. Beineke and R. J. Wilson (eds.), Topics in algebraic graph theory, volume 102 of Ency- clopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2004, doi:10.1017/CBO9780511529993. [3] E. R. Berlekamp, J. H. van Lint and J. J. Seidel, A strongly regular graph derived from the perfect ternary Golay code, in: A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971), 1973 pp. 25–30, doi:10.1016/ B978-0-7204-2262-7.50008-9. 6 Art Discrete Appl. Math. 4 (2021) #P2.01 [4] A. E. Brouwer, Database of strongly regular graphs, https://www.win.tue.nl/˜aeb/ graphs/srg/srgtab.html. [5] A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-regular graphs, volume 18 of Ergeb- nisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 1989, doi:10.1007/978-3-642-74341-2. [6] DistanceRegular.org, Berlekamp-van Lint-Seidel graph, https://www. distanceregular.org/graphs/berlekamp-Vanlint-seidel.html. [7] M. Erickson, S. Fernando, W. H. Haemers, D. Hardy and J. Hemmeter, Deza graphs: a gen- eralization of strongly regular graphs, J. Combin. Des. 7 (1999), 395–405, doi:10.1002/(SICI) 1520-6610(1999)7:6h395::AID-JCD1i3.3.CO;2-L. [8] S. Goryainov, W. H. Haemers, V. V. Kabanov and L. Shalaginov, Deza graphs with parameters (n, k, k 1, a) and = 1, J. Combin. Des. 27 (2019), 188–202, doi:10.1002/jcd.21644. [9] W. Haemers, Dual Seidel switching, Technical University Eindhoven, pp. 183–191, EUT Report 84-WSK-03, 1984, pagination: XV, 373, https://research.tilburguniversity.edu/en/publications/ d10600db-43eb-461b-a194-06a73f505339. [10] R. A. Wilson, P. Walsh, J. Tripp, I. Suleiman, R. A. Parker, S. P. Norton, S. Nickerson, S. Lin- ton, J. Bray and R. Abbott, Atlas of finite group representations, http://brauer.maths. qmul.ac.uk/Atlas/v3. ISSN 2590-9770 The Art of Discrete and Applied Mathematics 4 (2021) #P2.02 https://doi.org/10.26493/2590-9770.1340.364 (Also available at http://adam-journal.eu) On divisible design Cayley graphs ⇤ Vladislav V. Kabanov† Krasovskii Institute of Mathematics and Mechanics, S. Kovalevskaja st. 16, Yekaterinburg 620990, Russia, and Sobolev Institute of Mathematics, Acad. Koptyug prospect 4, Novosibirsk 630090, Russia Leonid Shalaginov Krasovskii Institute of Mathematics and Mechanics, S. Kovalevskaja st. 16, Yekaterinburg 620990, Russia, and Sobolev Institute of Mathematics, Acad. Koptyug prospect 4, Novosibirsk 630090, Russia, and Chelyabinsk State University, Brat’ev Kashirinyh st. 129, Chelyabinsk 454021, Russia Received 29 November 2019, accepted 16 February 2020, published online 10 March 2021 Abstract Let v, k, b, a be integers such that v > k b a 0. A Deza graph with parameters (v, k, b, a) is a k-regular graph on v vertices in which the number of common neighbors of any two distinct vertices takes two values a or b (a  b). A k-regular graph on v vertices is a divisible design graph with parameters (v, k,1,2,m, n) when its vertex set can be partitioned into m classes of size n, such that any two distinct vertices from the same class have 1 common neighbors, and any two vertices from different classes have 2 common neighbors. It is clear, that divisible design graphs are Deza graphs. It is shown that divisible design Cayley graphs arise only by means of divisible dif- ference sets relative to some subgroup. Construction of a special set in an affine group over a finite field is given and shown that this set is a divisible difference set and thus its development is a divisible design Cayley graph. Keywords: Deza graph, divisible design graph, divisible design, divisible different set, Cayley graph, affine group over a finite field. Math. Subj. Class.: 05B05, 05C51, 51E05 ⇤The work is supported by Mathematical Center in Akademgorodok, the agreement with Ministry of Science and High Education of the Russian Federation number 075-15-2019-1613. Both authors are also partially sup- ported by RFBR according to the research project 17-51-560008. The authors are grateful to Vladimir Trofimov, Sergey Goryainov, Elena Konstantinova and participants of workshop “New trends in algebraic graph theory”, which was organized by Mathematical Center in Akademgorodok, for useful discussions. †Corresponding author. E-mail addresses: vvk@imm.uran.ru (Vladislav V. Kabanov), 44sh@mail.ru (Leonid Shalaginov) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 2 Art Discrete Appl. Math. 4 (2021) #P2.02 1 Introduction Let v, k, b, a be integers such that v > k b a 0. A Deza graph with parameters (v, k, b, a) is a k-regular graph on v vertices in which the number of common neighbors of any two distinct vertices takes two values a or b (a  b). Deza graphs appeared as a generalization of strongly regular graphs in [7]. This is a wide class of graphs which includes not only strongly regular graphs but also regular (0,)-graphs [10], (0, 2)-graphs [5, 11] and divisible design graphs [6, 9]. A k-regular graph on v vertices is a divisible design graph with parameters (v, k,1,2,m, n) when its vertex set can be partitioned into m classes of size n, such that any two distinct vertices from the same class have 1 common neighbors, and any two vertices from different classes have 2 common neighbors. For a divisible design graph the partition into classes is called a canonical partition. Let G be a finite group with the identity element e. If S is a subset of G which is closed under inversion and does not contain e, then Cayley graph Cay(G,S) is a graph with the vertex set G and two vertices x, y are adjacent if and only if xy1 2 S. In this paper, some divisible design graphs are constructed that arise from finite groups in the form of Cayley graphs. The following theorem is the basis for the construction. Theorem 1.1. Let Cay(G,S) be a Deza graph with parameters (v, k, b, a). Let also A, B and {e} be a partition of G and SS1 be a multiset such that SS1 = aA+ bB+k{e}. If either A [ {e} or B [ {e} is a subgroup of G, then Cay(G,S) is a divisible design graph and the right cosets of this subgroup give a canonical partition of the graph. Conversely, if Cay(G,S) is a divisible design graph, then the class of its canonical partition which contains the identity of G is a subgroup of G and classes of the canonical partition of divisible design graph coincide with the cosets of this subgroup. Proof. Let N = A [ {e} be a subgroup of G and let G = Na1 [ · · · [Nar be a partition of G by the right cosets of N . For every g, h 2 N and 1  i  r, 1  j  r, the set of neighbors of hai is Shai and the set of neighbors of gaj is Sgaj . Thus, the set of common neighbors for hai and gaj coincides with Shai \Sgaj . The number |Shai \Sgaj | equals |{(s, s⇤)|shai = s⇤gaj}|, where s, s⇤ 2 S. Therefore, h = s1s⇤gaja1i . If i = j, then h = s1s⇤g and s1s⇤ 2 N . So there are exactly a such pairs if s1s⇤ 2 A and k such pairs if s1s⇤ = e. If i 6= j, then gaja1i /2 N and hence s1s⇤ /2 N . So s1s⇤ 2 B and there are exactly b such pairs. The case of N = B [ {e} is viewed in a similar way. Conversely, let Cay(G,S) be a divisible design graph and N be a class of the canonical partition of divisible design graph which contains the identity of G. It’s enough to prove that for any h, g 2 N we have hg1 2 N . Since h and g belong to the same class of the canonical partition of Cay(G,S), then |Sh \ Sg| = 1. The number of pairs (s, s⇤) such that sh = s⇤g is equal to 1. Thus, hg1 = s1s⇤ is repeated 1 times in SS1. Theorem 1.1 shows that Cayley divisible design graphs arise only by means of divis- ible difference sets relative to some subgroup. Let a finite group G of order mn have a subgroup N of order n. A subset S of G is called a divisible difference set with exceptional subgroup N if there are constants 1 and 2 such that every non-identity element of N can be expressed as a right quotient of elements in S in exactly 1 ways and every element in G \N can be expressed as a right quotient of elements in S in exactly 2 ways. In other words, if k = |S|, then SS1 = k{e}+ 1(N {e}) + 2(GN). V. V. Kabanov and L. Shalaginov: On divisible design Cayley graphs 3 2 Construction of divisible design Cayley graphs Let F be a finite field with qr elements, where q is a prime power and r > 1. Consider the group G of all 2⇥ 2 matrices ✓ 1 0 ↵ ◆ , where ↵ 2 F and 2 F \ {0}. It’s clear that G is a semi-direct product of two subgroups N and K, where N consists of all matrices ✓ 1 0 ↵ 1 ◆ , and ↵ 2 F, K consists of all matrices ✓ 1 0 0 ◆ , and 2 F \ {0}. define a bijection + between N and F as follows: +(a) = ↵ for any a 2 N , a = ✓ 1 0 ↵ 1 ◆ , ↵ 2 F. If a1 = ✓ 1 0 ↵1 1 ◆ and a2 = ✓ 1 0 ↵2 1 ◆ ↵1,↵2 2 F, then a1a2 = ✓ 1 0 ↵1 1 ◆✓ 1 0 ↵2 1 ◆ = ✓ 1 0 ↵1 + ↵2 1 ◆ . Thus, N is isomorphic to the additive group F+ which we can consider as a linear space of dimension r over Fq . Define a bijection ⇥ between K and F \ {0} as follows: ⇥(b) = for any b 2 K, b = ✓ 1 0 0 ◆ , 2 F \ {0}. Clearly, ⇥ is an isomorphism between K and the multiplicative group of F. Let K be generated by matrix f⇤ = ✓ 1 0 0 ⌧ ◆ , where ⌧ is a primitive element F. Also let H be a cyclic group generated by f = (f⇤)q1 = ✓ 1 0 0 ✓ ◆ , where ✓ = ⌧ q1. Thus, G = NH is a subgroup of G of index q 1 and the order of G is equal to q r(qr 1)/(q 1). Furthermore, N is a normal subgroup of order qr and index (qr 1)/(q 1) in G. By the formula of Gaussian binomial coefficients, the number of (r 1)-dimensional subspaces of N equals t, where t = (qr 1)/(q 1). Let M be the set of preimages of all these (r1)-dimensional subspaces of F+ in N under +. Since + is an isomorphism, then the set M consists of t subgroups of order qr1 from N . Denote by M one of the subgroups from M. Thus, M = {Mi | +(Mi) = +(M)⌧ i, i = 1, 2, . . . , t}. Let ' be a permutation on the set M = {M1,M2, . . . ,Mt}. As usual by f i(N \M'(i)) we denote the set {f ia : a 2 N \M'(i)}. Define a subset S of G in the following way: S = t[ i=1 f i(N \M'(i)). 4 Art Discrete Appl. Math. 4 (2021) #P2.02 It is obvious, that S is a generating set of G. In the following lemmas, we consider the question of when this set is closed under inversion. Lemma 2.1. Subset S is closed under inversion in G if and only if for all integers i the following condition holds +(M'(i))✓ i = +(M'(ti)). (⇤) Proof. Let s 2 S and s = f ia, for some integer i and a 2 N \ M'(i). Also, let a =✓ 1 0 ↵ 1 ◆ , for some ↵ 2 +(N \M'(i)) and f i = ✓ 1 0 0 ✓i ◆ . In such case, a1 = ✓ 1 0 ↵ 1 ◆ , for ↵ 2 +(N \M'(i)) and f i = ✓ 1 0 0 ✓ti ◆ . Then f i a = ✓ 1 0 0 ✓i ◆✓ 1 0 ↵ 1 ◆ = ✓ 1 0 ↵✓ i ✓ i ◆ , (f ia)1 = ✓ 1 0 ↵ 1 ◆✓ 1 0 0 ✓ti ◆ = ✓ 1 0 (↵✓i)✓ti 1 ◆✓ 1 0 0 ✓ti ◆ . Thus, s1 2 S if and only if ↵✓i 2 +(M'(ti)). Hence, S = S1 if and only if (⇤) holds. The interesting question to study all permutations for which the property (⇤) holds is done next. Let ' = ('1, . . . ,'t) be a permutation of the set {1, . . . , t}, where '(i) = 'i. Lemma 2.2. For any t > 2 there is at least one permutation ' satisfying +(M'(i))✓ i = +(M'(ti)) for all integers i. Proof. Let ' = ('1, . . . ,'t). If t is an odd integer, then ' = (1, t 1, t 3, . . . , 2, t, t 2, t 4, . . . , 3). If t is an even integer, then ' = (1, t 1, t 3, . . . , t/2 + 2, t/2 1, t/2 3, . . . , 2, t/2, t 2, t 4, . . . , . . . , t/2 + 1, t, t/2 2, t/2 4, . . . , 3). For example, if t = 7, then there are exactly three permutations (1, 6, 4, 2, 7, 5, 3), (1, 6, 3, 7, 2, 4, 5), and (1, 6, 2, 3, 5, 7, 4) which satisfy the condition (⇤) in Lemma 2.1. V. V. Kabanov and L. Shalaginov: On divisible design Cayley graphs 5 Construction 2.3. Let be a Cayley graph Cay(G,S) whose vertices are elements of the group G defined as above and S = t[ i=1 f i(N \M'(i)). In the next section we prove that if S satisfies the condition (⇤), then is a divisible design graph. Construction 2.3 gives us an infinite series of divisible design graphs which are Cayley graphs. Only the first graph among them is known and given in [9, Construction 4.20]. This divisible design Cayley graph with parameters (12, 6, 2, 3, 3, 4) is the line graph of the octahedron and can be obtained as a Cayley graph from the alternating group of degree 4 (See Example 4.1 in Section 4). 3 Main theorem The main goal of our article is to prove the following theorem. Theorem 3.1. Let be a Cayley graph from Construction 2.3. If S satisfies the condition (⇤), then is a divisible design graph with parameters (v, k,1,2,m, n), where v = qr(qr 1)/(q 1), k = qr1(qr 1), 1 = q r1(qr qr1 1), 2 = qr2(q 1)(qr 1), m = (qr 1)/(q 1), n = qr. Proof. It is clear, that is an undirected graph on v = qr(qr 1)/(q 1) vertices of valency k = |S| = t1[ i=0 f i(N \M'(i)) = t1X i=0 (qr qr1) = = (qr qr1)(qr 1)/(q 1) = qr1(qr 1). Calculate a number of common adjacent vertices for any two distinct vertices from any coset. Since is a Cayley graph, then it is enough to calculate this number for the identity element of G and any non-identity element from N . Let e be the identity element of G, a 6= e and a 2 N . It is important to note that a belongs to exactly t1 = (qr1 1)/(q 1) subgroups of N from M. Since (e) \ (a) = S \ Sa, then (e) \ (a) = ⇣t1[ i=0 f i(N \M'(i)) ⌘ \ ⇣t1[ i=0 f i(N \M'(i))a ⌘ = = ⇣ [ a2M'(i) f i(N \M'(i)) ⌘ [ ⇣ [ a/2M'(i) f i(N \M'(i)) \ f i(N \M'(i))a ⌘ . Hence |(e) \ (a)| = t11X i=0 (qr qr1) + t1X i=t1 (qr (q 2)qr1) = qr1(qr qr1 1). 6 Art Discrete Appl. Math. 4 (2021) #P2.02 Calculate a number of common adjacent vertices for any two vertices from any two distinct cosets. Since is a Cayley graph, then it is enough to calculate this number for the identity element e from G and any element g from Nf i, where i 6= 0 (mod t). Let g = fxag , x 6= 0 (mod t). We have fxag = [fx, a1g ]agfx, where [fx, a1g ] is the commutator of elements fx and a1g . It is easy to verify, that if bg = [fx, a1g ]ag , then bg 2 N . Since (e) \ (g) = S \ Sg, then (e) \ (g) = ⇣t1[ i=0 f i(N \M'(i)) ⌘ \ ⇣t1[ i=0 f i(N \M'(i))g ⌘ . Taking into account that N is a normal subgroup of G and bg 2 N we have f i(N \M'(i)) \ f j(N \M'(j))bgfx 6= ; if and only if i x = j. Thus, we have t1[ i=0 f i(N \M'(i)) \ f ix(N \M'(ix))fx = = t1[ i=0 f ix f x(N \M'(i)) \ (N \M'(ix))fx . If h 2 fx(N \M'(i)) \ (N \M'(ix))fx, then there are some ↵1 2 +(N \M'(i)), ↵2 2 +(N \M'(ix)) such that h = ✓ 1 0 0 ✓x ◆✓ 1 0 ↵1 1 ◆ = ✓ 1 0 ↵2 1 ◆✓ 1 0 0 ✓x ◆ . Therefore, ✓x↵1 = ↵2. Since bijection + is an isomorphism between N and F+, then |fx(N \M'(i)) \ (N \M'(ix))fx| = = | +(N \M'(i)) \ ✓x +((N \M'(ix)))| = = | +(N) \ +(M'(i)) [ ✓x +(M'(ix))| = = (qr 2qr1 + qr2). Thus, |(e) \ (g)| == t1X i=0 (qr 2qr1 + qr2) = = (qr 2qr1 + qr2)(qr 1)/(q 1) = = qr2(q 1)(qr 1). Hence, is a divisible design graph. V. V. Kabanov and L. Shalaginov: On divisible design Cayley graphs 7 4 Examples All examples in this section except Example 4.1 were found using computer search. Example 4.1. Divisible design graph with parameters (12, 6, 2, 3, 3, 4). There is the unique example of divisible design Cayley graph with parameters (12, 6, 2, 3, 3, 4) based on the alternating group Alt4. We can choose the set S = {(13)(24), (12)(34), (123), (132), (234), (243)} as its generating set. A fragment of the Cayley table of Alt4 below shows us the necessary properties. (13)(24) (12)(34) (123) (132) (234) (243) (13)(24) e (14)(23) (243) (124) (143) (123) (12)(34) (14)(23) e (134) (234) (132) (142) (123) (142) (243) (132) e (13)(24) (143) (132) (234) (143) e (123) (142) (12)(34) (234) (132) (124) (12)(34) (134) (243) e (243) (134) (123) (124) (13)(24) e (234) Example 4.2. Divisible design graphs with parameters (36, 24, 15, 16, 4, 9). It was found in [8] that there are three non-isomorphic divisible design graphs with parameters (36, 24, 15, 16, 4, 9). From our Construction 2.3 with permutations (1, 3, 4, 2) and (1, 3, 2, 4) we have two of that non-isomorphic divisible design graphs, which are based on subgroup of index 2 of AG(9). It is important to note that, if t is even, then t/2 and t can be in any place in ' according to the condition (⇤). Example 4.3. Divisible design graphs with parameters (56, 28, 12, 14, 7, 8). It was found in [8] that there are five non-isomorphic examples of divisible design graphs with parameters (56, 28, 12, 14, 7, 8) which are based on group AG(8). This group can be described as follows G = h f1, f2, f3, f4 i with defining group relationships f 7 1 = f 2 2 = f 2 3 = f 2 4 = e, f2 ⇤ f1 = f1 ⇤ f3, f3 ⇤ f1 = f1 ⇤ f4, f4 ⇤ f1 = f1 ⇤ f2 ⇤ f4. Below, we give the list of generating sets for these Cayley graphs. S1 = {f3, f2 ⇤f3, f3 ⇤f4, f2 ⇤f3 ⇤f4, f1 ⇤f4, f1 ⇤f2 ⇤f4, f1 ⇤f2 ⇤f3 ⇤f4, f1 ⇤f3 ⇤f4, f21 ⇤ f2, f 2 1 ⇤ f3, f21 ⇤ f4, f21 ⇤ f2 ⇤ f3 ⇤ f4, f31 ⇤ f2, f31 ⇤ f3, f31 ⇤ f2 ⇤ f4, f31 ⇤ f3 ⇤ f4, f41 ⇤ f2, f41 ⇤ f2 ⇤ f3, f41 ⇤ f2 ⇤ f4, f41 ⇤ f2 ⇤ f3 ⇤ f4, f51 ⇤ f2, f51 ⇤ f4, f51 ⇤ f2 ⇤ f3, f51 ⇤ f3 ⇤ f4, f61 ⇤ f3, f61 ⇤ f4, f 6 1 ⇤ f2 ⇤ f3, f61 ⇤ f2 ⇤ f4}; S2 = {f2, f4, f2 ⇤ f3, f3 ⇤ f4, f1, f1 ⇤ f3, f1 ⇤ f4, f1 ⇤ f3 ⇤ f4, f21 ⇤ f3, f21 ⇤ f4, f21 ⇤ f2 ⇤ f3, f 2 1 ⇤ f2 ⇤ f4, f31 , f31 ⇤ f2 ⇤ f4, f31 ⇤ f3 ⇤ f4, f31 ⇤ f2 ⇤ f3, f41 , f41 ⇤ f2, f41 ⇤ f4, f41 ⇤ f2 ⇤ 8 Art Discrete Appl. Math. 4 (2021) #P2.02 f4, f 5 1 ⇤ f2, f51 ⇤ f3, f51 ⇤ f2 ⇤ f4, f51 ⇤ f3 ⇤ f4, f61 , f61 ⇤ f2, f61 ⇤ f3, f61 ⇤ f2 ⇤ f3}; S3 = {f1, f3, f4, f21 , f1 ⇤ f3, f2 ⇤ f4, f3 ⇤ f4, f21 ⇤ f2, f21 ⇤ f3, f21 ⇤ f4, f1 ⇤ f2 ⇤ f4, f1 ⇤ f3 ⇤ f4, f31 ⇤ f3, f51 , f41 ⇤ f2, f41 ⇤ f4, f31 ⇤ f2 ⇤ f3, f31 ⇤ f2 ⇤ f4, f31 ⇤ f3 ⇤ f4, f61 , f51 ⇤ f2, f41 ⇤ f2 ⇤ f3, f41 ⇤ f2 ⇤ f4, f61 ⇤ f2, f61 ⇤ f4, f51 ⇤ f2 ⇤ f3, f51 ⇤ f3 ⇤ f4, f61 ⇤ f2 ⇤ f3}; S4 = {f1, f2, f3, f1 ⇤ f3, f1 ⇤ f4, f2 ⇤ f4, f3 ⇤ f4, f21 ⇤ f2, f21 ⇤ f3, f21 ⇤ f4, f1 ⇤ f3 ⇤ f4, f31 ⇤ f3, f 4 1 ⇤ f3, f41 ⇤ f4, f31 ⇤ f2 ⇤ f3, f31 ⇤ f3 ⇤ f4, f21 ⇤ f2 ⇤ f3 ⇤ f4, f61 , f51 ⇤ f2, f51 ⇤ f4, f41 ⇤ f2 ⇤ f3, f 4 1 ⇤ f2 ⇤ f4, f31 ⇤ f2 ⇤ f3 ⇤ f4, f61 ⇤ f2, f61 ⇤ f3, f51 ⇤ f2 ⇤ f3, f51 ⇤ f3 ⇤ f4, f61 ⇤ f2 ⇤ f3}; S5 = {f1, f2, f3, f4, f21 , f1 ⇤ f3, f1 ⇤ f4, f21 ⇤ f4, f1 ⇤ f3 ⇤ f4, f2 ⇤ f3 ⇤ f4, f31 ⇤ f3, f31 ⇤ f4, f 2 1 ⇤ f2 ⇤ f3, f51 , f41 ⇤ f2, f41 ⇤ f4, f31 ⇤ f2 ⇤ f3, f31 ⇤ f2 ⇤ f4, f21 ⇤ f2 ⇤ f3 ⇤ f4, f61 , f51 ⇤ f2, f 5 1 ⇤ f4, f41 ⇤ f2 ⇤ f3, f41 ⇤ f3 ⇤ f4, f61 ⇤ f2, f61 ⇤ f3, f51 ⇤ f2 ⇤ f4, f61 ⇤ f2 ⇤ f3}. Three of them are isomorphic to the graphs we have from our Construction 2.3 with permutations which pointed out after Lemma 2.2. Example 4.4. Divisible design graph with parameters (80, 60, 44, 45, 5, 16). There is at least one divisible design Cayley graph which is based on subgroup of index 3 of AG(42) that we have from our Construction 2.3 with permutations (1, 4, 2, 5, 3). This is the first example where q is not prime. 5 Conclusion remarks Any divisible design graph can be considered as a symmetric group-divisible design [2, 3, 4], the vertices of which are points, and the neighborhoods of the vertices are blocks. Such a design is called the neighborhood design. However, non-isomorphic graphs can correspond to isomorphic designs. Examples 4.2 and 4.3 of this article give us non-isomorphic divisible design graphs which produce isomorphic group divisible designs. If group-divisible design admit a symmetric incidence matrix with zero diagonal, then it corresponds to divisible design graph (see [9, Section 4.3]). There is a great possibility to construct divisible designs from groups. Let G be a group of order mn containing a subgroup N of order n. A k-subset D of G is called a divisible (m,n, k,1,2) difference set (divisible by cosets of subgroup N ) if the list of elements xy1 with x, y 2 D contains all non-identity elements in N exactly 1 times and all elements in G \N exactly 2 times. In case that N = {0}, the definition of a divisible difference set coincides with the definition of a difference set in the usual sense. In case that 1 = 0, the definition of a divisible difference set coincides with the definition of a relative difference set [2, 12, 13]. Divisible difference set D gives rise to a symmetric group-divisible design D with the set of blocks {Dg| g 2 G} and has the same parameters as D. This symmetric group- divisible design is called the development of D and admits G as a regular automorphism group (by right translation). Thus, symmetric group-divisible designs with a regular group G are equivalent to divisible difference sets in G. For having a symmetric incidence matrix with zero diagonal, the divisible difference set should be reversible (or equivalently, it must have a strong multiplier 1). There is more information on such difference sets in [1]. V. V. Kabanov and L. Shalaginov: On divisible design Cayley graphs 9 ORCID iDs Vladislav V. Kabanov https://orcid.org/0000-0001-7520-3302 Leonid Shalaginov https://orcid.org/0000-0001-6912-2493 References [1] K. T. Arasu, D. Jungnickel and A. Pott, Divisible difference sets with multiplier 1, J. Algebra 133 (1990), 35–62, doi:10.1016/0021-8693(90)90067-X. [2] T. Beth, D. Jungnickel and H. Lenz, Design theory. 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Pott, A survey on relative difference sets, in: Groups, difference sets, and the Monster (Columbus, OH, 1993), de Gruyter, Berlin, volume 4 of Ohio State Univ. Math. Res. Inst. Publ., pp. 195–232, 1996. [14] R.-H. Schulz and A. G. Spera, Divisible designs and groups, Geom. Dedicata 44 (1992), 147– 157, doi:10.1007/BF00182946. [15] A. G. Spera, Semi-regular divisible designs and Frobenius groups, Geom. Dedicata 42 (1992), 285–294, doi:10.1007/BF02414067. ISSN 2590-9770 The Art of Discrete and Applied Mathematics 4 (2021) #P2.03 https://doi.org/10.26493/2590-9770.1361.dd9 (Also available at http://adam-journal.eu) Torsion free equiaffine connections on three-dimensional homogeneous spaces Natalya P. Mozhey Belarusian State University of Informatics and Radioelectronics P. Brovki Street, 6, Minsk, 220013 Belarus Received 22 February 2020, accepted 9 May 2020, published online 10 March 2021 Abstract The aim of this paper is to describe equiaffine connections on three-dimensional homo- geneous spaces. The affine connection is equiaffine if it admits a parallel volume form. Only the case of spaces not admitting connections with nonzero torsion is considered. For such homogeneous spaces, it is determined under what conditions the connection is equiaffine (locally equiaffine). In addition, equiaffine (locally equiaffine) connections and Ricci tensors are written out in explicit form. In this work we use the algebraic approach for description of connections, methods of the theory of Lie groups, Lie algebras and ho- mogeneous spaces. Keywords: Equiaffine connection, homogeneous space, transformation group, Lie algebra, torsion tensor, Ricci tensor. Math. Subj. Class.: 53B05 1 Introduction The aim of this paper is to describe equiaffine connections on three-dimensional homo- geneous spaces. Only the case of spaces not admitting connections with nonzero torsion is considered. The case of affine connections is known (see [2]). The affine connection is equiaffine if it admits a parallel volume form (see [4]). For all such spaces, it is deter- mined under what conditions the connection is equiaffine (locally equiaffine). In addition, equiaffine (locally equiaffine) connections and Ricci tensors are written out in explicit form. Let (G,M) be a three-dimensional homogeneous space, where G is a Lie group acts transitively on the manifold M . We fix an arbitrary point o 2 M and denote by G = Go the stationary subgroup of o. Then we can correspond the pair (ḡ, g) of Lie algebras to (G,G), where ḡ is the Lie algebra of G and g is the subalgebra of ḡ corresponding to the subgroup E-mail address: mozheynatalya@mail.ru (Natalya P. Mozhey) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 2 Art Discrete Appl. Math. 4 (2021) #P2.03 G. The pair (ḡ, g) is said to be isotropy-faithful if its isotropic representation is injective. The classification all three-dimensional isotropically–faithful pairs (ḡ, g) with torsion-free connections only, is described in [2]. Let m = ḡ/g. Invariant affine connections on (G,M) are in one-to-one correspondence [3] with linear mappings ⇤ : ḡ ! gl(m) such that ⇤|g = and ⇤ is g-invariant. We call this mappings (invariant) affine connections on the pair (ḡ, g). If there exists at least one invariant connection on (ḡ, g) then this pair is isotropy- faithful [1]. The curvature and torsion tensors of the invariant affine connection ⇤ are given by the following formulas: R : m ^ m ! gl(m), (x1+g) ^ (x2+g) 7! [⇤(x1),⇤(x2)] ⇤([x1, x2]); T : m^m ! m, (x1+g)^(x2+g) 7! ⇤(x1)(x2+g)⇤(x2)(x1+g)[x1, x2]m. The connection ⇤ is torsion-free (or without torsion) if T = 0. In this case we have: R(x, y)z +R(y, z)x+R(z, x)y = 0 for all x, y, z 2 m (the first Bianchi identity). We define the Ricci tensor: Ric(y, z) = tr{x 7! R(x, y)z}. An affine connection ⇤ with zero torsion has symmetric Ricci tensor if and only if it is locally equiaffine [4]. Really, Ric(y, z)Ric(z, y) = tr{x 7! R(x, y)zR(x, z)y}. From the first Bianchi identity we obtain Ric(y, z)Ric(z, y)=tr{x 7!R(y, z)x}=trR(y, z). Then Ric(y, z)Ric(z, y)= tr(⇤(y)⇤(z)⇤(z)⇤(y))+tr⇤([y, z])=tr⇤([y, z]). Hence Ric is symmetric if and only if tr⇤([y, z])=0 for all y, z 2 ḡ. We say that the affine connection ⇤ is locally equiaffine if tr⇤([x, y])=0 for all x, y 2 ḡ (i.e. ⇤([ḡ, ḡ]) ⇢ sl(m)). By equiaffine connection we mean the (torsion-free) affine connection ⇤ such that tr⇤(x)=0 for all x 2 ḡ. In this case, it is obvious (g) ⇢ sl(m). We define (ḡ, g) by the commutation table of the Lie algebra ḡ. Here by {e1, ..., en} we denote a basis of ḡ (n = dim ḡ). We assume that the Lie algebra g is generated by e1, ..., en3. Let {u1=en2, u2=en1, u3=en} be a basis of m. We describe affine con- nection by ⇤(u1), ⇤(u2), ⇤(u3), curvature tensor R by R(u1, u2), R(u1, u3), R(u2, u3) and torsion tensor T by T (u1, u2), T (u1, u3), T (u2, u3). We say that the affine connection is trivial if ⇤(u1) = ⇤(u2) = ⇤(u3) = 0. To refer to the pair we use the notation d.n.m, where d is the dimension of the subalgebra, n is the number of the subalgebra of gl(3,R), m is the number of (ḡ, g) in [2]. The description of (torsion-free) equiaffine connections on three-dimensional homoge- neous spaces can be divided into the following parts: - classification of pairs that allow nontrivial locally equiaffine connections (the curva- ture tensor is only zero in Theorem 2.1; the curvature tensor is not only zero in Theorem 2.2); - classification of pairs with only trivial locally equiaffine connections (the curvature tensor is zero in Theorem 3.1; the curvature tensor is not zero in Theorem 3.2). The information about equiaffine (locally equiaffine) connections and Ricci tensors is contained in the proof of the Theorems 2.1, 2.2, 3.1 and 3.2. 2 Pairs of Lie algebras, admitting nontrivial locally equiaffine connec- tions 2.1 The curvature tensor is only zero Theorem 2.1. I. If the pair (ḡ, g) allows nontrivial equiaffine connections, the curvature and torsion tensors are only zero, then (ḡ, g) is equivalent to one and only one of the following pairs: N. P. Mozhey: Torsion free equiaffine connections 3 – ḡ is nonsolvable: 4.21.11. e1 e2 e3 e4 u1 u2 u3 e1 0 e2 µe3 (1 µ)e4 u1 0 µu3 e2 e2 0 e4 0 0 e2 + u1 0 e3 µe3 e4 0 0 0 2e3 u2 e4 (µ 1)e4 0 0 0 0 e4 e2 + u1 u1 u1 0 0 0 0 0 0 u2 0 e2 u1 2e3 e4 0 0 2u3 u3 µu3 0 u2 e2 u1 0 2u3 0 , µ = 1; – ḡ is solvable: 1.2.1 e1 u1 u2 u3 e1 0 u1 u2 µu3 u1 u1 0 0 0 u2 u2 0 0 0 u3 µu3 0 0 0 , =2/3, µ=1/3; 2.9.1. e1 e2 u1 u2 u3 e1 0 (1 µ)e2 u1 u2 µu3 e2 (µ 1)e2 0 0 0 u1 u1 u1 0 0 0 0 u2 u2 0 0 0 0 u3 µu3 u1 0 0 0 , =3/2, µ=1/2; =2/3, µ=1/3; =1/2, µ=3/2; 3.20.25, µ0 e1 e2 e3 u1 u2 u3 e1 0 (12µ)e2 (1µ)e3 u1 2µu2 µu3 e2 (2µ1)e2 0 0 0 u1 e3 e3 (µ1)e3 0 0 0 0 u1 u1 u1 0 0 0 0 0 u2 2µu2 u1 0 0 0 0 u3 µu3 e3 u1 0 0 0 , µ=1/3. The Ricci tensors are zero. II. Any pair (ḡ, g), allows nontrivial affine connections, the curvature and torsion ten- sors are only zero (i.e. if ḡ is nonsolvable then d.n.m = 6.3.2, 5.9.2, 4.19.2, 4.21.11 (µ6=0, 1,1/2), 3.6.2, 3.12.2, 3.13.6 (µ6=0, 1,1, 1/2), 3.28.2, 2.8.7 (6=0, 1,1, 1/2), if ḡ is solvable then d.n.m = 5.10.1 (=1/2, µ=0), 4.8.1 (=0, µ=1/2), 4.11.1 (µ=0,=1/2), 4.11.5, 3.7.1 (=1/2), 3.8.1 (+µ=1/2, µ=0, 1/2), 3.14.1 (µ6=0, 2), 3.19.17, 3.20.1 (=1/2 (µ6=0, 1/2); µ=1/2 (6=0, 1/2)), 3.20.25 (µ6=0), 3.20.26 (6=1/3, 1/4), 3.23.1 (=3/4), 3.29.1 (µ=1/2), 2.1.1 (=1/2), 2.8.1 (=1/2), 2.9.1 (=1/2 (µ 6=0,1/2, 1/4, 1/2); =2µ (µ 6=0, 1/4, 1/3, 1); µ=1/2 ( 6=1/2, 0, 1, 3/2)), 2.19.1 (=1/2), 2.19.5, 2.21.1 (=3/4), 1.2.1 (µ=2 (6=1/3, 1/4); µ=/2 (6=2); =1/2 (µ 6= 1/2)), 1.7.1 (=1/2), see [2]), admits locally equiaffine connections. Remark. In the cases 5.10.1 ( = 1/2, µ = 0), 3.8.1 ( = 0, µ = 1/2), 3.20.1 (=1/2 (µ 6=0, 1/2), µ=1/2 ( 6=0, 1/2)), 3.23.1 ( = 3/4), 3.29.1 (µ = 1/2), 2.9.1 (µ=1/2 (6=1/2, 0, 1, 3/2)), 2.19.1 ( = 1/2) the connection is trivial after basis replacement. 4 Art Discrete Appl. Math. 4 (2021) #P2.03 Proof. For the subalgebras g of gl(3,R) in [2] we find isotropy-faithful pairs (ḡ, g) and choose pairs, allows nontrivial affine, equiaffine (locally equiaffine) connections, such that the curvature and torsion tensors are zero for all connections. Let ḡ is nonsolvable, for example, the pair (ḡ, g) has the form 6.3.2 and ⇤(u1)= 0 @ p1,1 p1,2 p1,3 p2,1 p2,2 p2,3 p3,1 p3,2 p3,3 1 A, ⇤(u2)= 0 @ q1,1 q1,2 q1,3 q2,1 q2,2 q2,3 q3,1 q3,2 q3,3 1 A, ⇤(u3)= 0 @ r1,1 r1,2 r1,3 r2,1 r2,2 r2,3 r3,1 r3,2 r3,3 1 A, pi,j , qi,j , ri,j 2 R (i, j=1, 2, 3). ⇤|g is the isotropic representation of g, ⇤ is g-invariant ) [⇤(e2),⇤(u1)] = 0, p3,1 = p3,2 = p1,2 =0, p3,3 = p2,2. [⇤(e1),⇤(u1)] =⇤([e1, u1]) ) p1,3 = p2,1 = p2,3 = 0. [⇤(e5),⇤(u1)] = ⇤([e5, u1]) ) p2,2 = p1,1. If [⇤(e2),⇤(u2)] = 0 then q3,1 = q3,2 = q1,2 = 0, q3,3 = q2,2. [⇤(e1),⇤(u2)] = ⇤(u2), q1,1 = q2,2 = q2,3 = 0. [⇤(e3),⇤(u2)]=⇤(u3), r1,1=r1,3=r2,1=r2,2=r2,3=r3,2=r3,3=0, r3,1=q2,1, r1,2= q1,3. If [⇤(e4),⇤(u2)]=⇤(u2) then r1,2=0. [⇤(e5),⇤(u2)]=⇤(u1)+⇤(e1)+3⇤(e4), p1,1 = r3,1 = 2, tr⇤(3e4+u1) = 0 ) tr⇤([x, y]) = 0 for all x, y 2 ḡ, the connection is locally equiaffine and has the form, presented in the table, the curvature and torsion tensors are zero. In this case Ricci tensor is equal to zero too. We have tr⇤(e4) 6= 0 ) the connection is not equiaffine. In the cases 3.13.6 (µ = 1) and 2.8.7 ( = 1) the connection is equiaffine, but admitting nonzero torsion tensor. Similarly we obtain the results in the other cases: Pair Locally equiaffine connection 6.3.2 0 @ 2 0 0 0 2 0 0 0 2 1 A, 0 @ 0 0 0 2 0 0 0 0 0 1 A, 0 @ 0 0 0 0 0 0 2 0 0 1 A 5.9.2 3.12.2 3.13.6, µ6=0,1,1,1/2 0 @ 0 1 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 1 0 0 0 1 1 A, 0 @ 0 0 0 0 0 0 0 1 0 1 A 4.19.2. 0 @ 0 1 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 1 0 0 0 1 1 A, 0 @ 0 0 0 0 0 0 0 1 0 1 A 4.21.11, µ 6= 0, 1, 1/2 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 1 0 0 0 1 1 A, 0 @ 0 0 0 0 0 0 0 1 0 1 A 3.6.2 2.8.7,6=0,1,1,1/2 0 @ 1/2 0 0 0 0 0 0 0 1/2 1 A, 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 0 0 1/2 0 0 1 A 3.28.2 0 @ 0 1 0 0 0 0 0 0 0 1 A, 0 @ 0 2 0 0 1 0 0 0 1 1 A, 0 @ 0 0 0 0 0 0 0 1 0 1 A The connection is equiaffine only in the case 4.21.11, µ = 1. Let ḡ is solvable, for example, (ḡ, g) is 5.10.1 (=1/2, µ=0) then ⇤|g is the isotropic representation of g. ⇤ is g-invariant ) tr⇤([x, y])=0 for all x, y 2 ḡ and locally equiaffine connection there exist and has the form, presented in the table, the Ricci tensor, curvature and torsion tensors are equal to zero. In this case tr⇤(e1) 6= 0 and the connection is not equiaffine. Similarly we obtain the results in the other cases: N. P. Mozhey: Torsion free equiaffine connections 5 Pair Locally equiaffine connection 5.10.1,=1/2, µ=0 4.11.1, µ=0,=1/2 3.20.1, µ=1/2(6=0, 1/2) 3.23.1,=3/4 3.29.1, µ=1/2 3.8.1,=0, µ=1/2 2.9.1, µ=1/2(6=1/2, 0, 1, 3/2) 2.19.1,=1/2 2.21.1,=3/4 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 r1,3 0 0 0 0 0 0 1 A 4.8.1,=0, µ=1/2 3.8.1,=1/2, µ=0 2.8.1,=1/2 2.9.1,=2µ(µ6=0, 1/4, 1/3, 1/2, 1) 1.2.1, µ=/2(6=2) 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 0 r2,3 0 0 0 1 A 4.11.5 3.19.17 3.20.25, µ 6= 0 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 0 1 0 0 0 1 A 3.20.1,=1/2(µ6=0, 1/2) 3.7.1, = 1/2 2.1.1,=1/2 2.9.1,=1/2(µ6=0,1/2, 1/4, 1/2) 1.2.1,=1/2(µ6=1/2, 1) 1.7.1,=1/2 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 q1,2 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 0 0 0 0 0 1 A 3.14.1, µ 6= 0, 2 2.9.1, = 1, µ = 1/2 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 r1,3 0 0 r2,3 0 0 0 1 A 3.20.26, 6= 1/3, 1/4 (q1,2 = 0 6= 1/2) 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 q1,2 0 0 0 0 0 1 0 1 A, 0 @ 0 0 0 0 0 0 0 0 0 1 A 2.19.5 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 r1,3 0 0 1 0 0 0 1 A 1.2.1, µ = 2 ( 6= 1/3, 1/4) 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 0 0 0 q3,2 0 1 A, 0 @ 0 0 0 0 0 0 0 0 0 1 A 1.2.1, = 1/2, µ = 1 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 q1,2 0 0 0 0 0 q3,2 0 1 A, 0 @ 0 0 0 0 0 0 0 0 0 1 A We have equiaffine connections only in the cases 2.9.1, = 3/2, µ = 1/2; 1.2.1,= 2/3, µ=1/3 and 2.9.1,=2/3, µ=1/3; 3.20.25, µ=1/3; 2.9.1,=1/2, µ=3/2 (respectively). The Ricci tensors Ric(y, z)=tr{x 7! R(x, y)z} are equal to zero. 6 Art Discrete Appl. Math. 4 (2021) #P2.03 2.2 The curvature tensor is not only zero Theorem 2.2. I. There are no pairs (ḡ, g), admitting nontrivial equiaffine connections with nonzero curvature tensor and only zero torsion tensor. II. Any pair (ḡ, g), allows nontrivial affine connections, the curvature tensor is not only zero, the torsion tensor is only zero (i.e. if ḡ is nonsolvable then d.n.m=4.21.11 (µ=1/2), 3.13.6 (µ=1/2), 2.8.7 (=1/2), if ḡ is solvable then d.n.m=3.13.2 (µ=1/2, 1/4), 3.20.4 (=1/4), 3.20.26 (=1/4), 3.20.27, 2.9.1 (=1/2, µ=1/4), 2.9.3 (µ=1/4), 1.2.1 (=1/4, µ=1/2), see [2]), admits locally equiaffine connections. The Ricci tensors are zero. Proof. Just as earlier, in case, for example, 3.13.2 (µ = 1/2, 1/4) we have tr⇤([x, y])=0 for all x, y 2 ḡ, the torsion tensor and Ricci tensor are zero and locally equiaffine connec- tion has the form, presented in the table. The connection is equiaffine if tr⇤(e1) = 0 ) µ = 2, but in this case 0 < µ < 1 ) the pair does not allow equiaffine connections. In the case 3.20.4 the connection is equiaffine if =2, but < 1/3. Similarly we obtain the results in other cases: – ḡ is solvable: Pair Locally equiaffine connection 3.13.2, µ = 1/4 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 0 r2,3 0 0 0 1 A 3.13.2, µ = 1/2 3.20.4, = 1/4 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 r1,3 0 0 0 0 0 0 1 A 3.20.26, = 1/4 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 0 0 0 1 0 1 A, 0 @ 0 0 r1,3 0 0 0 0 0 0 1 A 3.20.27 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 0 0 0 1 0 1 A, 0 @ 0 0 0 0 0 0 0 0 0 1 A 2.9.1, = 1/2, µ = 1/4 2.9.3, µ = 1/4 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 q1,2 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 0 r2,3 0 0 0 1 A 1.2.1, = 1/4, µ = 1/2 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 0 0 0 q3,2 0 1 A, 0 @ 0 0 r1,3 0 0 0 0 0 0 1 A – ḡ is nonsolvable: Pair Locally equiaffine connection 4.21.11, µ = 1/2 0 @ 0 0 0 0 0 0 0 0 0 1 A , 0 @ 0 0 0 0 1 0 0 0 1 1 A , 0 @ 0 0 r1,3 0 0 0 0 1 0 1 A 3.13.6, µ = 1/2 0 @ 0 1 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 1 0 0 0 1 1 A, 0 @ 0 0 r1,3 0 0 0 0 1 0 1 A 2.8.7, = 1/2 0 @ 1/2 0 0 0 0 0 0 0 1/2 1 A, 0 @ 0 0 0 0 0 0 0 0 0 1 A, 0 @ 0 0 0 0 0 r2,3 1/2 0 0 1 A In these cases, equiaffine connections does not exist. The Ricci tensors are equal to zero. N. P. Mozhey: Torsion free equiaffine connections 7 3 Pairs of Lie algebras with only trivial locally equiaffine connections 3.1 The curvature tensor is only zero Theorem 3.1. I. If the pair (ḡ, g) admits only trivial equiaffine connection, the curvature and torsion tensors are zero, then (ḡ, g) is equivalent to one of the pairs: – ḡ is nonsolvable: 8.1.1(sl(3,R)); 6.2.1 e1 e2 e3 e4 e5 e6 u1 u2 u3 e1 0 0 0 (1)e4 0 (1)e6 u1 u2 u3 e2 0 0 2e3 e4 2e5 e6 u1 u2 0 e3 0 2e3 0 0 e2 e4 0 u1 0 e4 (1)e4 e4 0 0 e6 0 0 0 u1 e5 0 2e5 e2 e6 0 0 u2 0 0 e6 (1)e6 e6 e4 0 0 0 0 0 u2 u1 u1 u1 0 0 u2 0 0 0 0 u2 u2 u2 u1 0 0 0 0 0 0 u3 u3 0 0 u1 0 u2 0 0 0 , =1/2; 6.4.1 e1 e2 e3 e4 e5 e6 u1 u2 u3 e1 0 0 0 0 (1)e5 (1)e6 u1 u2 u3 e2 0 0 2e3 2e4 e5 e6 0 u2 u3 e3 0 2e3 0 e2 e6 0 0 0 u2 e4 0 2e4 e2 0 0 e5 0 u3 0 e5 (1)e5 e5 e6 0 0 0 0 u1 0 e6 (1)e6 e6 0 e5 0 0 0 0 u1 u1 u1 0 0 0 0 0 0 0 0 u2 u2 u2 0 u3 u1 0 0 0 0 u3 u3 u3 u2 0 0 u1 0 0 0 , =1/2; 4.2.1. e1 e2 e3 e4 u1 u2 u3 e1 0 0 0 0 u1 u2 u3 e2 0 0 2e3 2e4 u1 u2 0 e3 0 2e3 0 e2 0 u1 0 e4 0 2e4 e2 0 u2 0 0 u1 u1 u1 0 u2 0 0 0 u2 u2 u2 u1 0 0 0 0 u3 u3 0 0 0 0 0 0 , = 1/2; – ḡ is solvable: 5.10.1 ( = µ = 1), 4.8.1 ( = µ = 1), 4.9.1 ( = 0, µ = 2), 4.11.1 (=µ=1), 4.14.1 (=0, µ=2), 4.21.1 (µ=1 ( 6=1,3/2)), 3.8.1 ( = µ = 1), 3.13.1 ( = µ 1 (µ 6= 0, 1/2,1/3,1)), 3.16.1 ( = 2µ), 3.20.1 (=µ 1 ( 6= 0, 1/2,1,3/2)), 3.22.1 (=2µ ( 6= 0)), 3.23.1 (= 0), 3.27.1 (=1/2), 3.29.1 (µ=2), 2.2.1 (=µ=1), 2.4.1 (=0, µ=2), 2.9.1 (=µ 1 ( 6= 1/2, 0,1,2/3,3/2)), 2.16.1 (=1/2), 2.19.1 (=2), 1.2.1 (=µ 1 (µ 6=1/3,3/2, 0,2/3)), 1.4.1 (=2µ ( 6=0)), 1.7.1 (=2). The Ricci tensors are zero. II. Any pair (ḡ, g) that admits only trivial affine connection, the curvature and torsion tensors are zero (i.e. if ḡ is nonsolvable then d.n.m = 9.1.1, 8.1.1, 7.1.1, 7.2.1, 6.2.1, 8 Art Discrete Appl. Math. 4 (2021) #P2.03 6.3.1, 6.4.1(6=1/2), 5.1.1, 4.2.1(6=1/2), 4.3.1, 4.5.1, if ḡ is solvable then d.n.m = 6.5.1, 5.4.1, 5.5.1, 5.6.1, 5.7.1, 5.8.1, 5.9.1, 5.10.1 (2 + µ2 6= 0, ( 1)2 + (µ + 1)2 6= 0, (1/2)2+µ2 6=0), 4.4.1, 4.6.1, 4.7.1, 4.8.1 ((+1)2+(µ1)2 6=0, 2+(µ1/2)2 6=0, 2 + µ2 6= 0), 4.9.1 (2 + µ2 6= 0), 4.11.1 (2 + µ2 6= 0, (µ + 1)2 + ( 1)2 6= 0, µ2 + ( 1/2)2 6=0), 4.12.1, 4.13.1, 4.14.1 ((µ 2)2 + 2 6=0), 4.15.1, 4.16.1, 4.17.1, 4.18.1, 4.19.1, 4.20.1 ( 6=0,1), 4.21.1 (µ 6=0, µ 6=1/2, µ 6=1 ), 4.22.1, 3.1.1, 3.2.1, 3.6.1, 3.7.1 ( 6=0, 1/2), 3.8.1 (2 + (µ 1/2)2 6=0, 2 + µ2 6=0, ( 1/2)2 + µ2 6=0, ( + 1)2 + (µ 1)2 6=0, ( 1)2 + (µ + 1)2 6=0), 3.9.1, 3.10.1, 3.11.1, 3.12.1, 3.13.1 (µ 6= 0, µ 6= 1 , µ 6= 1/2, µ 6= 1, µ 6= /2), 3.16.1, 3.17.1 ( 6= 0), 3.18.1, 3.19.1 ( 6= 0,1), 3.20.1 ( 6= 0, 6= 1/2, µ 6= 0, µ 6= 1/2, µ 6= 1 ), 3.21.1 ( 6= 0), 3.22.1 ( 6= 2µ), 3.23.1 ( 6= 2/3, 1/2, 3/4, 3.24.1, 3.26.1, 3.27.1 ( 6= 0, 1/2), 3.28.1, 3.29.1 (µ 6=0, 1/2), 3.30.1, 3.31.1, 2.1.1 ( 6=0, 1/2), 2.2.1 ((1)2+(µ1)2 6=0), 2.3.1, 2.4.1 (2 + µ2 6=0, 2 + (µ 2)2 6=0), 2.5.1, 2.6.1, 2.8.1 ( 6=0, 1/2, 1,1), 2.9.1 ( 6=1/2, 6=0, 6=1 µ, 6=2µ, 6=µ+ 1, µ 6=0, µ 6=1/2), 2.10.1, 2.11.1, 2.12.1, 2.14.1, 2.16.1 ( 6= 0, 1/2), 2.19.1 ( 6= 0, 1/2), 2.21.1 ( 6= 0, 1/2, 2/3, 3/4), 2.22.1, 1.2.1 (µ 6= + 1, µ 6=2, µ 6=1 , µ 6=/2, 6=1/2), 1.4.1 (µ 6=2), 1.7.1 ( 6=0, 2, 1/2), 1.9.1, see [2]), admits the locally equiaffine connection. Proof. If, for example, (ḡ, g) is the space 8.1.1 (sl(3,R)), ⇤|g=, ⇤ is g-invariant ) ⇤(u1)=⇤(u2)=⇤(u3)=0, then the torsion and Ricci tensors are zero, tr⇤(ei)=0, i = ¯1, 8, (⇤(ei) 2 sl(3,R)) ) the connection is equiaffine (and locally equiaffine). In the other cases are similarly. 3.2 The curvature tensor is not zero for some connections Theorem 3.2. I. If the pair (ḡ, g) admits only trivial equiaffine connection, the curvature tensor is not zero, the torsion tensor is zero, then (ḡ, g) is equivalent to one and only one of the pairs 4.21.2. e1 e2 e3 e4 u1 u2 u3 e1 0 (1)e2 (31)/2e3 (1+)/2e4 u1 u2 (1)/2u3 e2 (1)e2 0 e4 0 0 u1 0 e3 (13)/2e3 e4 0 0 0 0 u2 e4 (1+)/2e4 0 0 0 0 0 u1 u1 u1 0 0 0 0 0 0 u2 u2 u1 0 0 0 0 e4 u3 (1)/2u3 0 u2 u1 0 e4 0 , =3; 3.13.4, 1<µ<0 e1 e2 e3 u1 u2 u3 e1 0 (1+µ)e2 (1 µ)e3 u1 (1+2µ)u2 µu3 e2 (µ+1)e2 0 0 0 0 u2 e3 (µ1)e3 0 0 0 0 u1 u1 u1 0 0 0 0 e2 u2 (2µ+1)u2 0 0 0 0 0 u3 µu3 u2 u1 e2 0 0 , µ = 2/3; N. P. Mozhey: Torsion free equiaffine connections 9 3.20.5, µ1/3 e1 e2 e3 u1 u2 u3 e1 0 2µe2 (1µ)e3 u1 (12µ)u2 µu3 e2 2µe2 0 0 0 u1 0 e3 (µ1)e3 0 0 0 0 u1 u1 u1 0 0 0 0 0 u2 (2µ1)u2 u1 0 0 0 e3 u3 µu3 0 u1 0 e3 0 , µ = 2; 2.9.3. e1 e2 u1 u2 u3 e1 0 (1 µ)e2 u1 (1 2µ)u2 µu3 e2 (µ 1)e2 0 0 0 u1 u1 u1 0 0 0 0 u2 (2µ 1)u2 0 0 0 e2 u3 µu3 u1 0 e2 0 , µ = 2. In these cases ḡ is solvable. II. If the pair (ḡ, g), admitting only trivial affine connection (with nonzero curvature tensor and zero torsion tensor), does not admit locally equiaffine connection, then (ḡ, g) is equivalent to one of the pairs: 2.9.12. e1 e2 u1 u2 u3 e1 0 e2 u1 2u2 2u3 e2 e2 0 0 0 u1 u1 u1 0 0 e2 0 u2 2u2 0 e2 0 e1 u3 2u3 u1 0 e1 0 ; 3.8.8. e1 e2 e3 u1 u2 u3 e1 0 0 e3 u1 0 0 e2 0 0 e3 0 u2 u3 e3 e3 e3 0 0 0 u1 u1 u1 0 0 0 e3 0 u2 0 u2 0 e3 0 2e2 e1 u3 0 u3 u1 0 e1 2e2 0 . III. Any pair (ḡ, g) that admits only trivial affine connection, the curvature tensor is not zero, the torsion tensor is zero, except 2.9.12 and 3.8.8 (i.e. if ḡ is nonsolvable then d.n.m = 4.11.2, 4.13.2, 4.13.3, 2.1.2, 2.3.2, 2.3.3, if ḡ is solvable then d.n.m = 5.10.2, 4.8.10, 4.11.4, 4.20.2, 4.21.2 ( 6= 1), 3.8.9, 3.13.2 (µ 6= 0, 1/2, 1/4), 3.13.4, 3.14.2, 3.19.16, 3.20.4 ( 6= 0, 1/4), 3.20.5 (µ 6= 1/2), 3.23.2, 3.27.2, 2.8.6, 2.9.3 (µ 6= 0, 1/2, 1/4), 2.16.2, see [2]) admits the locally equiaffine connection. The Ricci tensors has the form (in the other cases Ricci tensors are zero): Pair Ricci tensor Pair Ricci tensor Pair Ricci tensor 4.11.2 0 @ 0 0 0 0 0 2 0 2 0 1 A 4.13.2 0 @ 0 0 0 0 2 0 0 0 2 1 A 4.13.3 0 @ 0 0 0 0 2 0 0 0 2 1 A 2.1.2 0 @ 0 1 0 1 0 0 0 0 0 1 A 2.3.2 0 @ 1 0 0 0 1 0 0 0 0 1 A 2.3.3 0 @ 1 0 0 0 1 0 0 0 0 1 A 10 Art Discrete Appl. Math. 4 (2021) #P2.03 Proof. In the case 2.9.12 we have ⇤ is g-invariant ) ⇤(u1) = ⇤(u2) = ⇤(u3) = 0, Ricci tensor has the form 0 @ 0 0 0 0 0 3 0 2 0 1 A and Ricci tensor is not symmetric. Also we have tr⇤(e1) 6= 0 ) the connection is not locally equiaffine (and equiaffine too). In the case 3.8.8 Ricci tensor as in the case 2.9.12, also we have tr⇤(e1 2e2) 6=0 ) the connection is not locally equiaffine (and equiaffine too). In other cases the connection is locally equiaffine (tr⇤([x, y])=0 for all x, y 2 ḡ), the Ricci tensors have the form, presented in the theorem, and Ricci tensors are symmetric. So for all three-dimensional homogeneous spaces, not admitting connections with non- zero torsion tensor, it is determined under what conditions the connection is equiaffine (locally equiaffine). In addition, equiaffine (locally equiaffine) connections and Ricci ten- sors are written out in explicit form. For example, there are only two spaces (not admitting connections with nonzero torsion tensor) that admit affine connections, but do not admit locally equiaffine connections. There are no pairs, admitting nontrivial equiaffine connec- tions with nonzero curvature tensor and only zero torsion tensor. In this work we use the algebraic approach for description of connections, methods of the theory of Lie groups, Lie algebras and homogeneous spaces. The results can find applications in mathematics and physics, since many fundamental problems in these fields are reduced to the study of invariant objects on homogeneous spaces. ORCID iDs Natalya P. Mozhey https://orcid.org/0000-0001-9237-7208 References [1] S. Kobayashi, Transformation groups in differential geometry, Classics in Mathematics, Springer-Verlag, Berlin, 1995, doi:10.1007/978-3-642-61981-6, reprint of the 1972 edition. [2] N. P. Mozhey, Torsion free affine connections on three-dimensional homogeneous spaces, Sib. Èlektron. Mat. Izv. 14 (2017), 280–295, doi:10.17377/semi.2017.14.026. [3] K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33–65, doi:10.2307/2372398. [4] K. Nomizu and T. Sasaki, Affine differential geometry, volume 111 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1994, geometry of affine immersions. ISSN 2590-9770 The Art of Discrete and Applied Mathematics 4 (2021) #P2.04 https://doi.org/10.26493/2590-9770.1353.c0e (Also available at http://adam-journal.eu) Symmetrical 2-extensions of the 3-dimensional grid. I Kirill V. Kostousov* Krasovskii Institute of Mathematics and Mechanics, S. Kovalevskaya Str, 16, 620108, Yekaterinburg, Russia Received 24 January 2020, accepted 30 May 2020, published online 10 March 2021 Abstract For a positive integer d, a connected graph is a symmetrical 2-extension of the d- dimensional grid ⇤d if there exists a vertex-transitive group G of automorphisms of and its imprimitivity system with blocks of size 2 such that there exists an isomorphism ' of the quotient graph / onto ⇤d. The tuple (, G,,') with specified components is called a realization of the symmetrical 2-extension of the grid ⇤d. Two realizations (1, G1, 1,'1) and (2, G2,2,'2) are called equivalent if there exists an isomorphism of the graph 1 onto 2 which maps 1 onto 2. V.I. Trofimov proved that, up to equivalence, there are only finitely many realizations of symmetrical 2-extensions of ⇤d for each positive integer d. E.A. Konovalchik and K.V. Kostousov found all, up to equivalence, realizations of symmetrical 2-extensions of the grid ⇤2. In this work we found all, up to equivalence, realizations (, G,,') of symmetrical 2-extensions of the grid ⇤3 for which only the trivial automorphism of preserves all blocks of . Namely we prove that there are 5573 such realizations, and that among corresponding graphs there are 5350 pairwise non- isomorphic. Keywords: Symmetrical extensions of a graph, d-dimensional grid. Math. Subj. Class.: 20H15 1 Introduction Recall that, for a positive integer d, a d-dimensional grid ⇤d is a graph whose vertices are integer tuples (a1, . . . ad) and two vertices (a01, . . . , a0d) and (a 00 1 , . . . , a 00 d ) are adjacent if and only if |a01 a001 |+ . . .+ |a0d a00d | = 1. According to [6] for a finite graph , define *Our work was performed using ‘Uran’ supercomputer of IMM UB RAS, Yekaterinburg, Russia. E-mail address: kkostousov@gmail.com (Kirill V. Kostousov) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 2 Art Discrete Appl. Math. 4 (2021) #P2.04 a connected graph to be a symmetrical extension of ⇤d by if there exists a vertex- transitive group G of automorphisms of and an imprimitivity system of G on V () such that subgraphs of generated by blocks of are isomorphic to and there exists an isomorphism of / (i.e., of factor-graph of by partition of its vertex set) onto ⇤d. A tuple (, G,,') with specified components is called a realization of symmetrical extension of the grid ⇤d by the graph . For a positive integer q, a graph is called a symmetrical q-extension of the grid ⇤d, if is a symmetrical extension of the grid ⇤d by some graph such that |V ()| = q. In this situation the tuple (, G,,') with specified components is called a realization of the symmetrical q-extension of the grid ⇤d, and we say that is a graph of this realization. Along with purely mathematical interest, symmetrical q-extensions of the grid ⇤d for small d 1 and q > 1 are iteresting for crystallography and some physical theories (see [5]). For crystallography, symmetrical 2-extensions of grids ⇤d are of the most interest. They naturally arise when considering “molecular” crystals whose “molecules” consist of two “atoms” or, more generally, have a distinguished axis. It is natural to consider realizations of symmetric q-extensions of the grid ⇤d up to equivalence defined as follows (see [5]). We call two such realizations R1 = (1, G1, 1,'1) and R2 = (2, G2,2,'2) equivalent and write R1 ⇠ R2 if there exists an isomor- phism of the graph 1 to the graph 2 which maps 1 onto 2. We say that the realization (, G,,') of the symmetrical q-extension of the grid ⇤d is maximal if G = Aut() is the group of all automorphisms of the graph which preserve the partition . It is clear that each realization of the symmetrical q-extension of the grid ⇤d has an equivalent maximal realization (unique up to equivalence). V.I. Trofimov proved that, up to equivalence, for an arbitrary positive integer d, there is only a finite number of realizations of symmetrical 2-extensions of d-dimensional grid (see [7, Theorem 2]). An algorithm for constructing these extensions is also proposed in [7]. Using this algorithm, in [3] and [4] were found all, up to equivalence, realizations of symmetrical 2-extensions of the grid ⇤2 (162 realizations). Among the graphs of these realizations, there are exactly 152 pairwise nonisomorphic graphs. For an arbitrary realization (, G,,') of the symmetrical 2-extension of the grid ⇤d and an arbitrary pair of adjacent vertices B1, B2 of the graph /, the set of edges of the graph , one end of which lies in B1 and the other in B2, will be called a connection. The following types of connections are possible: type 1 means a single edge; type 2|| means two non-adjacent edges; type 2V means two adjacent edges; type 3 means three edges; type 4 means full connection (4 edges). A realization that necessarily has connections of types not equal to 2|| and 4 will be called a realization of class I. A realization that have connections only of types 2|| and 4 (maybe only of one type) will called a realizations of class II. By Proposition 4 from [7], realizations of class I are exactly the realizations of symmetrical 2-extensions of the grid ⇤d such that only a trivial automorphism of their graph fixes all blocks. All 162 realizations of symmetrical 2-extensions of the grid ⇤2 are distributed in classes I and II as follows: 87 realizations of class I (see [3]) and 75 realizations of class II (see [4]). This paper is devoted to the description of all, up to equivalence, realizations of symmetrical 2-extensions of the grid ⇤3 of class I. A realization of symmetrical extension of the grid ⇤d by the graph K2 (full graph on two vertices) will be called a it saturated realization of the symmetrical 2-extension of the grid ⇤d. Accordingly, a realization of symmetrical extension of the grid ⇤d by the K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 3 graph complemented to K2 will be call a it non-saturated realization of the symmetrical 2-extension of the grid ⇤d. In this paper, we have proved that, up to equivalence, there are 5573 realizations of symmetrical 2-extensions of the grid ⇤3 of class I, among which 2872 are saturated and 2701 are non-saturated (see Theorem 1 and Corollary 1). Among the graphs of saturated realizations of symmetrical 2-extensions of the grid ⇤3 of class I there are exactly 2792 pairwise nonisomorphic; among the graphs of non-saturated realizations of class I there are exactly 2594 pairwise nonisomorphic; and among all graphs of realizations of class I there are 5350 pairwise nonisomorphic (see Corollary 2). In Sec. 3, we give the description of all, up to equivalence, realizations of symmetrical 2-extensions of the grid ⇤3 of class I (Theorem 1 and Corollary 1). This is obtained using the approach from [7] implemented in GAP [2] (Algorithms 1 and 2 from [3]). Sec. 2 contains preliminary results. 2 Preliminaries Using GAP [2], we have listed all conjugacy classes of vertex-transitive subgroups of the group Aut(⇤3). It turned out that there are 786 such classes. The details are as follows. Each vertex-transitive group of automorphisms of ⇤3 is generated by the stabilizer in this group of the vertex (0, 0, 0) and six elements of this group that translate the vertex (0, 0, 0) to the vertices adjacent to it. Let S0 be a stabilizer of (0, 0, 0) in Aut(⇤3), N1 = S0tx, N2 = t1x , N3 = S0ty, N4 = t1x , N5 = S0tz, N6 = t 1 z , and N = N1[N2[ ...[N6. We look over all subgroups S of S0, up to conjugation in S0. For every S, using backtracking, we search for all minimal subsets N 0 of N such that |N 0 \Ni|  1 for all i = 1, ..., 6 and hS,N 0i \Ni 6= ; for all i = 1, ..., 6. For every found N 0, if hS,N 0i is a proper subgroup of Aut(⇤3), then we put this group into the resulting list H. At the end, we thin out the list L up to conjugation in Aut(⇤3). It turned out that |H| = 786, H = H1, . . . , H786 . These groups are given in Table 2 below by their generating systems. The following notation is used for certain automor- phisms of the grid ⇤3: rx : (x, y, z) 7! (x,z, y), ry : (x, y, z) 7! (z, y,x), rz : (x, y, z) 7! (y,x, z), mx : (x, y, z) 7! (x, y, z), my : (x, y, z) 7! (x,y, z), mz : (x, y, z) 7! (x, y,z), i : (x, y, z) 7! (x,y,z), tx : (x, y, z) 7! (x+ 1, y, z), ty : (x, y, z) 7! (x, y + 1, z), tz : (x, y, z) 7! (x, y, z + 1), where x, y, z 2 Z. Remark 2.1. In the natural embedding of the grid ⇤3 in the Euclidean space R3, each auto- morphism g 2 Aut(⇤3) is induced by the only isometry g̃ of this space. The isometries that induce the above automorphisms of ⇤3 have the following geometric meaning: r̃x, r̃y, r̃z are rotations by the angle ⇡2 around coordinate axes x, y, and z respectively, m̃x, m̃y, m̃z are reflections relative coordinate planes, ĩ — central symmetry about the origin, t̃x, t̃y, t̃z are translations by 1 along axes x, y, and z respectively. Using GAP, we constructed and tested 786 stabilizers {H(0,0,0) : H 2 H} for con- jugacy in Aut(⇤3)(0,0,0). It turned out that, up to conjugation, there are only 33 such 4 Art Discrete Appl. Math. 4 (2021) #P2.04 stabilizers. We give them in Table 1 by their generators indicated in column 3. For each of the 33 groups, the abstract group structure is given in column 2. T a b l e 1 Stabilizers of vertex (0, 0, 0) in vertex-transitive subgroups of Aut(⇤3) up to conjugation in Aut(⇤3)(0,0,0) Group structure Generators 1 1 1 2 C2 hii 3 C2 hmzi 4 C2 hr2zrxi 5 C2 hr2zi 6 C2 hmzrxi 7 C3 hr1y r1z i 8 C2 ⇥ C2 hr2y, r2zi 9 C2 ⇥ C2 hi, r2zi 10 C2 ⇥ C2 hmzr1x , r2xi 11 C2 ⇥ C2 hmx, r2yrxi 12 C2 ⇥ C2 hi, r2zrxi 13 C2 ⇥ C2 hr2yrx, r2zrxi 14 C2 ⇥ C2 hmx, r2zi 15 C4 hr1x , r2xi 16 C4 hmxr1x i 17 C6 hi,mxr1y r1x i 18 S3 hmxry, r1y r1z i 19 S3 hr2yrz, r2zrxi 20 C2 ⇥ C2 ⇥ C2 hi, r2yrx, r2zrxi 21 C2 ⇥ C2 ⇥ C2 hi, r2y, r2zi 22 C4 ⇥ C2 hi,mxr1x i 23 D8 hmxr1x , r2zrxi 24 D8 hr2z , r2zrxi 25 D8 hmy,mzr1x , r2xi 26 D8 hmxr1x , r2zi 27 A4 hryr1x , r2y, r2zi 28 D12 hi, r2yrz, r2zrxi 29 C2 ⇥D8 hi, r2z , r2zrxi 30 C2 ⇥A4 hi,mxr1y r1x , r2y, r2zi 31 S4 hmxr1x ,mxrz, r2zi 32 S4 hr2yrz, r2z , r2zrxi 33 C2 ⇥ S4 hi, r2yrz, r2z , r2zrxi Each group H 2 H is identified with some space group and, therefore, has a point group P(H) and a translation basis (see [1]). Using GAP, we verified that the set of point groups {P(H) : H 2 H} is equal, up to conjugation in P(Aut(⇤3)), to the set of 33 stabilizers given in Table 1. In column 1 of Table 2 below, we give the set of groups H defined by their generators. In column 2, for each group H 2 H, we give the of the group from Table 1 conjugate to the stabilizer H(0,0,0) in Aut(⇤3)(0,0,0). In column 3, for K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 5 each group H 2 H, we give the of the group from Table 1 conjugate to the point group P(H) in P(Aut(⇤3)). In column 4, for each group H 2 H, we give its translation basis. The groups in Table 2 are sorted lexicographically first by in column 2 and then by in column 3. T a b l e 2 Representatives of conjugacy classes for vertex-transitive subgroups of Aut(⇤3) H H0,0,0 P (H) Translation basis of H H1 = htx, t 1 y , t 1 z i 1 1 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H2 = hitz, it 1 z , tx, t 1 y i 1 2 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H3 = hity, it 1 y , itz, it 1 z , txi 1 2 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H4 = hitx, it 1 x , ity, it 1 y , itz, it 1 z i 1 2 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H5 = hmxt 1 y , tx, t 1 z i 1 3 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H6 = hmyty,myt 1 y , tx, t 1 z i 1 3 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H7 = hmyty,myt 1 y ,myt 1 z , txi 1 3 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H8 = hmytx,myt 1 z , t 1 y i 1 3 [ [ 1, 0, 1 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H9 = hmytx,myty,myt 1 y ,myt 1 z i 1 3 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H10 = hr 2 x ryty, r 2 x ryt 1 y , tx, t 1 z i 1 4 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H11 = hr 2 z rxt 1 y , r 2 z rxtz, txi 1 4 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H12 = hryr 2 x tx, ryr 2 x t 1 x , ryr 2 x ty, ryr 2 x t 1 y i 1 4 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H13 = hr 2 x ty, r 2 x t 1 y , tx, t 1 z i 1 5 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H14 = hr 2 z t 1 z , tx, t 1 y i 1 5 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H15 = hr 2 x ty, r 2 x t 1 y , r 2 x tz, r 2 x t 1 z , txi 1 5 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H16 = hr 2 z ty, r 2 z t 1 y , r 2 z t 1 z , txi 1 5 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H17 = hr 2 y tx, r 2 y t 1 x , r 2 y t 1 y , r 2 y tz, r 2 y t 1 z i 1 5 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H18 = hmxr 1 y t 1 y , tx, t 1 z i 1 6 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H19 = hmzr 1 x t 1 y ,mzr 1 x tz, txi 1 6 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H20 = hmzr 1 x tx,mzr 1 x t 1 y ,mzr 1 x tzi 1 6 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H21 = hryrxt 1 x , ryrxt 1 z , r 1 y r 1 z txi 1 7 [ [ 1, 0, 2 ], [ 0, 1, 2 ], [ 0, 0, 3 ] ] H22 = hr 2 x t 1 y , r 2 z ty, tx, t 1 z i 1 8 [ [ 1, 0, 0 ], [ 0, 4, 0 ], [ 0, 0, 1 ] ] H23 = hr 2 x ty, r 2 x t 1 y , r 2 z t 1 z , txi 1 8 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H24 = hr 2 x ty, r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , txi 1 8 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H25 = hr 2 x ty, r 2 x t 1 y , r 2 y tx, r 2 y t 1 x , t 1 z i 1 8 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H26 = hr 2 y t 1 y , r 2 z t 1 z , txi 1 8 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H27 = hr 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 z ty, txi 1 8 [ [ 1, 0, 0 ], [ 0, 2, 1 ], [ 0, 0, 2 ] ] H28 = hr 2 x ty, r 2 x t 1 y , r 2 x tz, r 2 x t 1 z , r 2 y tx, r 2 y t 1 x i 1 8 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H29 = hr 2 y tx, r 2 y t 1 x , r 2 z ty, r 2 z t 1 y , r 2 z t 1 z i 1 8 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H30 = hr 2 y tx, r 2 y t 1 x , r 2 y t 1 y , r 2 z t 1 z i 1 8 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H31 = hr 2 x ty, r 2 x t 1 y , r 2 x tz, r 2 x t 1 z , r 2 y tx, r 2 z t 1 x i 1 8 [ [ 2, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H32 = hr 2 x tz, r 2 x t 1 z , r 2 y tx, r 2 y t 1 x , r 2 z ty, r 2 z t 1 y i 1 8 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H33 = hr 2 x ty, r 2 x t 1 y , r 2 y tx, r 2 y t 1 x , r 2 z t 1 z i 1 8 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H34 = hr 2 x tx, r 2 y t 1 y , r 2 z t 1 z i 1 8 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H35 = hity, r 2 x t 1 y , tx, t 1 z i 1 9 [ [ 1, 0, 0 ], [ 0, 4, 0 ], [ 0, 0, 1 ] ] H36 = hity,myt 1 y , tx, t 1 z i 1 9 [ [ 1, 0, 0 ], [ 0, 4, 0 ], [ 0, 0, 1 ] ] H37 = hity, it 1 z , r 2 x t 1 y , r 2 x tz, txi 1 9 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 4 ] ] H38 = hit 1 z ,myty,myt 1 y , r 2 y tz, txi 1 9 [ [ 1, 0, 0 ], [ 0, 1, 2 ], [ 0, 0, 4 ] ] H39 = hmxt 1 y , r 2 x tz, r 2 x t 1 z , txi 1 9 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H40 = hitz, it 1 z , r 2 x ty, r 2 x t 1 y , txi 1 9 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H41 = hitz, it 1 z ,myty,myt 1 y , txi 1 9 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H42 = hmyty,myt 1 y , r 2 y tz, r 2 y t 1 z , txi 1 9 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H43 = hitz, it 1 z ,mzt 1 y , txi 1 9 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H44 = hmzt 1 y , r 2 z t 1 z , txi 1 9 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H45 = hitz, it 1 z ,mytx, t 1 y i 1 9 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H46 = hity, it 1 y , r 2 z t 1 z , txi 1 9 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H47 = hity, it 1 y , r 2 y tz, r 2 y t 1 z , txi 1 9 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H48 = hity,mxt 1 z , r 2 x t 1 y , txi 1 9 [ [ 1, 0, 0 ], [ 0, 2, 1 ], [ 0, 0, 2 ] ] H49 = hity,myt 1 y , r 2 y tz, r 2 y t 1 z , txi 1 9 [ [ 1, 0, 0 ], [ 0, 2, 1 ], [ 0, 0, 2 ] ] H50 = hmyty,myt 1 y , r 2 y tx, r 2 y t 1 x , r 2 y tz, r 2 y t 1 z i 1 9 [ [ 1, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H51 = hmyty,myt 1 y ,myt 1 z , r 2 y tx, r 2 y t 1 x i 1 9 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H52 = hitz, it 1 z ,mytx,myty,myt 1 y i 1 9 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H53 = hity, it 1 y , itz, it 1 z ,mztxi 1 9 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H54 = hmztx, r 2 z ty, r 2 z t 1 y , r 2 z t 1 z i 1 9 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H55 = hitz, it 1 z ,mztx,mzt 1 y i 1 9 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H56 = hmztx,mzt 1 y , r 2 z t 1 z i 1 9 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H57 = hitx, it 1 x , ity, it 1 y , r 2 z t 1 z i 1 9 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] 6 Art Discrete Appl. Math. 4 (2021) #P2.04 H H0,0,0 P (H) Translation basis of H H58 = hitx, it 1 x , ity, it 1 y , r 2 y tz, r 2 y t 1 z i 1 9 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H59 = hitx, it 1 x , ity, it 1 y ,mztz,mzt 1 z i 1 9 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H60 = hitx, it 1 x , r 2 z ty, r 2 z t 1 y , r 2 z t 1 z i 1 9 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H61 = hitx, it 1 x , r 2 x ty, r 2 x t 1 y , r 2 x tz, r 2 x t 1 z i 1 9 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H62 = hitx, it 1 y ,mztz,mzt 1 z , r 2 z t 1 x , r 2 z tyi 1 9 [ [ 1, 1, 0 ], [ 0, 2, 1 ], [ 0, 0, 2 ] ] H63 = hity,myt 1 y , r 2 y tx, r 2 y t 1 x , r 2 y tz, r 2 y t 1 z i 1 9 [ [ 1, 0, 1 ], [ 0, 2, 1 ], [ 0, 0, 2 ] ] H64 = hit 1 z ,mytx,myty,myt 1 y , r 2 y tzi 1 9 [ [ 1, 0, 2 ], [ 0, 1, 2 ], [ 0, 0, 4 ] ] H65 = hitz, it 1 z ,myty,myt 1 y , r 2 y tx, r 2 y t 1 x i 1 9 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H66 = hity, it 1 y ,mztx, r 2 z t 1 z i 1 9 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H67 = hitz, it 1 z ,mztx, r 2 z ty, r 2 z t 1 y i 1 9 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H68 = hmzr 1 x tz, r 2 x t 1 y , r 2 x t 1 z , txi 1 10 [ [ 1, 0, 0 ], [ 0, 1, 3 ], [ 0, 0, 4 ] ] H69 = hmzr 1 x tx, r 2 x ty, r 2 x t 1 y , r 2 x tz, r 2 x t 1 z i 1 10 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H70 = hmzrxt 1 y ,mzrxt 1 z ,mzr 1 x txi 1 10 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H71 = hmxrytx,mxryt 1 x , r 2 y t 1 y i 1 10 [ [ 1, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H72 = hmxryt 1 y ,mxr 1 y t 1 x , r 2 y tx, r 2 y tzi 1 10 [ [ 1, 0, 3 ], [ 0, 1, 2 ], [ 0, 0, 4 ] ] H73 = hmyty, r 2 x ryt 1 y , tx, t 1 z i 1 11 [ [ 1, 0, 0 ], [ 0, 4, 0 ], [ 0, 0, 1 ] ] H74 = hmzr 1 x t 1 y , r 2 z rxtz, txi 1 11 [ [ 1, 0, 0 ], [ 0, 1, 3 ], [ 0, 0, 4 ] ] H75 = hmzr 1 x tx, r 2 z rxt 1 y , r 2 z rxtzi 1 11 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H76 = hmxt 1 y ,mxt 1 z ,mzr 1 x txi 1 11 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H77 = hmztx,mzt 1 y , r 2 y rztz, r 2 y rzt 1 z i 1 11 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H78 = hmxtx,mxt 1 x , r 2 z rxt 1 y , r 2 z rxtzi 1 11 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H79 = hmxrytx,mxryt 1 x , ryr 2 x ty, ryr 2 x t 1 y i 1 11 [ [ 1, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H80 = hmxrytx,mxryt 1 x ,myty,myt 1 y i 1 11 [ [ 1, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H81 = hmxtx,mzr 1 x t 1 y ,mzr 1 x tz, r 2 z rxt 1 x i 1 11 [ [ 2, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H82 = hmxtx,mxt 1 x ,mzr 1 x tz, r 2 z rxt 1 y i 1 11 [ [ 1, 0, 2 ], [ 0, 1, 3 ], [ 0, 0, 4 ] ] H83 = hity, r 2 x ryt 1 y , tx, t 1 z i 1 12 [ [ 1, 0, 0 ], [ 0, 4, 0 ], [ 0, 0, 1 ] ] H84 = hity, itz,mzr 1 x t 1 y , txi 1 12 [ [ 1, 0, 0 ], [ 0, 1, 3 ], [ 0, 0, 4 ] ] H85 = hit 1 y , it 1 z , r 2 z rxtz, txi 1 12 [ [ 1, 0, 0 ], [ 0, 1, 3 ], [ 0, 0, 4 ] ] H86 = hit 1 y , itz,mzr 1 x tx, r 2 y rxt 1 z i 1 12 [ [ 1, 0, 2 ], [ 0, 1, 1 ], [ 0, 0, 4 ] ] H87 = hitx, it 1 y , it 1 z , r 2 z rxt 1 x , r 2 z rxtzi 1 12 [ [ 1, 0, 1 ], [ 0, 1, 3 ], [ 0, 0, 4 ] ] H88 = hity, it 1 y , itz, it 1 z ,mzr 1 x txi 1 12 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H89 = hmzr 1 x tx, r 2 y rxt 1 y , r 2 y rxt 1 z i 1 12 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H90 = hitx, it 1 x , ity, it 1 y , r 2 y rztz, r 2 y rzt 1 z i 1 12 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H91 = hitx, it 1 x , r 2 z rxt 1 y , r 2 z rxtzi 1 12 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H92 = hitx, it 1 x ,mzr 1 x t 1 y ,mzr 1 x tzi 1 12 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H93 = hmxrytx,mxryt 1 x , r 2 x ryty, r 2 x ryt 1 y i 1 12 [ [ 1, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H94 = hitx,mzrxt 1 y ,mzrxt 1 z , r 2 z rxt 1 x i 1 12 [ [ 2, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H95 = hitx, ity,mxrzt 1 x , r 2 x rztz, r 2 x rzt 1 z i 1 12 [ [ 1, 1, 1 ], [ 0, 2, 1 ], [ 0, 0, 2 ] ] H96 = hr 2 x ryt 1 y , ryr 2 x ty, tx, t 1 z i 1 13 [ [ 1, 0, 0 ], [ 0, 4, 0 ], [ 0, 0, 1 ] ] H97 = hr 2 x t 1 y , r 2 x tz, r 2 y rxt 1 z , txi 1 13 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 4 ] ] H98 = hryr 2 x tx, ryr 2 x t 1 x , r 2 y t 1 y i 1 13 [ [ 1, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H99 = hr 2 x ryty, r 2 x ryt 1 y , ryr 2 x tx, ryr 2 x t 1 x i 1 13 [ [ 1, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H100 = hr 2 x ty, r 2 x t 1 y , r 2 x tz, r 2 x t 1 z , r 2 y rxtx, r 2 y rxt 1 x i 1 13 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H101 = hr 2 x ryt 1 y , ryr 2 x ty, r 2 y tx, r 2 y t 1 x , r 2 y tz, r 2 y t 1 z i 1 13 [ [ 1, 0, 1 ], [ 0, 2, 1 ], [ 0, 0, 2 ] ] H102 = hr 2 x ryt 1 x , ryr 2 x ty, ryr 2 x t 1 y , r 2 y tx, r 2 y tzi 1 13 [ [ 1, 0, 3 ], [ 0, 1, 2 ], [ 0, 0, 4 ] ] H103 = hmyty, r 2 x t 1 y , tx, t 1 z i 1 14 [ [ 1, 0, 0 ], [ 0, 4, 0 ], [ 0, 0, 1 ] ] H104 = hmxt 1 y ,mzt 1 z , r 2 y tz, txi 1 14 [ [ 1, 0, 0 ], [ 0, 1, 2 ], [ 0, 0, 4 ] ] H105 = hmxt 1 y ,myt 1 z , txi 1 14 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H106 = hmxt 1 y ,mztz,mzt 1 z , txi 1 14 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H107 = hmyt 1 z , r 2 x ty, r 2 x t 1 y , txi 1 14 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H108 = hmztz,mzt 1 z , r 2 x ty, r 2 x t 1 y , txi 1 14 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H109 = hmyty,myt 1 y , r 2 z t 1 z , txi 1 14 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H110 = hmyty,myt 1 y ,mztz,mzt 1 z , txi 1 14 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H111 = hmzt 1 y , r 2 y tz, r 2 y t 1 z , txi 1 14 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H112 = hmyt 1 z ,mzt 1 y , txi 1 14 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H113 = hmzt 1 y , r 2 y tx, r 2 y t 1 x , t 1 z i 1 14 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H114 = hmytx, r 2 z t 1 z , t 1 y i 1 14 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H115 = hmyty,mztz,mzt 1 z , r 2 x t 1 y , txi 1 14 [ [ 1, 0, 0 ], [ 0, 2, 1 ], [ 0, 0, 2 ] ] H116 = hmzt 1 y , r 2 y tx, r 2 y t 1 x , r 2 y tz, r 2 y t 1 z i 1 14 [ [ 1, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H117 = hmzt 1 y ,mztz,mzt 1 z , r 2 y tx, r 2 y t 1 x i 1 14 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H118 = hmytx,myty,myt 1 y , r 2 z t 1 z i 1 14 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H119 = hmxt 1 z ,mytx,myty,myt 1 y i 1 14 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H120 = hmytx,myty,myt 1 y ,mztz,mzt 1 z i 1 14 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H121 = hmytx,myt 1 z ,mzt 1 y i 1 14 [ [ 1, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H122 = hmztx, r 2 y t 1 y , r 2 y tz, r 2 y t 1 z i 1 14 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H123 = hmztx,mzt 1 y , r 2 y tz, r 2 y t 1 z i 1 14 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H124 = hmxtx,mxt 1 x , r 2 z ty, r 2 z t 1 y , r 2 z t 1 z i 1 14 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 7 H H0,0,0 P (H) Translation basis of H H125 = hmxt 1 x ,mzt 1 y ,mztz,mzt 1 z , r 2 y txi 1 14 [ [ 2, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H126 = hmxt 1 z ,mzt 1 y , r 2 y tx, r 2 y t 1 x i 1 14 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H127 = hmxt 1 z ,mztx, r 2 y t 1 y i 1 14 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H128 = hmyty,myt 1 y ,mztx, r 2 x tz, r 2 x t 1 z i 1 14 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H129 = hmxtx,mxt 1 x ,mztz,mzt 1 z , r 2 y t 1 y i 1 14 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H130 = hryt 1 y , tx, t 1 z i 1 15 [ [ 1, 0, 0 ], [ 0, 4, 0 ], [ 0, 0, 1 ] ] H131 = hrxt 1 z , r 1 x t 1 y , txi 1 15 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H132 = hr 1 x tx, r 2 x ty, r 2 x t 1 y , r 2 x tz, r 2 x t 1 z i 1 15 [ [ 2, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H133 = hr 1 y tx, r 1 y t 1 x , r 2 y t 1 y i 1 15 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H134 = hmxrxt 1 z ,mxr 1 x t 1 y , txi 1 16 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H135 = hmzrztx,mzrzt 1 x , r 2 z t 1 z i 1 16 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H136 = hryr 2 x tx, ryr 2 x ty, r 2 y rzt 1 x , r 2 y rztzi 1 19 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 6 ] ] H137 = hit 1 y , itz,mzr 1 x tx, r 2 x ty, r 2 x t 1 z i 1 20 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 4 ] ] H138 = hit 1 y , itz,mzrxt 1 z ,mzr 1 x txi 1 20 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 4 ] ] H139 = hmzrxt 1 z ,mzr 1 x tx, r 2 y rxt 1 y i 1 20 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 4 ] ] H140 = hitx, it 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 z t 1 x , r 2 z tyi 1 20 [ [ 1, 1, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H141 = hmxryt 1 x , r 2 x ryty, r 2 x ryt 1 y , ryr 2 x txi 1 20 [ [ 1, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H142 = hmyt 1 y , r 2 x ryty, r 2 y tx, r 2 y t 1 x , r 2 y tz, r 2 y t 1 z i 1 20 [ [ 1, 0, 1 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H143 = hit 1 x ,mxtx,mzr 1 x t 1 y ,mzr 1 x tzi 1 20 [ [ 4, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H144 = hmxtx,mzrxt 1 y ,mzrxt 1 z , r 2 z rxt 1 x i 1 20 [ [ 4, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H145 = hitx, r 2 x ty, r 2 x t 1 y , r 2 x tz, r 2 x t 1 z , r 2 z rxt 1 x i 1 20 [ [ 4, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H146 = hitx,mzr 1 x t 1 y ,mzr 1 x tz, r 2 z rxt 1 x i 1 20 [ [ 4, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H147 = hmxrytx,mxryt 1 x , r 2 x ryt 1 y , ryr 2 x tyi 1 20 [ [ 1, 0, 1 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H148 = hmxryt 1 y , r 2 x ryt 1 x , r 2 y tx, r 2 y tzi 1 20 [ [ 1, 0, 3 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H149 = hmyty,myt 1 y , r 2 x ryt 1 x , r 2 y tx, r 2 y tzi 1 20 [ [ 1, 0, 3 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H150 = hmxr 1 y t 1 x , ryr 2 x ty, ryr 2 x t 1 y , r 2 y tx, r 2 y tzi 1 20 [ [ 1, 0, 3 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H151 = hmxr 1 y t 1 x ,myty,myt 1 y , r 2 y tx, r 2 y tzi 1 20 [ [ 1, 0, 3 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H152 = hitx, ity, r 2 x rztz, r 2 x rzt 1 z , r 2 y rzt 1 x i 1 20 [ [ 1, 3, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H153 = hitx, ity,mztz,mzt 1 z , r 2 y rzt 1 x i 1 20 [ [ 1, 3, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H154 = hitx, ity,mxrzt 1 x ,mztz,mzt 1 z i 1 20 [ [ 1, 3, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H155 = hmxr 1 y t 1 x ,myty, ryr 2 x t 1 y , r 2 y tx, r 2 y tzi 1 20 [ [ 1, 0, 3 ], [ 0, 2, 2 ], [ 0, 0, 4 ] ] H156 = hity, it 1 y ,mztx, r 2 x rztz, r 2 y rzt 1 z i 1 20 [ [ 1, 1, 2 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H157 = hmxt 1 y ,mztz, r 2 x t 1 z , txi 1 21 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H158 = hitz,mxt 1 y ,mzt 1 z , txi 1 21 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H159 = hity, r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , txi 1 21 [ [ 1, 0, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H160 = hmyty, r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , txi 1 21 [ [ 1, 0, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H161 = hmxt 1 z , r 2 x t 1 y , r 2 z ty, txi 1 21 [ [ 1, 0, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H162 = hity,mztz,mzt 1 z , r 2 x t 1 y , txi 1 21 [ [ 1, 0, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H163 = hmztz,mzt 1 z , r 2 x t 1 y , r 2 z ty, txi 1 21 [ [ 1, 0, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H164 = hit 1 z ,myty,myt 1 y ,mztz, txi 1 21 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H165 = hmyt 1 y , r 2 y tz, r 2 y t 1 z , r 2 z ty, txi 1 21 [ [ 1, 0, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H166 = hmyty,myt 1 y ,mztz, r 2 y t 1 z , txi 1 21 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H167 = hity, r 2 x t 1 y , r 2 y tx, r 2 y t 1 x , t 1 z i 1 21 [ [ 2, 0, 0 ], [ 0, 4, 0 ], [ 0, 0, 1 ] ] H168 = hit 1 z ,mytx, r 2 x tz, t 1 y i 1 21 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 4 ] ] H169 = hmyt 1 y ,mztz, r 2 y t 1 z , r 2 z ty, txi 1 21 [ [ 1, 0, 0 ], [ 0, 2, 2 ], [ 0, 0, 4 ] ] H170 = hmyt 1 y , r 2 y tx, r 2 y t 1 x , r 2 y tz, r 2 y t 1 z , r 2 z tyi 1 21 [ [ 1, 0, 1 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H171 = hmxt 1 x ,myty,myt 1 y ,myt 1 z , r 2 y txi 1 21 [ [ 4, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H172 = hit 1 x ,mxtx,myty,myt 1 y ,myt 1 z i 1 21 [ [ 4, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H173 = hit 1 z ,mytx,myty,myt 1 y , r 2 x tzi 1 21 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H174 = hmytx,myty,myt 1 y , r 2 x tz, r 2 y t 1 z i 1 21 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H175 = hitx, r 2 x ty, r 2 x t 1 y , r 2 x tz, r 2 x t 1 z , r 2 z t 1 x i 1 21 [ [ 4, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H176 = hmztz,mzt 1 z , r 2 x t 1 y , r 2 y tx, r 2 y t 1 x , r 2 z tyi 1 21 [ [ 1, 2, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H177 = hity,mxt 1 z ,myt 1 y , r 2 y tx, r 2 y t 1 x i 1 21 [ [ 1, 2, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H178 = hity,myt 1 y ,mztz,mzt 1 z , r 2 y tx, r 2 y t 1 x i 1 21 [ [ 1, 2, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H179 = hit 1 y ,mztx, r 2 y tz, r 2 y t 1 z , r 2 z tyi 1 21 [ [ 1, 2, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H180 = hit 1 y ,mxt 1 z ,mztx, r 2 z tyi 1 21 [ [ 1, 2, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H181 = hity,mztz,mzt 1 z , r 2 x t 1 y , r 2 y tx, r 2 y t 1 x i 1 21 [ [ 1, 2, 1 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H182 = hmxt 1 z ,myt 1 y , r 2 x ty, r 2 y tx, r 2 y t 1 x i 1 21 [ [ 1, 2, 1 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H183 = hmyt 1 y ,mztz,mzt 1 z , r 2 y tx, r 2 y t 1 x , r 2 z tyi 1 21 [ [ 1, 2, 1 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H184 = hit 1 y ,mztx, r 2 x ty, r 2 y tz, r 2 y t 1 z i 1 21 [ [ 1, 2, 1 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H185 = hmxt 1 z ,mztx, r 2 x ty, r 2 z t 1 y i 1 21 [ [ 1, 2, 1 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H186 = hmxtx,mxt 1 x ,mztz,mzt 1 z , r 2 x ty, r 2 z t 1 y i 1 21 [ [ 1, 2, 1 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H187 = hmyt 1 z , r 2 x ty, r 2 x t 1 y , r 2 y tx, r 2 y t 1 x i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H188 = hmztz,mzt 1 z , r 2 x ty, r 2 x t 1 y , r 2 y tx, r 2 y t 1 x i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H189 = hmyty,myt 1 y , r 2 y tx, r 2 y t 1 x , r 2 z t 1 z i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H190 = hmxt 1 z ,myty,myt 1 y , r 2 y tx, r 2 y t 1 x i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H191 = hmyty,myt 1 y , r 2 x tz, r 2 x t 1 z , r 2 y tx, r 2 y t 1 x i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] 8 Art Discrete Appl. Math. 4 (2021) #P2.04 H H0,0,0 P (H) Translation basis of H H192 = hmyty,myt 1 y ,mztz,mzt 1 z , r 2 y tx, r 2 y t 1 x i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H193 = hitz, it 1 z ,mzt 1 y , r 2 y tx, r 2 y t 1 x i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H194 = hmzt 1 y , r 2 y tx, r 2 y t 1 x , r 2 z t 1 z i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H195 = hmzt 1 y , r 2 x tz, r 2 x t 1 z , r 2 y tx, r 2 y t 1 x i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H196 = hmyt 1 z ,mzt 1 y , r 2 y tx, r 2 y t 1 x i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H197 = hitz, it 1 z ,mytx,mzt 1 y i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H198 = hmytx,mzt 1 y , r 2 z t 1 z i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H199 = hmytx,mzt 1 y , r 2 y tz, r 2 y t 1 z i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H200 = hmxt 1 z ,mytx,mzt 1 y i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H201 = hity, it 1 y ,mztx, r 2 y tz, r 2 y t 1 z i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H202 = hity, it 1 y ,mxt 1 z ,mztxi 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H203 = hity, it 1 y ,mztx, r 2 x tz, r 2 x t 1 z i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H204 = hmztx, r 2 x tz, r 2 x t 1 z , r 2 z ty, r 2 z t 1 y i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H205 = hitz, it 1 z ,mztx, r 2 y t 1 y i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H206 = hmztx, r 2 y t 1 y , r 2 z t 1 z i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H207 = hmztx, r 2 x tz, r 2 x t 1 z , r 2 y t 1 y i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H208 = hitz, it 1 z ,myty,myt 1 y ,mztxi 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H209 = hmyty,myt 1 y ,mztx, r 2 z t 1 z i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H210 = hmxt 1 z ,myty,myt 1 y ,mztxi 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H211 = hitx, it 1 x , r 2 y tz, r 2 y t 1 z , r 2 z ty, r 2 z t 1 y i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H212 = hitx, it 1 x , r 2 x tz, r 2 x t 1 z , r 2 z ty, r 2 z t 1 y i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H213 = hitx, it 1 x , r 2 y t 1 y , r 2 z t 1 z i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H214 = hitx, it 1 x , r 2 x tz, r 2 x t 1 z , r 2 y t 1 y i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H215 = hitx, it 1 x ,mztz,mzt 1 z , r 2 y t 1 y i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H216 = hitx, it 1 x ,mztz,mzt 1 z , r 2 x ty, r 2 x t 1 y i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H217 = hitx, it 1 x ,myty,myt 1 y ,mztz,mzt 1 z i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H218 = hmxtx,mxt 1 x , r 2 y t 1 y , r 2 z t 1 z i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H219 = hmxtx,mxt 1 x ,myty,myt 1 y ,mztz,mzt 1 z i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H220 = hmyty,mztz,mzt 1 z , r 2 x t 1 y , r 2 y tx, r 2 y t 1 x i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 1 ], [ 0, 0, 2 ] ] H221 = hmxt 1 z ,myt 1 y , r 2 y tx, r 2 y t 1 x , r 2 z tyi 1 21 [ [ 2, 0, 0 ], [ 0, 2, 1 ], [ 0, 0, 2 ] ] H222 = hmxt 1 x ,myty,myt 1 y ,mztz,mzt 1 z , r 2 y txi 1 21 [ [ 2, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H223 = hmxt 1 x ,myt 1 z ,mzt 1 y , r 2 y txi 1 21 [ [ 2, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H224 = hmztx, r 2 x ty, r 2 y tz, r 2 y t 1 z , r 2 z t 1 y i 1 21 [ [ 2, 0, 0 ], [ 0, 2, 1 ], [ 0, 0, 2 ] ] H225 = hitx,mztz,mzt 1 z , r 2 x ty, r 2 x t 1 y , r 2 z t 1 x i 1 21 [ [ 2, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H226 = hitx,myty,myt 1 y ,mztz,mzt 1 z , r 2 z t 1 x i 1 21 [ [ 2, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H227 = hit 1 x ,mxtx,myty,myt 1 y ,mztz,mzt 1 z i 1 21 [ [ 2, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H228 = hit 1 x ,mxtx,myt 1 z ,mzt 1 y i 1 21 [ [ 2, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H229 = hmxt 1 x ,myty,mztz,mzt 1 z , r 2 x t 1 y , r 2 y txi 1 21 [ [ 2, 0, 1 ], [ 0, 2, 1 ], [ 0, 0, 2 ] ] H230 = hmxrxt 1 z , r 1 x t 1 y , txi 1 22 [ [ 1, 0, 0 ], [ 0, 2, 2 ], [ 0, 0, 4 ] ] H231 = hmxr 1 x t 1 y ,mxtx,mxt 1 x , rxt 1 z i 1 22 [ [ 1, 0, 2 ], [ 0, 2, 2 ], [ 0, 0, 4 ] ] H232 = hitx, it 1 x ,mxrxt 1 z ,mxr 1 x t 1 y i 1 22 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H233 = hitx, it 1 x , rxt 1 z , r 1 x t 1 y i 1 22 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H234 = hmxrxt 1 z ,mxr 1 x t 1 y ,mxtx,mxt 1 x i 1 22 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H235 = hmyty,myt 1 y , r 1 y tx, r 1 y t 1 x i 1 22 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H236 = hit 1 x ,mxtx, rxt 1 z , r 1 x t 1 y i 1 22 [ [ 2, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H237 = hity, it 1 y ,mxt 1 z , r 1 x txi 1 22 [ [ 2, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H238 = hmxrxt 1 z ,mxr 1 x t 1 y , r 1 x txi 1 22 [ [ 2, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H239 = hmxrxt 1 z ,myt 1 y ,mztz, txi 1 23 [ [ 1, 0, 0 ], [ 0, 2, 2 ], [ 0, 0, 4 ] ] H240 = hmxt 1 z ,mztx, r 2 x ryt 1 y , ryr 2 x tyi 1 23 [ [ 1, 2, 1 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H241 = hmxtx,mxt 1 x ,mztz,mzt 1 z , r 2 x ryt 1 y , ryr 2 x tyi 1 23 [ [ 1, 2, 1 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H242 = hmxt 1 z ,mztx, r 2 x ryty, r 2 x ryt 1 y i 1 23 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H243 = hmxtx,mxt 1 x ,mztz,mzt 1 z , r 2 x ryty, r 2 x ryt 1 y i 1 23 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H244 = hmzrztx,mzrzt 1 x , r 2 y rztz, r 2 y rzt 1 z i 1 23 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H245 = hmzrztx,mzrzt 1 x , r 2 x rztz, r 2 x rzt 1 z i 1 23 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H246 = hmxt 1 x ,myt 1 y ,mzrztx, r 2 x rztz, r 2 x rzt 1 z i 1 23 [ [ 2, 0, 1 ], [ 0, 2, 1 ], [ 0, 0, 2 ] ] H247 = hr 2 x ryt 1 y , r 2 z ty, tx, t 1 z i 1 24 [ [ 1, 0, 0 ], [ 0, 8, 0 ], [ 0, 0, 1 ] ] H248 = hr 1 x t 1 y , r 2 y t 1 z , r 2 z ty, txi 1 24 [ [ 1, 0, 0 ], [ 0, 2, 2 ], [ 0, 0, 4 ] ] H249 = hr 1 x tx, r 2 z ty, r 2 z t 1 y , r 2 z t 1 z i 1 24 [ [ 4, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H250 = hr 1 x tx, r 2 z rxt 1 y , r 2 z rxtzi 1 24 [ [ 4, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H251 = hr 2 x ty, ryr 2 x tx, ryr 2 x t 1 x , r 2 z t 1 y i 1 24 [ [ 1, 0, 1 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H252 = hr 2 x rztz, r 2 y rzt 1 z , r 2 y tx, r 2 y t 1 x , r 2 y t 1 y i 1 24 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H253 = hr 2 x ty, r 2 x t 1 y , r 2 x tz, r 2 x t 1 z , r 2 y rxtx, r 2 y t 1 x i 1 24 [ [ 4, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H254 = hr 2 x ryt 1 y , ryr 2 x ty, r 1 y tx, r 1 y t 1 x i 1 24 [ [ 1, 2, 1 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H255 = hr 2 x ty, r 2 x t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 y tx, r 2 y t 1 x i 1 24 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H256 = hr 2 x ryty, r 2 x ryt 1 y , r 1 y tx, r 1 y t 1 x i 1 24 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H257 = hryr 2 x ty, ryr 2 x t 1 y , r 1 y tx, r 1 y t 1 x i 1 24 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H258 = hr 2 y rxtx, r 2 y rxt 1 x , r 2 y t 1 y , r 2 z t 1 z i 1 24 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 9 H H0,0,0 P (H) Translation basis of H H259 = hr 2 x t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 y tx, r 1 z t 1 x i 1 24 [ [ 2, 0, 1 ], [ 0, 2, 1 ], [ 0, 0, 2 ] ] H260 = hmyt 1 y ,mztz, rxt 1 z , txi 1 25 [ [ 1, 0, 0 ], [ 0, 2, 2 ], [ 0, 0, 4 ] ] H261 = hmyt 1 y ,mzr 1 x tx,mztz, rxt 1 z i 1 25 [ [ 1, 0, 2 ], [ 0, 2, 2 ], [ 0, 0, 4 ] ] H262 = hmyty,myt 1 y ,mzr 1 x tx,mztz,mzt 1 z i 1 25 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H263 = hmzr 1 x tx, rxt 1 z , r 1 x t 1 y i 1 25 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H264 = hmzr 1 x tx, rxt 1 y , r 1 x tzi 1 25 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H265 = hmyt 1 z ,mzr 1 x tx,mzt 1 y i 1 25 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H266 = hmyty,myt 1 y ,mztz,mzt 1 z , r 1 x txi 1 25 [ [ 2, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H267 = hmyt 1 z ,mzt 1 y , r 1 x txi 1 25 [ [ 2, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H268 = hmzr 1 z t 1 x , r 2 x t 1 y , r 2 y tx, t 1 z i 1 26 [ [ 2, 2, 0 ], [ 0, 4, 0 ], [ 0, 0, 1 ] ] H269 = hmzrztx,mzrzt 1 x , r 2 x tz, r 2 y t 1 z i 1 26 [ [ 1, 1, 2 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H270 = hmxrzt 1 z , r 2 x ty, r 2 x t 1 y , r 2 y tx, r 2 y t 1 x i 1 26 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H271 = hmxrxt 1 z ,mxr 1 x t 1 y ,mzr 1 x txi 1 26 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H272 = hmxrxt 1 y ,mxr 1 x tz,mzr 1 x txi 1 26 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H273 = hmzr 1 x tx, r 2 y t 1 y , r 2 z t 1 z i 1 26 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H274 = hmxrzt 1 z ,mzr 1 z t 1 x , r 2 x t 1 y , r 2 y txi 1 26 [ [ 2, 0, 1 ], [ 0, 2, 1 ], [ 0, 0, 2 ] ] H275 = hmxt 1 z ,mztx, r 2 x ryty, r 2 z t 1 y i 1 29 [ [ 1, 4, 1 ], [ 0, 8, 0 ], [ 0, 0, 2 ] ] H276 = hmxtx,mxt 1 x ,mztz,mzt 1 z , r 2 x ryty, r 2 z t 1 y i 1 29 [ [ 1, 4, 1 ], [ 0, 8, 0 ], [ 0, 0, 2 ] ] H277 = hmxtx,myty,myt 1 y ,mztz,mzt 1 z , r 2 z rxt 1 x i 1 29 [ [ 4, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H278 = hmxtx, rxt 1 z , r 1 x t 1 y , r 2 z rxt 1 x i 1 29 [ [ 4, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H279 = hmxtx, rxt 1 y , r 1 x tz, r 2 z rxt 1 x i 1 29 [ [ 4, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H280 = hmxtx,myt 1 z ,mzt 1 y , r 2 z rxt 1 x i 1 29 [ [ 4, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H281 = hit 1 y ,mxt 1 z ,mztx, r 2 x rytyi 1 29 [ [ 2, 0, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H282 = hitx,myty,myt 1 y ,mztz,mzt 1 z , r 2 z rxt 1 x i 1 29 [ [ 4, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H283 = hitx, rxt 1 z , r 1 x t 1 y , r 2 z rxt 1 x i 1 29 [ [ 4, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H284 = hitx, rxt 1 y , r 1 x tz, r 2 z rxt 1 x i 1 29 [ [ 4, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H285 = hmxt 1 x ,myty, r 2 x t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 y txi 1 29 [ [ 2, 2, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H286 = hmxrzt 1 z ,mxt 1 x ,myty, r 2 x t 1 y , r 2 y txi 1 29 [ [ 2, 2, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H287 = hmzr 1 z t 1 x , r 2 x t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 y txi 1 29 [ [ 2, 2, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H288 = hmzr 1 z t 1 x ,mztz,mzt 1 z , r 2 x t 1 y , r 2 y txi 1 29 [ [ 2, 2, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H289 = hmxrzt 1 z , r 2 x t 1 y , r 2 y tx, r 1 z t 1 x i 1 29 [ [ 2, 2, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H290 = hmztz,mzt 1 z , r 2 x t 1 y , r 2 y tx, r 1 z t 1 x i 1 29 [ [ 2, 2, 0 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H291 = hmxr 1 x t 1 y ,mzr 1 x tx, rxt 1 z i 1 29 [ [ 2, 0, 0 ], [ 0, 2, 2 ], [ 0, 0, 4 ] ] H292 = hmxr 1 x t 1 y ,myty,mzr 1 x tx,mzt 1 z i 1 29 [ [ 2, 0, 0 ], [ 0, 2, 2 ], [ 0, 0, 4 ] ] H293 = hmxr 1 x t 1 y ,mxtx,mxt 1 x ,myty,mzt 1 z i 1 29 [ [ 2, 0, 0 ], [ 0, 2, 2 ], [ 0, 0, 4 ] ] H294 = hmxt 1 x ,mzt 1 z , ryr 2 x ty, ryr 2 x t 1 y , r 1 y txi 1 29 [ [ 2, 0, 2 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H295 = hmxt 1 x ,myty,myt 1 y ,mzt 1 z , r 1 y txi 1 29 [ [ 2, 0, 2 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H296 = hryr 2 x ty, ryr 2 x t 1 y , r 1 y myt 1 x , r 1 y txi 1 29 [ [ 2, 0, 2 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H297 = hit 1 z ,mzr 1 z t 1 x , r 2 x rztz, r 2 x t 1 y , r 2 y txi 1 29 [ [ 2, 0, 2 ], [ 0, 2, 2 ], [ 0, 0, 4 ] ] H298 = hmxtx,myt 1 y ,mztz, rxt 1 z , r 2 z rxt 1 x i 1 29 [ [ 2, 0, 2 ], [ 0, 2, 2 ], [ 0, 0, 4 ] ] H299 = hi, itx, it 1 y , it 1 z i 2 2 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H300 = hi, itx, it 1 y , r 2 z t 1 z i 2 9 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H301 = hi, itx, it 1 y , r 2 y t 1 z i 2 9 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H302 = hi, itx, r 2 z t 1 y , r 2 z t 1 z i 2 9 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H303 = hi, itx,mxt 1 y ,mxt 1 z i 2 9 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H304 = hi, r 2 z tx, r 2 z t 1 y , r 2 z t 1 z i 2 9 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H305 = hi, itx, it 1 y , r 2 y rzt 1 z i 2 12 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H306 = hi, itx, r 2 z rxt 1 y i 2 12 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H307 = hi, r 2 z rxtx, r 2 z rxt 1 y i 2 12 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H308 = hi, r 2 y rzt 1 z , r 2 z tx, r 2 z t 1 y i 2 20 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H309 = hi, r 2 y rxt 1 y , r 2 z rxtxi 2 20 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H310 = hi,mxtx, r 2 z rxt 1 y i 2 20 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H311 = hi, itx, r 2 y t 1 z , r 2 z t 1 y i 2 21 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H312 = hi, itx,mxt 1 z , r 2 z t 1 y i 2 21 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H313 = hi, itx, r 2 y t 1 y , r 2 z t 1 z i 2 21 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H314 = hi, itx,mxt 1 z , r 2 y t 1 y i 2 21 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H315 = hi, r 2 y t 1 z , r 2 z tx, r 2 z t 1 y i 2 21 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H316 = hi, r 2 y t 1 y , r 2 z tx, r 2 z t 1 z i 2 21 [ [ 1, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H317 = hi, r 2 y t 1 y , r 2 y t 1 z , r 2 z txi 2 21 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H318 = hi,mxt 1 z , r 2 y t 1 y , r 2 z txi 2 21 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H319 = hi,mxt 1 y , r 2 y t 1 z , r 2 z txi 2 21 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H320 = hi,mxtx, r 2 y t 1 y , r 2 z t 1 z i 2 21 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H321 = hi, itx,mxr 1 x t 1 y i 2 22 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H322 = hi,mxr 1 x t 1 y ,mxtxi 2 22 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H323 = hi,mxt 1 z , r 2 x ryt 1 y , r 2 z txi 2 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H324 = hi,mxr 1 x t 1 y , r 2 z rxtxi 2 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H325 = hi,mxrxt 1 y , r 2 z rxtxi 2 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] 10 Art Discrete Appl. Math. 4 (2021) #P2.04 H H0,0,0 P (H) Translation basis of H H326 = hi, r 2 y t 1 y , r 2 z rxtx, r 2 z t 1 z i 2 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H327 = hmz,mztx,mzt 1 y ,mzt 1 z i 3 3 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H328 = hitx, it 1 x ,mz,mzt 1 y ,mzt 1 z i 3 9 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H329 = hit 1 z ,mz,mztx,mzt 1 y i 3 9 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H330 = hitx, it 1 x , ity, it 1 y ,mz,mzt 1 z i 3 9 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H331 = hitx, it 1 x , it 1 z ,mz,mzt 1 y i 3 9 [ [ 1, 0, 1 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H332 = hitx, it 1 x , ity, it 1 y , it 1 z ,mzi 3 9 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H333 = hmz,mztx,mzt 1 y , r 2 y rzt 1 z i 3 11 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H334 = hmz,mzt 1 z , r 2 y rztx, r 2 y rzt 1 x i 3 11 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H335 = hmz, r 2 y rztx, r 2 y rzt 1 x , r 2 y rzt 1 z i 3 11 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H336 = hmz,mzt 1 y ,mzt 1 z , r 2 y tx, r 2 y t 1 x i 3 14 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H337 = hmz,mzt 1 y ,mzt 1 z , r 2 x txi 3 14 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H338 = hmz,mztx,mzt 1 y , r 2 y t 1 z i 3 14 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H339 = hmz,mzt 1 z , r 2 y tx, r 2 y t 1 x , r 2 y t 1 y i 3 14 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H340 = hmz,mzt 1 y , r 2 y tx, r 2 y t 1 x , r 2 y t 1 z i 3 14 [ [ 1, 0, 1 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H341 = hmz,mzt 1 y , r 2 x tx, r 2 x t 1 z i 3 14 [ [ 1, 0, 1 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H342 = hmz, r 2 y tx, r 2 y t 1 x , r 2 y t 1 y , r 2 y t 1 z i 3 14 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H343 = hitx, ity,mz,mzt 1 z , r 2 y rzt 1 x i 3 20 [ [ 1, 3, 0 ], [ 0, 4, 0 ], [ 0, 0, 1 ] ] H344 = hitx, it 1 x , ity, it 1 y ,mz, r 2 y rzt 1 z i 3 20 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H345 = hit 1 z ,mz, r 2 y rztx, r 2 y rzt 1 x i 3 20 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H346 = hmz, r 2 x rzt 1 z , r 2 y rztx, r 2 y rzt 1 x i 3 20 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H347 = hitx, ity,mz, r 2 x rzt 1 z , r 2 y rzt 1 x i 3 20 [ [ 1, 1, 1 ], [ 0, 2, 1 ], [ 0, 0, 2 ] ] H348 = hitx,mz,mzt 1 y ,mzt 1 z , r 2 y t 1 x i 3 21 [ [ 4, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H349 = hitx, it 1 x ,mz,mzt 1 z , r 2 y t 1 y i 3 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H350 = hitx, it 1 x ,mz,mzt 1 z , r 2 x ty, r 2 x t 1 y i 3 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H351 = hitx, it 1 x ,mz,mzt 1 y , r 2 y t 1 z i 3 21 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H352 = hitx, it 1 x ,mz,mzt 1 y , r 2 x t 1 z i 3 21 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H353 = hmz,mzt 1 z , r 2 x ty, r 2 x t 1 y , r 2 y tx, r 2 y t 1 x i 3 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H354 = hit 1 z ,mz,mzt 1 y , r 2 y tx, r 2 y t 1 x i 3 21 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H355 = hmz,mzt 1 y , r 2 x t 1 z , r 2 y tx, r 2 y t 1 x i 3 21 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H356 = hmz,mzt 1 z , r 2 x tx, r 2 y t 1 y i 3 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H357 = hit 1 z ,mz,mzt 1 y , r 2 x txi 3 21 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H358 = hmz,mzt 1 y , r 2 x tx, r 2 y t 1 z i 3 21 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H359 = hitx,mz,mzt 1 z , r 2 x ty, r 2 x t 1 y , r 2 y t 1 x i 3 21 [ [ 2, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H360 = hitx,mz,mzt 1 y , r 2 x t 1 z , r 2 y t 1 x i 3 21 [ [ 2, 0, 1 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H361 = hitx, it 1 x , ity, it 1 y ,mz, r 2 y t 1 z i 3 21 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H362 = hitx, it 1 x , it 1 z ,mz, r 2 y t 1 y i 3 21 [ [ 1, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H363 = hitx, it 1 x ,mz, r 2 y t 1 y , r 2 y t 1 z i 3 21 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H364 = hitx, it 1 x , it 1 z ,mz, r 2 x ty, r 2 x t 1 y i 3 21 [ [ 1, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H365 = hitx, it 1 x ,mz, r 2 x ty, r 2 x t 1 y , r 2 x t 1 z i 3 21 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H366 = hit 1 z ,mz, r 2 y tx, r 2 y t 1 x , r 2 y t 1 y i 3 21 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H367 = hmz, r 2 x t 1 z , r 2 y tx, r 2 y t 1 x , r 2 y t 1 y i 3 21 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H368 = hmz, r 2 x ty, r 2 x t 1 y , r 2 y tx, r 2 y t 1 x , r 2 y t 1 z i 3 21 [ [ 1, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H369 = hmz, r 2 x tx, r 2 y t 1 y , r 2 y t 1 z i 3 21 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H370 = hitx,mz, r 2 x ty, r 2 x t 1 y , r 2 x t 1 z , r 2 y t 1 x i 3 21 [ [ 2, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H371 = hitx, it 1 x ,mz, r 2 x t 1 z , r 2 y t 1 y i 3 21 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H372 = hitx, it 1 x ,mz, r 2 x ty, r 2 x t 1 y , r 2 y t 1 z i 3 21 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H373 = hit 1 z ,mz, r 2 x ty, r 2 x t 1 y , r 2 y tx, r 2 y t 1 x i 3 21 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H374 = hit 1 z ,mz, r 2 x tx, r 2 y t 1 y i 3 21 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H375 = hmz,mzr 1 z tx,mzr 1 z t 1 x ,mzt 1 z i 3 22 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H376 = hit 1 z ,mz,mzr 1 z tx,mzr 1 z t 1 x i 3 22 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H377 = hmz,mzr 1 z t 1 x ,mzt 1 z , r 2 x t 1 y , r 2 y txi 3 29 [ [ 2, 2, 0 ], [ 0, 4, 0 ], [ 0, 0, 1 ] ] H378 = hmz, r 2 x ty, r 2 x t 1 y , r 2 y rzt 1 z , r 2 y tx, r 2 y t 1 x i 3 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H379 = hmz,mzr 1 z tx,mzr 1 z t 1 x , r 2 y rzt 1 z i 3 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H380 = hmz,mzr 1 z tx,mzr 1 z t 1 x , r 2 x rzt 1 z i 3 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H381 = hmz, r 2 x tx, r 2 y rzt 1 z , r 2 y t 1 y i 3 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H382 = hmz,mzr 1 z t 1 x , r 2 x t 1 y , r 2 y rzt 1 z , r 2 y txi 3 29 [ [ 2, 0, 1 ], [ 0, 2, 1 ], [ 0, 0, 2 ] ] H383 = hr 2 z rx, r 2 z rxtx, r 2 z rxt 1 y i 4 4 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H384 = hmxtx, r 2 z rx, r 2 z rxt 1 y i 4 11 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H385 = hmxt 1 y , r 2 z rx, r 2 z rxtxi 4 11 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H386 = hmxtx,mxt 1 y , r 2 z rxi 4 11 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H387 = hitx, r 2 z rx, r 2 z rxt 1 y i 4 12 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H388 = hit 1 y , it 1 z , r 2 z rx, r 2 z rxtxi 4 12 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H389 = hitx, it 1 y , it 1 z , r 2 z rxi 4 12 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H390 = hr 2 y rxtx, r 2 z rx, r 2 z rxt 1 y i 4 13 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H391 = hr 2 y rxt 1 y , r 2 y rxt 1 z , r 2 z rx, r 2 z rxtxi 4 13 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H392 = hr 2 y rxtx, r 2 y rxt 1 y , r 2 y rxt 1 z , r 2 z rxi 4 13 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 11 H H0,0,0 P (H) Translation basis of H H393 = hr 2 x ryt 1 y , r 2 y rztx, r 2 z rxi 4 19 [ [ 1, 0, 2 ], [ 0, 1, 2 ], [ 0, 0, 3 ] ] H394 = hit 1 y , r 2 y rxt 1 z , r 2 z rx, r 2 z rxtxi 4 20 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 4 ] ] H395 = hit 1 y ,mxtx, r 2 y rxt 1 z , r 2 z rxi 4 20 [ [ 1, 0, 2 ], [ 0, 1, 1 ], [ 0, 0, 4 ] ] H396 = hitx, r 2 y rxt 1 y , r 2 y rxt 1 z , r 2 z rxi 4 20 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H397 = hitx,mxt 1 y , r 2 z rxi 4 20 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H398 = hit 1 y , it 1 z , r 2 y rxtx, r 2 z rxi 4 20 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H399 = hmxt 1 y , r 2 y rxtx, r 2 z rxi 4 20 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H400 = hit 1 y , it 1 z ,mxtx, r 2 z rxi 4 20 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H401 = hmxtx, r 2 y rxt 1 y , r 2 y rxt 1 z , r 2 z rxi 4 20 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H402 = hmxrxt 1 z ,mxr 1 x t 1 y , r 2 z rx, r 2 z rxtxi 4 23 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H403 = hmxrxt 1 y , r 2 z rx, r 2 z rxtxi 4 23 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H404 = hmxrxt 1 z ,mxr 1 x t 1 y , r 2 y rxtx, r 2 z rxi 4 23 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H405 = hmxrxt 1 y , r 2 y rxtx, r 2 z rxi 4 23 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H406 = hr 2 z rx, r 2 z rxt 1 y , r 2 z txi 4 24 [ [ 4, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H407 = hr 2 y t 1 z , r 2 z rx, r 2 z rxtx, r 2 z t 1 y i 4 24 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H408 = hr 2 y t 1 y , r 2 z rx, r 2 z rxtxi 4 24 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H409 = hr 2 y rxt 1 y , r 2 y rxt 1 z , r 2 z rx, r 2 z txi 4 24 [ [ 2, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H410 = hr 2 y rxtx, r 2 y t 1 z , r 2 z rx, r 2 z t 1 y i 4 24 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H411 = hr 2 y rxtx, r 2 y t 1 y , r 2 z rxi 4 24 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H412 = hmxrxt 1 z , r 2 z rx, r 2 z rxtx, r 2 z t 1 y i 4 29 [ [ 1, 0, 0 ], [ 0, 2, 2 ], [ 0, 0, 4 ] ] H413 = hmxrxt 1 z ,mxtx, r 2 z rx, r 2 z t 1 y i 4 29 [ [ 1, 0, 2 ], [ 0, 2, 2 ], [ 0, 0, 4 ] ] H414 = hitx, r 2 y t 1 z , r 2 z rx, r 2 z t 1 y i 4 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H415 = hitx,mxrxt 1 z ,mxr 1 x t 1 y , r 2 z rxi 4 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H416 = hitx,mxrxt 1 y , r 2 z rxi 4 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H417 = hitx, r 2 y t 1 y , r 2 z rxi 4 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H418 = hmxtx, r 2 y t 1 z , r 2 z rx, r 2 z t 1 y i 4 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H419 = hmxrxt 1 z ,mxr 1 x t 1 y ,mxtx, r 2 z rxi 4 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H420 = hmxrxt 1 y ,mxtx, r 2 z rxi 4 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H421 = hmxtx, r 2 y t 1 y , r 2 z rxi 4 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H422 = hmxrxt 1 z ,mxr 1 x t 1 y , r 2 z rx, r 2 z txi 4 29 [ [ 2, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H423 = hmxrxt 1 y , r 2 z rx, r 2 z txi 4 29 [ [ 2, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H424 = hr 2 z , r 2 z tx, r 2 z t 1 y , r 2 z t 1 z i 5 5 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H425 = hr 2 y tz, r 2 y t 1 z , r 2 z , r 2 z tx, r 2 z t 1 y i 5 8 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H426 = hr 2 y t 1 y , r 2 z , r 2 z tx, r 2 z t 1 z i 5 8 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H427 = hr 2 y t 1 y , r 2 y tz, r 2 y t 1 z , r 2 z , r 2 z txi 5 8 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H428 = hr 2 y tx, r 2 y t 1 y , r 2 z , r 2 z t 1 z i 5 8 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H429 = hr 2 y tx, r 2 y t 1 y , r 2 y tz, r 2 y t 1 z , r 2 z i 5 8 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H430 = hitx, r 2 z , r 2 z t 1 y , r 2 z t 1 z i 5 9 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H431 = hitz, it 1 z , r 2 z , r 2 z tx, r 2 z t 1 y i 5 9 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H432 = hitx, it 1 y , r 2 z , r 2 z t 1 z i 5 9 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H433 = hitx, itz, it 1 z , r 2 z , r 2 z t 1 y i 5 9 [ [ 1, 0, 1 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H434 = hitx, it 1 y , itz, it 1 z , r 2 z i 5 9 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H435 = hmxrzt 1 z , r 2 z , r 2 z tx, r 2 z t 1 y i 5 10 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H436 = hmxrztx, r 2 z , r 2 z t 1 z i 5 10 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H437 = hmxrztx,mxrzt 1 z , r 2 z i 5 10 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H438 = hr 2 y rztz, r 2 y rzt 1 z , r 2 z , r 2 z tx, r 2 z t 1 y i 5 13 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H439 = hr 2 y rztx, r 2 z , r 2 z t 1 z i 5 13 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H440 = hr 2 y rztx, r 2 y rztz, r 2 y rzt 1 z , r 2 z i 5 13 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H441 = hmxt 1 z , r 2 z , r 2 z tx, r 2 z t 1 y i 5 14 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H442 = hmxt 1 y , r 2 z , r 2 z tx, r 2 z t 1 z i 5 14 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H443 = hmxt 1 y ,mxt 1 z , r 2 z , r 2 z txi 5 14 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H444 = hmxtx,mxt 1 y , r 2 z , r 2 z t 1 z i 5 14 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H445 = hmxtx,mxt 1 y ,mxt 1 z , r 2 z i 5 14 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H446 = hr 1 z t 1 z , r 2 z , r 2 z tx, r 2 z t 1 y i 5 15 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H447 = hr 1 z tx, r 2 z , r 2 z t 1 z i 5 15 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H448 = hr 1 z tx, r 1 z t 1 z , r 2 z i 5 15 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H449 = hmzr 1 z tz,mzr 1 z t 1 z , r 2 z , r 2 z tx, r 2 z t 1 y i 5 16 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H450 = hmzr 1 z tx, r 2 z , r 2 z t 1 z i 5 16 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H451 = hmzr 1 z tx,mzr 1 z tz,mzr 1 z t 1 z , r 2 z i 5 16 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H452 = hit 1 z , r 2 y rztz, r 2 z , r 2 z tx, r 2 z t 1 y i 5 20 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 4 ] ] H453 = hitx, it 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 z i 5 20 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H454 = hitx, it 1 y ,mxrzt 1 z , r 2 z i 5 20 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H455 = hitz, it 1 z , r 2 y rztx, r 2 z i 5 20 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H456 = hmxrzt 1 z , r 2 y rztx, r 2 z i 5 20 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H457 = hitz, it 1 z ,mxrztx, r 2 z i 5 20 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H458 = hmxrztx, r 2 y rztz, r 2 y rzt 1 z , r 2 z i 5 20 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H459 = hit 1 z ,mxrztx, r 2 y rztz, r 2 z i 5 20 [ [ 1, 0, 2 ], [ 0, 1, 2 ], [ 0, 0, 4 ] ] H460 = hit 1 z , r 2 y tz, r 2 z , r 2 z tx, r 2 z t 1 y i 5 21 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 4 ] ] 12 Art Discrete Appl. Math. 4 (2021) #P2.04 H H0,0,0 P (H) Translation basis of H H461 = hit 1 z ,mxt 1 y , r 2 y tz, r 2 z , r 2 z txi 5 21 [ [ 1, 0, 0 ], [ 0, 1, 2 ], [ 0, 0, 4 ] ] H462 = hitx, r 2 y tz, r 2 y t 1 z , r 2 z , r 2 z t 1 y i 5 21 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H463 = hitx,mxt 1 z , r 2 z , r 2 z t 1 y i 5 21 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H464 = hitx, r 2 y t 1 y , r 2 z , r 2 z t 1 z i 5 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H465 = hitx,mxt 1 y , r 2 z , r 2 z t 1 z i 5 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H466 = hitz, it 1 z , r 2 y t 1 y , r 2 z , r 2 z txi 5 21 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H467 = hmxt 1 z , r 2 y t 1 y , r 2 z , r 2 z txi 5 21 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H468 = hitz, it 1 z ,mxt 1 y , r 2 z , r 2 z txi 5 21 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H469 = hmxt 1 y , r 2 y tz, r 2 y t 1 z , r 2 z , r 2 z txi 5 21 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H470 = hmxt 1 y , r 2 y tx, r 2 z , r 2 z t 1 z i 5 21 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H471 = hitx, it 1 y , r 2 y tz, r 2 y t 1 z , r 2 z i 5 21 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H472 = hitx, it 1 y ,mxt 1 z , r 2 z i 5 21 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H473 = hitx, itz, it 1 z , r 2 y t 1 y , r 2 z i 5 21 [ [ 1, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H474 = hitx, r 2 y t 1 y , r 2 y tz, r 2 y t 1 z , r 2 z i 5 21 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H475 = hitx, itz, it 1 z ,mxt 1 y , r 2 z i 5 21 [ [ 1, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H476 = hitx,mxt 1 y ,mxt 1 z , r 2 z i 5 21 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H477 = hitz, it 1 z , r 2 y tx, r 2 y t 1 y , r 2 z i 5 21 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H478 = hmxt 1 z , r 2 y tx, r 2 y t 1 y , r 2 z i 5 21 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H479 = hmxt 1 y , r 2 y tx, r 2 y tz, r 2 y t 1 z , r 2 z i 5 21 [ [ 1, 0, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H480 = hmxt 1 y ,mxt 1 z , r 2 y tx, r 2 z i 5 21 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H481 = hitz, it 1 z ,mxtx,mxt 1 y , r 2 z i 5 21 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H482 = hmxtx,mxt 1 y , r 2 y tz, r 2 y t 1 z , r 2 z i 5 21 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H483 = hit 1 z ,mxtx,mxt 1 y , r 2 y tz, r 2 z i 5 21 [ [ 1, 0, 2 ], [ 0, 1, 2 ], [ 0, 0, 4 ] ] H484 = hitx,mxt 1 z , r 2 y t 1 y , r 2 z i 5 21 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H485 = hitx,mxt 1 y , r 2 y tz, r 2 y t 1 z , r 2 z i 5 21 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H486 = hitz, it 1 z ,mxt 1 y , r 2 y tx, r 2 z i 5 21 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H487 = hit 1 z ,mzr 1 z tz, r 2 z , r 2 z tx, r 2 z t 1 y i 5 22 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 4 ] ] H488 = hitx, it 1 y ,mzr 1 z tz,mzr 1 z t 1 z , r 2 z i 5 22 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H489 = hitx, it 1 y , r 1 z t 1 z , r 2 z i 5 22 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H490 = hitz, it 1 z ,mzr 1 z tx, r 2 z i 5 22 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H491 = hmzr 1 z tx, r 1 z t 1 z , r 2 z i 5 22 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H492 = hitz, it 1 z , r 1 z tx, r 2 z i 5 22 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H493 = hmzr 1 z tz,mzr 1 z t 1 z , r 1 z tx, r 2 z i 5 22 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H494 = hit 1 z ,mzr 1 z tz, r 1 z tx, r 2 z i 5 22 [ [ 1, 0, 2 ], [ 0, 1, 2 ], [ 0, 0, 4 ] ] H495 = hmzr 1 z tz, r 2 y rzt 1 z , r 2 z , r 2 z tx, r 2 z t 1 y i 5 23 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 4 ] ] H496 = hmxtx,mxt 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 z i 5 23 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H497 = hmxtx,mxt 1 y ,mzr 1 z tz,mzr 1 z t 1 z , r 2 z i 5 23 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H498 = hmxt 1 z , r 2 y rztx, r 2 z i 5 23 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H499 = hmzr 1 z tz,mzr 1 z t 1 z , r 2 y rztx, r 2 z i 5 23 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H500 = hmxt 1 z ,mzr 1 z tx, r 2 z i 5 23 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H501 = hmzr 1 z tx, r 2 y rztz, r 2 y rzt 1 z , r 2 z i 5 23 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H502 = hmxtx,mxt 1 y ,mzr 1 z tz, r 2 y rzt 1 z , r 2 z i 5 23 [ [ 1, 0, 2 ], [ 0, 1, 2 ], [ 0, 0, 4 ] ] H503 = hr 2 y rztz, r 2 y t 1 z , r 2 z , r 2 z tx, r 2 z t 1 y i 5 24 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 4 ] ] H504 = hr 2 x ryt 1 y , r 2 z , r 2 z tx, r 2 z t 1 z i 5 24 [ [ 1, 0, 0 ], [ 0, 4, 0 ], [ 0, 0, 1 ] ] H505 = hr 2 y rztz, r 2 y rzt 1 z , r 2 y tx, r 2 y t 1 y , r 2 z i 5 24 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H506 = hr 2 y tx, r 2 y t 1 y , r 1 z t 1 z , r 2 z i 5 24 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H507 = hr 2 y rztx, r 2 y tz, r 2 y t 1 z , r 2 z i 5 24 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H508 = hr 2 y rztx, r 1 z t 1 z , r 2 z i 5 24 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H509 = hr 2 y tz, r 2 y t 1 z , r 1 z tx, r 2 z i 5 24 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H510 = hr 2 y rztz, r 2 y rzt 1 z , r 1 z tx, r 2 z i 5 24 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H511 = hr 2 y t 1 y , r 2 y tz, r 2 y t 1 z , r 2 z , r 2 z rxtxi 5 24 [ [ 2, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H512 = hr 2 y rztz, r 2 y t 1 z , r 1 z tx, r 2 z i 5 24 [ [ 1, 0, 2 ], [ 0, 1, 2 ], [ 0, 0, 4 ] ] H513 = hmxrzt 1 z ,mxtx,mxt 1 y , r 2 z i 5 25 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H514 = hmxtx,mxt 1 y , r 1 z t 1 z , r 2 z i 5 25 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H515 = hmxrztx,mxt 1 z , r 2 z i 5 25 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H516 = hmxrztx, r 1 z t 1 z , r 2 z i 5 25 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H517 = hmxt 1 z , r 1 z tx, r 2 z i 5 25 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H518 = hmxrzt 1 z , r 1 z tx, r 2 z i 5 25 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H519 = hmzr 1 z tz, r 2 y t 1 z , r 2 z , r 2 z tx, r 2 z t 1 y i 5 26 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 4 ] ] H520 = hmxrzt 1 z , r 2 y tx, r 2 y t 1 y , r 2 z i 5 26 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H521 = hmzr 1 z tz,mzr 1 z t 1 z , r 2 y tx, r 2 y t 1 y , r 2 z i 5 26 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H522 = hmxrztx, r 2 y tz, r 2 y t 1 z , r 2 z i 5 26 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H523 = hmxrztx,mzr 1 z tz,mzr 1 z t 1 z , r 2 z i 5 26 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H524 = hmzr 1 z tx, r 2 y tz, r 2 y t 1 z , r 2 z i 5 26 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H525 = hmxrzt 1 z ,mzr 1 z tx, r 2 z i 5 26 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H526 = hmxrztx,mzr 1 z tz, r 2 y t 1 z , r 2 z i 5 26 [ [ 1, 0, 2 ], [ 0, 1, 2 ], [ 0, 0, 4 ] ] H527 = hit 1 z ,mxtx,mxt 1 y , r 2 y rztz, r 2 z i 5 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H528 = hit 1 z ,mxtx,mxt 1 y ,mzr 1 z tz, r 2 z i 5 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 13 H H0,0,0 P (H) Translation basis of H H529 = hmxtx,mxt 1 y , r 2 y rztz, r 2 y t 1 z , r 2 z i 5 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H530 = hmxtx,mxt 1 y ,mzr 1 z tz, r 2 y t 1 z , r 2 z i 5 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H531 = hit 1 z ,mxrztx, r 2 y tz, r 2 z i 5 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H532 = hit 1 z ,mxrztx,mzr 1 z tz, r 2 z i 5 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H533 = hmxrztx, r 2 y rztz, r 2 y t 1 z , r 2 z i 5 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H534 = hmxrztx,mzr 1 z tz, r 2 y rzt 1 z , r 2 z i 5 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H535 = hit 1 z , r 2 y tz, r 1 z tx, r 2 z i 5 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H536 = hit 1 z , r 2 y rztz, r 1 z tx, r 2 z i 5 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H537 = hmzr 1 z tz, r 2 y t 1 z , r 1 z tx, r 2 z i 5 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H538 = hmzr 1 z tz, r 2 y rzt 1 z , r 1 z tx, r 2 z i 5 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H539 = hitx,mzr 1 z tz, r 2 y rzt 1 z , r 2 y t 1 y , r 2 z i 5 29 [ [ 1, 1, 2 ], [ 0, 2, 0 ], [ 0, 0, 4 ] ] H540 = hitx,mxt 1 z , r 2 x ryt 1 y , r 2 z i 5 29 [ [ 1, 2, 1 ], [ 0, 4, 0 ], [ 0, 0, 2 ] ] H541 = hitz, it 1 z ,mxt 1 y , r 2 z , r 2 z rxtxi 5 29 [ [ 2, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H542 = hmzrx,mzrxtx,mzrxt 1 y i 6 6 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H543 = hmzrx,mzrxt 1 y , r 2 x txi 6 10 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H544 = hmzrx,mzrxtx, r 2 x t 1 y , r 2 x tzi 6 10 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H545 = hmzrx, r 2 x tx, r 2 x t 1 y , r 2 x tzi 6 10 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H546 = hmzrx,mzrxt 1 y , r 2 y rxtx, r 2 y rxt 1 x i 6 11 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H547 = hmzrx,mzrxtx, r 2 y rxt 1 y i 6 11 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H548 = hmzrx, r 2 y rxtx, r 2 y rxt 1 x , r 2 y rxt 1 y i 6 11 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H549 = hitx, it 1 x ,mzrx,mzrxt 1 y i 6 12 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H550 = hit 1 y , itz,mzrx,mzrxtxi 6 12 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H551 = hitx, it 1 x , it 1 y , itz,mzrxi 6 12 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H552 = hitx,mzrx,mzrxt 1 y , r 2 y rxt 1 x i 6 20 [ [ 4, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H553 = hit 1 y ,mzrx,mzrxtx, r 2 x tzi 6 20 [ [ 1, 0, 0 ], [ 0, 1, 3 ], [ 0, 0, 4 ] ] H554 = hitx, it 1 x ,mzrx, r 2 y rxt 1 y i 6 20 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H555 = hitx, it 1 x ,mzrx, r 2 x t 1 y , r 2 x tzi 6 20 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H556 = hit 1 y , itz,mzrx, r 2 y rxtx, r 2 y rxt 1 x i 6 20 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H557 = hmzrx, r 2 x t 1 y , r 2 x tz, r 2 y rxtx, r 2 y rxt 1 x i 6 20 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H558 = hit 1 y , itz,mzrx, r 2 x txi 6 20 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H559 = hmzrx, r 2 x tx, r 2 y rxt 1 y i 6 20 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H560 = hitx,mzrx, r 2 x t 1 y , r 2 x tz, r 2 y rxt 1 x i 6 20 [ [ 2, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H561 = hit 1 y ,mzrx, r 2 x tz, r 2 y rxtx, r 2 y rxt 1 x i 6 20 [ [ 1, 0, 2 ], [ 0, 1, 3 ], [ 0, 0, 4 ] ] H562 = hmyt 1 y ,mzrx,mzrxtx, r 1 x tzi 6 25 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H563 = hmzrx,mzrxtx, r 1 x t 1 y i 6 25 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H564 = hmyt 1 y ,mzrx, r 1 x tz, r 2 x txi 6 25 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H565 = hmzrx, r 1 x t 1 y , r 2 x txi 6 25 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H566 = hmxr 1 x tz,mzrx,mzrxtx, r 2 z t 1 y i 6 26 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H567 = hmxr 1 x t 1 y ,mzrx,mzrxtxi 6 26 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H568 = hmxr 1 x tz,mzrx, r 2 x tx, r 2 z t 1 y i 6 26 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H569 = hmxr 1 x t 1 y ,mzrx, r 2 x txi 6 26 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H570 = hmzrx,mzrxtx, r 1 x tz, r 2 z t 1 y i 6 29 [ [ 1, 0, 0 ], [ 0, 2, 2 ], [ 0, 0, 4 ] ] H571 = hmzrx, r 1 x tz, r 2 y rxtx, r 2 y rxt 1 x , r 2 z t 1 y i 6 29 [ [ 1, 0, 2 ], [ 0, 2, 2 ], [ 0, 0, 4 ] ] H572 = hitx, it 1 x ,mxr 1 x tz,mzrx, r 2 z t 1 y i 6 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H573 = hitx, it 1 x ,mxr 1 x t 1 y ,mzrxi 6 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H574 = hitx, it 1 x ,myt 1 y ,mzrx, r 1 x tzi 6 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H575 = hitx, it 1 x ,mzrx, r 1 x t 1 y i 6 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H576 = hmxr 1 x tz,mzrx, r 2 y rxtx, r 2 y rxt 1 x , r 2 z t 1 y i 6 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H577 = hmxr 1 x t 1 y ,mzrx, r 2 y rxtx, r 2 y rxt 1 x i 6 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H578 = hmyt 1 y ,mzrx, r 1 x tz, r 2 y rxtx, r 2 y rxt 1 x i 6 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H579 = hmzrx, r 1 x t 1 y , r 2 y rxtx, r 2 y rxt 1 x i 6 29 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H580 = hitx,myt 1 y ,mzrx, r 1 x tz, r 2 y rxt 1 x i 6 29 [ [ 2, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H581 = hitx,mzrx, r 1 x t 1 y , r 2 y rxt 1 x i 6 29 [ [ 2, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H582 = hryrx, r 1 y r 1 z txi 7 7 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H583 = hitx, it 1 x , ryrxi 7 17 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H584 = hmxrytx, ryrxi 7 18 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H585 = hryrx, r 2 z rxtx, r 2 z rxt 1 x i 7 19 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H586 = hryrx, r 2 z tx, r 2 z t 1 x i 7 27 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H587 = hryrx, ryr 1 x txi 7 27 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H588 = hitx, ryrx, r 2 z rxt 1 x i 7 28 [ [ 1, 0, 3 ], [ 0, 1, 3 ], [ 0, 0, 4 ] ] H589 = hmxtx,mxt 1 x , ryrxi 7 30 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H590 = hmxryrxtx, ryrxi 7 30 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H591 = hmzr 1 z tx, ryrxi 7 31 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H592 = hryrx, r 2 y rxtx, r 2 y rxt 1 x i 7 32 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H593 = hmxt 1 x , ryrx, r 2 y rxtxi 7 33 [ [ 2, 0, 2 ], [ 0, 2, 2 ], [ 0, 0, 4 ] ] H594 = hr 2 x , r 2 z , r 2 z tx, r 2 z t 1 y , r 2 z t 1 z i 8 8 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H595 = hitx, r 2 x , r 2 z , r 2 z t 1 y , r 2 z t 1 z i 8 21 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H596 = hitx, it 1 y , r 2 x , r 2 z , r 2 z t 1 z i 8 21 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] 14 Art Discrete Appl. Math. 4 (2021) #P2.04 H H0,0,0 P (H) Translation basis of H H597 = hitx, it 1 y , it 1 z , r 2 x , r 2 z i 8 21 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H598 = hr 2 x , r 2 y rzt 1 z , r 2 z , r 2 z tx, r 2 z t 1 y i 8 24 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H599 = hr 2 x , r 2 z , r 2 z rxt 1 y , r 2 z txi 8 24 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H600 = hr 2 x , r 2 z , r 2 z rxtx, r 2 z rxt 1 y i 8 24 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H601 = hmxrzt 1 z , r 2 x , r 2 z , r 2 z tx, r 2 z t 1 y i 8 26 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H602 = hmxr 1 x t 1 y , r 2 x , r 2 z , r 2 z txi 8 26 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H603 = hmxr 1 x tx,mxr 1 x t 1 y , r 2 x , r 2 z i 8 26 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H604 = hitx, it 1 y , r 2 x , r 2 y rzt 1 z , r 2 z i 8 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H605 = hitx, it 1 y ,mxrzt 1 z , r 2 x , r 2 z i 8 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H606 = hitx, r 2 x , r 2 z , r 2 z rxt 1 y i 8 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H607 = hitx,mxr 1 x t 1 y , r 2 x , r 2 z i 8 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H608 = hmxr 1 x t 1 y , r 2 x , r 2 z , r 2 z rxtxi 8 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H609 = hmxr 1 x tx, r 2 x , r 2 z , r 2 z rxt 1 y i 8 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H610 = hi, itx, it 1 y , it 1 z , r 2 z i 9 9 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H611 = hi, itx, it 1 y , r 2 y rzt 1 z , r 2 z i 9 20 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H612 = hi, it 1 z , r 2 y rztx, r 2 z i 9 20 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H613 = hi, r 2 y rztx, r 2 y rzt 1 z , r 2 z i 9 20 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H614 = hi, itx, it 1 y , r 2 y t 1 z , r 2 z i 9 21 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H615 = hi, itx, it 1 z , r 2 y t 1 y , r 2 z i 9 21 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H616 = hi, itx, r 2 y t 1 y , r 2 y t 1 z , r 2 z i 9 21 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H617 = hi, it 1 z , r 2 y tx, r 2 y t 1 y , r 2 z i 9 21 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H618 = hi, r 2 y tx, r 2 y t 1 y , r 2 y t 1 z , r 2 z i 9 21 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H619 = hi, itx, it 1 y ,mzr 1 z t 1 z , r 2 z i 9 22 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H620 = hi, it 1 z ,mzr 1 z tx, r 2 z i 9 22 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H621 = hi,mzr 1 z tx,mzr 1 z t 1 z , r 2 z i 9 22 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H622 = hi, r 2 y rzt 1 z , r 2 y tx, r 2 y t 1 y , r 2 z i 9 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H623 = hi,mzr 1 z t 1 z , r 2 y tx, r 2 y t 1 y , r 2 z i 9 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H624 = hi, r 2 y rztx, r 2 y t 1 z , r 2 z i 9 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H625 = hi,mzr 1 z t 1 z , r 2 y rztx, r 2 z i 9 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H626 = hi,mzr 1 z tx, r 2 y t 1 z , r 2 z i 9 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H627 = hi,mzr 1 z tx, r 2 y rzt 1 z , r 2 z i 9 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H628 = hmzrx, r 2 x , r 2 x tx, r 2 x t 1 y i 10 10 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H629 = hitx, it 1 x ,mzrx, r 2 x , r 2 x t 1 y i 10 20 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H630 = hit 1 y ,mzrx, r 2 x , r 2 x txi 10 20 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H631 = hitx, it 1 x , it 1 y ,mzrx, r 2 x i 10 20 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H632 = hmytx,mzrx, r 2 x , r 2 x t 1 y i 10 25 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H633 = hmyt 1 y ,mzrx, r 2 x , r 2 x txi 10 25 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H634 = hmytx,myt 1 y ,mzrx, r 2 x i 10 25 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H635 = hmzrx, r 2 x , r 2 x t 1 y , r 2 z tx, r 2 z t 1 x i 10 26 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H636 = hmzrx, r 2 x , r 2 x tx, r 2 z t 1 y i 10 26 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H637 = hmzrx, r 2 x , r 2 z tx, r 2 z t 1 x , r 2 z t 1 y i 10 26 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H638 = hitx,mzrx, r 2 x , r 2 x t 1 y , r 2 z t 1 x i 10 29 [ [ 4, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H639 = hitx, it 1 x ,mzrx, r 2 x , r 2 z t 1 y i 10 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H640 = hitx, it 1 x ,myt 1 y ,mzrx, r 2 x i 10 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H641 = hit 1 y ,mzrx, r 2 x , r 2 z tx, r 2 z t 1 x i 10 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H642 = hmyt 1 y ,mzrx, r 2 x , r 2 z tx, r 2 z t 1 x i 10 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H643 = hit 1 y ,mytx,mzrx, r 2 x i 10 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H644 = hmytx,mzrx, r 2 x , r 2 z t 1 y i 10 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H645 = hitx,myt 1 y ,mzrx, r 2 x , r 2 z t 1 x i 10 29 [ [ 2, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H646 = hmx,mzrx, r 2 y rxtx, r 2 y rxt 1 y i 11 11 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H647 = hitx,mx,mzrx, r 2 y rxt 1 y i 11 20 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H648 = hit 1 y , itz,mx,mzrx, r 2 y rxtxi 11 20 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H649 = hitx, it 1 y , itz,mx,mzrxi 11 20 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H650 = hmx,mxr 1 x tz,mzrx, r 2 y rxtx, r 2 z t 1 y i 11 29 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H651 = hmx,mxr 1 x t 1 y ,mzrx, r 2 y rxtxi 11 29 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H652 = hitx,mx,mxr 1 x tz,mzrx, r 2 z t 1 y i 11 29 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H653 = hitx,mx,mxr 1 x t 1 y ,mzrxi 11 29 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H654 = hi, itx, it 1 y ,mzrxi 12 12 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H655 = hi, it 1 y ,mzrx, r 2 y rxtxi 12 20 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H656 = hi, itx,mzrx, r 2 y rxt 1 y i 12 20 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H657 = hi,mzrx, r 2 y rxtx, r 2 y rxt 1 y i 12 20 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H658 = hi, itx,mzrx, r 2 z t 1 y i 12 29 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H659 = hi, itx,mxr 1 x t 1 y ,mzrxi 12 29 [ [ 1, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H660 = hi,mzrx, r 2 y rxtx, r 2 z t 1 y i 12 29 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H661 = hi,mxr 1 x t 1 y ,mzrx, r 2 y rxtxi 12 29 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H662 = hr 2 x , r 2 z rx, r 2 z rxtx, r 2 z rxt 1 y i 13 13 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H663 = hitx, r 2 x , r 2 z rx, r 2 z rxt 1 y i 13 20 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 15 H H0,0,0 P (H) Translation basis of H H664 = hit 1 y , r 2 x , r 2 z rx, r 2 z rxtxi 13 20 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H665 = hitx, it 1 y , r 2 x , r 2 z rxi 13 20 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H666 = hmxr 1 x tx, r 2 x , r 2 z rx, r 2 z rxt 1 y i 13 23 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H667 = hmxr 1 x t 1 y , r 2 x , r 2 z rx, r 2 z rxtxi 13 23 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H668 = hmxr 1 x tx,mxr 1 x t 1 y , r 2 x , r 2 z rxi 13 23 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H669 = hr 2 x , r 2 z rx, r 2 z rxt 1 y , r 2 z txi 13 24 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H670 = hr 2 x , r 2 z rx, r 2 z rxtx, r 2 z t 1 y i 13 24 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H671 = hr 2 x , r 2 z rx, r 2 z tx, r 2 z t 1 y i 13 24 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H672 = hitx, r 2 x , r 2 z rx, r 2 z t 1 y i 13 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H673 = hitx,mxr 1 x t 1 y , r 2 x , r 2 z rxi 13 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H674 = hit 1 y , r 2 x , r 2 z rx, r 2 z txi 13 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H675 = hmxr 1 x t 1 y , r 2 x , r 2 z rx, r 2 z txi 13 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H676 = hit 1 y ,mxr 1 x tx, r 2 x , r 2 z rxi 13 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H677 = hmxr 1 x tx, r 2 x , r 2 z rx, r 2 z t 1 y i 13 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H678 = hmx, r 2 z , r 2 z tx, r 2 z t 1 y , r 2 z t 1 z i 14 14 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H679 = hitx,mx, r 2 z , r 2 z t 1 y , r 2 z t 1 z i 14 21 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H680 = hitz, it 1 z ,mx, r 2 z , r 2 z tx, r 2 z t 1 y i 14 21 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H681 = hitx, it 1 y ,mx, r 2 z , r 2 z t 1 z i 14 21 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H682 = hitx, itz, it 1 z ,mx, r 2 z , r 2 z t 1 y i 14 21 [ [ 1, 0, 1 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H683 = hitx, it 1 y , itz, it 1 z ,mx, r 2 z i 14 21 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H684 = hmx, r 2 y rztz, r 2 y rzt 1 z , r 2 z , r 2 z tx, r 2 z t 1 y i 14 23 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H685 = hmx, r 2 y rztx, r 2 z , r 2 z t 1 z i 14 23 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H686 = hmx, r 2 y rztx, r 2 y rztz, r 2 y rzt 1 z , r 2 z i 14 23 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H687 = hmx,mxrzt 1 z , r 2 z , r 2 z tx, r 2 z t 1 y i 14 25 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H688 = hmx,mxrztx, r 2 z , r 2 z t 1 z i 14 25 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 1 ] ] H689 = hmx,mxrztx,mxrzt 1 z , r 2 z i 14 25 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H690 = hit 1 z ,mx, r 2 y rztz, r 2 z , r 2 z tx, r 2 z t 1 y i 14 29 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 4 ] ] H691 = hitx, it 1 y ,mx, r 2 y rztz, r 2 y rzt 1 z , r 2 z i 14 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H692 = hitx, it 1 y ,mx,mxrzt 1 z , r 2 z i 14 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H693 = hitz, it 1 z ,mx, r 2 y rztx, r 2 z i 14 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H694 = hmx,mxrzt 1 z , r 2 y rztx, r 2 z i 14 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H695 = hitz, it 1 z ,mx,mxrztx, r 2 z i 14 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H696 = hmx,mxrztx, r 2 y rztz, r 2 y rzt 1 z , r 2 z i 14 29 [ [ 1, 1, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H697 = hit 1 z ,mx,mxrztx, r 2 y rztz, r 2 z i 14 29 [ [ 1, 0, 2 ], [ 0, 1, 2 ], [ 0, 0, 4 ] ] H698 = hr 1 x , r 2 x , r 2 x tx, r 2 x t 1 y i 15 15 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H699 = hitx, it 1 x , r 1 x , r 2 x , r 2 x t 1 y i 15 22 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H700 = hit 1 y , r 1 x , r 2 x , r 2 x txi 15 22 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H701 = hitx, it 1 x , it 1 y , r 1 x , r 2 x i 15 22 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H702 = hr 1 x , r 2 x , r 2 x t 1 y , r 2 z tx, r 2 z t 1 x i 15 24 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H703 = hr 1 x , r 2 x , r 2 x tx, r 2 z t 1 y i 15 24 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H704 = hr 1 x , r 2 x , r 2 z tx, r 2 z t 1 x , r 2 z t 1 y i 15 24 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H705 = hmytx, r 1 x , r 2 x , r 2 x t 1 y i 15 25 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H706 = hmyt 1 y , r 1 x , r 2 x , r 2 x txi 15 25 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H707 = hmytx,myt 1 y , r 1 x , r 2 x i 15 25 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H708 = hitx, r 1 x , r 2 x , r 2 x t 1 y , r 2 z t 1 x i 15 29 [ [ 4, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H709 = hitx, it 1 x , r 1 x , r 2 x , r 2 z t 1 y i 15 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H710 = hitx, it 1 x ,myt 1 y , r 1 x , r 2 x i 15 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H711 = hit 1 y , r 1 x , r 2 x , r 2 z tx, r 2 z t 1 x i 15 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H712 = hmyt 1 y , r 1 x , r 2 x , r 2 z tx, r 2 z t 1 x i 15 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H713 = hit 1 y ,mytx, r 1 x , r 2 x i 15 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H714 = hmytx, r 1 x , r 2 x , r 2 z t 1 y i 15 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H715 = hitx,myt 1 y , r 1 x , r 2 x , r 2 z t 1 x i 15 29 [ [ 2, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H716 = hmxrx,mxr 1 x tx,mxr 1 x t 1 y , r 2 x i 16 16 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H717 = hitx,mxrx,mxr 1 x t 1 y , r 2 x i 16 22 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H718 = hit 1 y ,mxrx,mxr 1 x tx, r 2 x i 16 22 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H719 = hitx, it 1 y ,mxrx, r 2 x i 16 22 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H720 = hmxrx,mxr 1 x t 1 y , r 2 x , r 2 z rxtxi 16 23 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H721 = hmxrx,mxr 1 x tx, r 2 x , r 2 z rxt 1 y i 16 23 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H722 = hmxrx, r 2 x , r 2 z rxtx, r 2 z rxt 1 y i 16 23 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H723 = hmxrx,mxr 1 x t 1 y , r 2 x , r 2 z txi 16 26 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H724 = hmxrx,mxr 1 x tx, r 2 x , r 2 z t 1 y i 16 26 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H725 = hmxrx, r 2 x , r 2 z tx, r 2 z t 1 y i 16 26 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H726 = hitx,mxrx, r 2 x , r 2 z t 1 y i 16 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H727 = hitx,mxrx, r 2 x , r 2 z rxt 1 y i 16 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H728 = hit 1 y ,mxrx, r 2 x , r 2 z txi 16 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H729 = hmxrx, r 2 x , r 2 z rxt 1 y , r 2 z txi 16 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H730 = hit 1 y ,mxrx, r 2 x , r 2 z rxtxi 16 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] 16 Art Discrete Appl. Math. 4 (2021) #P2.04 H H0,0,0 P (H) Translation basis of H H731 = hmxrx, r 2 x , r 2 z rxtx, r 2 z t 1 y i 16 29 [ [ 2, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H732 = hi, itx, ryrxi 17 17 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H733 = hi, ryrx, r 2 z rxtxi 17 28 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H734 = hi, ryrx, r 2 z txi 17 30 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H735 = hi,mxtx, ryrxi 17 30 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H736 = hi, ryrx, r 2 y rxtxi 17 33 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H737 = hmxrytx,mzrx, ryrxi 18 18 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H738 = hitx, it 1 x ,mzrx, ryrxi 18 28 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H739 = hmzrx, ryrx, ryr 1 x txi 18 31 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H740 = hmzrx, ryrx, r 2 y rxtx, r 2 y rxt 1 x i 18 33 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H741 = hryrx, r 2 z rx, r 2 z rxtxi 19 19 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H742 = hitx, ryrx, r 2 z rxi 19 28 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H743 = hryrx, r 2 y rxtx, r 2 z rxi 19 32 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H744 = hmxtx, ryrx, r 2 z rxi 19 33 [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H745 = hi, itx, it 1 y ,mzrx, r 2 x i 20 20 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H746 = hi, it 1 y ,mzrx, r 2 x , r 2 z txi 20 29 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H747 = hi, itx,mzrx, r 2 x , r 2 z t 1 y i 20 29 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H748 = hi,mzrx, r 2 x , r 2 z tx, r 2 z t 1 y i 20 29 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H749 = hi, itx, it 1 y , it 1 z , r 2 x , r 2 z i 21 21 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H750 = hi, itx, it 1 y , r 2 x , r 2 y rzt 1 z , r 2 z i 21 29 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 2 ] ] H751 = hi, itx, r 2 x , r 2 z , r 2 z rxt 1 y i 21 29 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H752 = hi, r 2 x , r 2 z , r 2 z rxtx, r 2 z rxt 1 y i 21 29 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H753 = hi, itx, it 1 y ,mxrx, r 2 x i 22 22 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H754 = hi, it 1 y ,mxrx, r 2 x , r 2 z txi 22 29 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H755 = hi, itx,mxrx, r 2 x , r 2 z t 1 y i 22 29 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H756 = hi,mxrx, r 2 x , r 2 z tx, r 2 z t 1 y i 22 29 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H757 = hmxrx,mz, r 2 x , r 2 z rxtx, r 2 z rxt 1 y i 23 23 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H758 = hitx,mxrx,mz, r 2 x , r 2 z rxt 1 y i 23 29 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H759 = hit 1 y ,mxrx,mz, r 2 x , r 2 z rxtxi 23 29 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H760 = hitx, it 1 y ,mxrx,mz, r 2 x i 23 29 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H761 = hr 2 x , r 2 z , r 2 z rx, r 2 z tx, r 2 z t 1 y i 24 24 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H762 = hitx, r 2 x , r 2 z , r 2 z rx, r 2 z t 1 y i 24 29 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H763 = hit 1 y , r 2 x , r 2 z , r 2 z rx, r 2 z txi 24 29 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H764 = hitx, it 1 y , r 2 x , r 2 z , r 2 z rxi 24 29 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H765 = hmz,mzrx, r 2 x , r 2 x tx, r 2 x t 1 y i 25 25 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H766 = hitx, it 1 x ,mz,mzrx, r 2 x , r 2 x t 1 y i 25 29 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H767 = hit 1 y ,mz,mzrx, r 2 x , r 2 x txi 25 29 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H768 = hitx, it 1 x , it 1 y ,mz,mzrx, r 2 x i 25 29 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H769 = hmzrx, r 2 x , r 2 z , r 2 z tx, r 2 z t 1 y i 26 26 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H770 = hitx,mzrx, r 2 x , r 2 z , r 2 z t 1 y i 26 29 [ [ 2, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H771 = hit 1 y ,mzrx, r 2 x , r 2 z , r 2 z txi 26 29 [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H772 = hitx, it 1 y ,mzrx, r 2 x , r 2 z i 26 29 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H773 = hr 2 x , ryrx, r 2 z , r 2 z txi 27 27 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H774 = hitx, r 2 x , ryrx, r 2 z i 27 30 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H775 = hmxr 1 x tx, r 2 x , ryrx, r 2 z i 27 31 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H776 = hr 2 x , ryrx, r 2 z , r 2 z rxtxi 27 32 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H777 = hi, itx,mzrx, ryrxi 28 28 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H778 = hi,mzrx, ryrx, r 2 y rxtxi 28 33 [ [ 1, 1, 1 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ] H779 = hi, itx, it 1 y ,mzrx, r 2 x , r 2 z i 29 29 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H780 = hi, itx, r 2 x , ryrx, r 2 z i 30 30 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H781 = hi, r 2 x , ryrx, r 2 z , r 2 z rxtxi 30 33 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H782 = hmzrx, r 2 x , ryrx, r 2 z , r 2 z txi 31 31 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H783 = hitx,mzrx, r 2 x , ryrx, r 2 z i 31 33 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H784 = hr 2 x , ryrx, r 2 z , r 2 z rx, r 2 z txi 32 32 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] H785 = hitx, r 2 x , ryrx, r 2 z , r 2 z rxi 32 33 [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 2 ] ] H786 = hi, itx,mzrx, r 2 x , ryrx, r 2 z i 33 33 [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] According to [5], we say that the realization R = (, G,,') of symmetrical 2- extension of ⇤3 satisfies the [px, py, pz]-periodicity condition, where px, py, pz are positive integers, if there exist g1, g2, g3 2 Aut() such that [g1, g2] = 1, [g2, g3] = 1, [g1, g3] = 1 and 'g1'1 = tpxx , 'g2'1 = t py y , 'g3'1 = tpzz . 3 Main result We have done a computer implementation of the proposed in [7] approach, which can be called a coordinatization of symmetrical extensions of graphs. According to it, the K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 17 realization of symmetrical 2-extension of the grid ⇤3 of class I can be defined by a triple H,L,X , where H is a vertex-transitive subgroup of Aut(⇤3), L is subgroup of index 2 of the stabilizer of the vertex (0, 0, 0) in H , and X is some subset of elements of H mapping the vertex (0, 0, 0) of ⇤3 to some its adjacent vertices (see [3]for details). Below we give adaptations of Algorithm 1 and Algorithm 2 from [3] to ⇤3: Algorithm 1. Generation of all saturated realizations of symmetrical 2-extensions of ⇤3. Output: A list of realizations RHi,Li,Pi , i = 1, ..., n. Description. Look over all groups H from HI. For each such group let K = H(0,0). Choose elements h1, ..., h6 2 H such that {h1(0, 0, 0), ..., h6(0, 0, 0)} = {(1, 0, 0), (1, 0, 0), (0, 1, 0), (0,1, 0), (0, 0, 1), (0, 0,1)}. Look over all subgroups L of K of index 2. For each such group L, choose g 2 K, such that K = L [ gL. Look over all subsets N of the set {h1L, ..., h6L, h1gL, ..., h6gL} such that N is invariant relative left multiplication by elements from L and such that hL 2 N imply h1L 2 N . For each of such set N let P = {{L, gL} [ {{L,L1} : L1 2 N}} and we obtain the graph H,L,P . If between the block {L, gL} and the blocks {hjL, hjgL}, j = 1, ..., 6, there are connections of a type non-equal to 2|| or 4, then the realization (H,L,P ,H/L(H),H,K,L, '̃H,K,L) is of type I, and we put it into the output list. Let RHi,Li,Pi , i = 1, 2, be two realizations generated by the Algorithm 1. Further we describe an algorithm which tests them for equivalency. Let Ki = (Hi)(0,0) for i = 1, 2. Sets of cosets H1/K1 and H2/K2 are essentially identified with the grid ⇤2. If the realizations are equivalent, then there exists an isomorphism between them which preserves blocks, and therefore, can be extended to some automorphism g 2 Aut(⇤2). Vertex symmetry of the extension H2,L2,P2 implies that can be multiplied by some automorphism of the extension H2,L2,P2 , which preserves blocks, in such a way that the resulting isomorphism between H1,L1,P1 and H2,L2,P2 , while mapping H1/K1 onto H2/K2, will give some element g̃ already from Aut(⇤2)(0,0). An element g̃ takes one of 48 possible values. Algorithm 2, described below, allows to check whether two realizations are equivalent or not, under the assumption that g̃ = 1. In general case, to test two realizations RHi,Li,Pi , i = 1, 2, for equivalency we must look over all g̃ 2 Aut(⇤2)(0,0), and for each g̃ execute algorithm 2 to check realizations RH1,L1,P1 and g̃1(RH2,L2,P2) for equivalence. Algorithm 2. Test two realizations for equivalency, under assumption g̃ = 1. Input: Realizations RHi,Li,Pi = (Hi,Li,Pi , Gi,i,'i), i = 1, 2. Output: Test result (’yes’ or ’no’). Description. Let (n1, n2, n3) be the lexicographically minimal triple of positive integers such that both implementations satisfy the condition [n1, n2] - periodicity. Let Fi, i = 1, 2, be a subgraph of Hi,Li,Pi generated by the vertex set ' 1 i (0, 0, 0) [ ... [ '1 i (n1 1, 0, 0) [ ... [ '1i (0, n2 1, 0) [ ... [ ' 1 i (n1 1, n2 1, 0) [ ... [ ' 1 i (0, 0, n31)[...['1i (n11, 0, n31)[...[' 1 i (0, n21, n31)[...['1i (n1 1, n2 1, n3 1) of ⇤3. Comparing the blocks '11 (k, l,m) and ' 1 2 (k, l,m) for all k 2 {0, ..., n1}, l 2 {0, ..., n2},m 2 {0, ..., n3}, we can construct 2n1n2n3 correspondences of the vertices of the subgraphs F1 and F2. If among them there is a correspondence defining an isomorphism of the subgraphs F1 and F2 with additional matching on the boundary, so that this correspondence in periodicity can be extended to an isomorphism of the graphs Hi,Li,Pi , i = 1, 2, then these realizations are equivalent, and if not, then nonequivalent. Looking over 2n1n2n3 correspondences of the vertices of subgraphs is accelerated by using a backtracking. 18 Art Discrete Appl. Math. 4 (2021) #P2.04 We applied these algorithms and get the following results. A list of saturated realiza- tions generated by Algorithm 1 and thinned out by Algorithm 2, contains 2872 realizations given in Table 4 below. When executing Algorithm 2, the realization with maximal by inclusion group Hi was selected in each class of equivalent realizations. Due to this, the realizations in the resulting list are maximal. We split the set of all realizations of the symmetrical 2-extensions of ⇤3 of the class I into subclasses defined by the types of connections in the neighborhood of vertex. In the first column of Table 3 below, we give all occurring combinations of connection types in the neighborhood of vertex (59 combinations). Combinations are of the form x1x2 y1y2 z1z2, where x1 is the type of the first connection by the first direction (the grid ⇤3, and therefore a 2-extension, has three directions along coordinate axes), x2 is the type of the second con- nection by the first direction, y1 is the type of the first connection by the second direction, y2 is the type of the second connection by the second direction, z1 is the type of the first connection by the third direction, z2 is the type of the second connection by the third di- rection. Here, for each extension, the numbering of directions and connections within the direction is performed so that the combination turned out to be lexicographically minimal. In the pictures of combinations in the first column of Table 3 the first direction is shown horizontally (first connection to the left, second to the right), the second direction is shown vertically (bottom connection is first, top connection is second), the third direction is shown diagonally (bottom-left connection is first, top-right connection is second). For each com- bination of connection types, the remaining columns in Table 3 contain pictures of all found corresponding extensions of vertex neighborhood up to equivalence. In these pictures, the edges in blocks are not shown because we use these types of vertex neighborhood both for saturated and non-saturated realizations. T a b l e 3 Vertex neighborhood extensions for 2-extensions of ⇤3 of class I Connection types Vertex neighborhood extensions 1 1 1 1 1 1dd d d d d d 11 11 11 r rrr rr r r r r r r r r D DD D DD ⌦ ⌦ ⌦ ⌦ ⌦ ⌦ 1A r rrr rr r r r r r r r r 1B 1 1 1 1 2|| 2||dd d d d d d 11 11 2||2|| r rrr rr r r r r r r r r D DD D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 2A r rrr rr r r r r r r r r 2B K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 19 Connection types Vertex neighborhood extensions 1 1 1 1 2|| 4dd d d d d d 11 11 2||4 r rrr rr r r r r r r r r ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 3A r rrr rr r r r r r r r r ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 3B 1 1 1 1 3 3dd d d d d d 11 11 33 r rrr rr r r r r r r r r ``̀ ``̀ ⌅ ⌅⌅ ⌅ ⌅⌅ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ 4A r rrr rr r r r r r r r r ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 4B 1 1 1 1 4 4dd d d d d d 11 11 44 r rrr rr r r r r r r r r D DD D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 5A r rrr rr r r r r r r r r ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 5B 1 1 1 2|| 1 2||dd d d d d d 11 12|| 12|| r rrr rr r r r r r r r r 6 1 1 1 3 1 3dd d d d d d 11 13 13 r rrr rr r r r r r r r r ⌅ ⌅⌅ D DD ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 7A r rrr rr r r r r r r r r ⌅ ⌅⌅ D DD ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 7B 1 1 1 4 1 4dd d d d d d 11 14 14 r rrr rr r r r r r r r r ⌅ ⌅⌅ D DD ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 8 20 Art Discrete Appl. Math. 4 (2021) #P2.04 Connection types Vertex neighborhood extensions 1 1 2|| 2|| 2|| 2||dd d d d d d 11 2||2|| 2||2|| r rrr rr r r r r r r r r 9 1 1 2|| 2|| 2|| 4dd d d d d d 11 2||2|| 2||4 r rrr rr r r r r r r r r ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 10 1 1 2|| 2|| 3 3dd d d d d d 11 2||2|| 33 r rrr rr r r r r r r r r ``̀ ``̀D DD ⌅ ⌅⌅ ⌅ ⌅⌅ D DD ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ 11 1 1 2|| 2|| 4 4dd d d d d d 11 2||2|| 44 r rrr rr r r r r r r r r ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 12 1 1 2|| 3 2|| 3dd d d d d d 11 2||3 2||3 r rrr rr r r r r r r r r ⌅ ⌅⌅ D DD ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 13 1 1 2|| 4 2|| 4dd d d d d d 11 2||4 2||4 r rrr rr r r r r r r r r ⌅ ⌅⌅ D DD ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 14 K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 21 Connection types Vertex neighborhood extensions 1 1 2|| 4 3 3dd d d d d d 11 2||4 33 r rrr rr r r r r r r r r ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 15 1 1 2|| 4 4 4dd d d d d d 11 2||4 44 r rrr rr r r r r r r r r ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 16 1 1 2V 2V 2V 2Vdd d d d d d 11 2V 2V 2V 2V r rrr rr r r r r r r r r D DD D DD ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ 17A r rrr rr r r r r r r r r D DD D DD ⌦ ⌦ ⌦ ⌦ ⌦ ⌦ 17B 1 1 3 3 3 3dd d d d d d 11 33 33 r rrr rr r r r r r r r r ``̀ ``̀ ⌅ ⌅⌅ ⌅ ⌅⌅ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ 18A r rrr rr r r r r r r r r D DD ⌅ ⌅⌅ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 18B 1 1 3 3 4 4dd d d d d d 11 33 44 r rrr rr r r r r r r r r ``̀ ``̀ ⌅ ⌅⌅ ⌅ ⌅⌅ ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 19 1 1 3 4 3 4dd d d d d d 11 34 34 r rrr rr r r r r r r r r D DD ⌅ ⌅⌅ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 20 22 Art Discrete Appl. Math. 4 (2021) #P2.04 Connection types Vertex neighborhood extensions 1 1 4 4 4 4dd d d d d d 11 44 44 r rrr rr r r r r r r r r D DD ⌅ ⌅⌅ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 21 1 2|| 1 2|| 2|| 2||dd d d d d d 12|| 12|| 2||2|| r rrr rr r r r r r r r r 22 1 2|| 1 2|| 2|| 4dd d d d d d 12|| 12|| 2||4 r rrr rr r r r r r r r r ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 23 1 2|| 1 2|| 3 3dd d d d d d 12|| 12|| 33 r rrr rr r r r r r r r r ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 24 1 2|| 1 2|| 4 4dd d d d d d 12|| 12|| 44 r rrr rr r r r r r r r r ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 25 1 2V 1 2V 2V 2Vdd d d d d d 12V 12V 2V 2V r rrr rr r r r r r r r r ⌅ ⌅⌅ ⌅ ⌅⌅ ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ 26A r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌦ ⌦ ⌦ 26B r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ D DD ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 26C K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 23 Connection types Vertex neighborhood extensions 1 3 1 3 2|| 2||dd d d d d d 13 13 2||2|| r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ D DD 27A r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ D DD 27B 1 3 1 3 2|| 4dd d d d d d 13 13 2||4 r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ D DD ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 28A r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ D DD ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 28B 1 3 1 3 3 3dd d d d d d 13 13 33 r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 29A r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 29B 1 3 1 3 4 4dd d d d d d 13 13 44 r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 30A r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 30B 1 4 1 4 2|| 2||dd d d d d d 14 14 2||2|| r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ D DD 31 1 4 1 4 2|| 4dd d d d d d 14 14 2||4 r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ D DD ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 32 24 Art Discrete Appl. Math. 4 (2021) #P2.04 Connection types Vertex neighborhood extensions 1 4 1 4 3 3dd d d d d d 14 14 33 r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 33 1 4 1 4 4 4dd d d d d d 14 14 44 r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 34 2|| 2|| 2|| 2|| 3 3dd d d d d d 2||2|| 2||2|| 33 r rrr rr r r r r r r r r ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 35 2|| 2|| 2|| 3 2|| 3dd d d d d d 2||2|| 2||3 2||3 r rrr rr r r r r r r r r ⌅ ⌅⌅ D DD ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 36 2|| 2|| 2|| 4 3 3dd d d d d d 2||2|| 2||4 33 r rrr rr r r r r r r r r ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 37 2|| 2|| 2V 2V 2V 2Vdd d d d d d 2||2|| 2V 2V 2V 2V r rrr rr r r r r r r r r D DD D DD ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ 38A r rrr rr r r r r r r r r D DD D DD ⌦ ⌦ ⌦ ⌦ ⌦ ⌦ 38B K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 25 Connection types Vertex neighborhood extensions 2|| 2|| 3 3 3 3dd d d d d d 2||2|| 33 33 r rrr rr r r r r r r r r ``̀ ``̀ ⌅ ⌅⌅ ⌅ ⌅⌅ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ 39A r rrr rr r r r r r r r r D DD ⌅ ⌅⌅ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 39B 2|| 2|| 3 3 4 4dd d d d d d 2||2|| 33 44 r rrr rr r r r r r r r r D DD ⌅ ⌅⌅ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 40 2|| 2|| 3 4 3 4dd d d d d d 2||2|| 34 34 r rrr rr r r r r r r r r D DD ⌅ ⌅⌅ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 41 2|| 2V 2|| 2V 2V 2Vdd d d d d d 2||2V 2||2V 2V 2V r rrr rr r r r r r r r r ``̀ D DD ⌅ ⌅⌅ ⌅ ⌅⌅ ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ 42A r rrr rr r r r r r r r r ``̀ ``̀D DD ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌦ ⌦ ⌦ 42B 2|| 3 2|| 3 2|| 4dd d d d d d 2||3 2||3 2||4 r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ D DD ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 43 2|| 3 2|| 3 3 3dd d d d d d 2||3 2||3 33 r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 44 26 Art Discrete Appl. Math. 4 (2021) #P2.04 Connection types Vertex neighborhood extensions 2|| 3 2|| 3 4 4dd d d d d d 2||3 2||3 44 r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 45 2|| 4 2|| 4 3 3dd d d d d d 2||4 2||4 33 r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 46 2|| 4 2V 2V 2V 2Vdd d d d d d 2||4 2V 2V 2V 2V r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ ⌅ ⌅⌅ ⌦ ⌦ ⌦ ⌦ ⌦ ⌦ 47A r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌦ ⌦ ⌦ 47B 2|| 4 3 3 3 3dd d d d d d 2||4 33 33 r rrr rr r r r r r r r r ``̀ D DD ⌅ ⌅⌅ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 48A r rrr rr r r r r r r r r ``̀ D DD ⌅ ⌅⌅ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 48B 2|| 4 3 3 4 4dd d d d d d 2||4 33 44 r rrr rr r r r r r r r r ``̀ D DD ⌅ ⌅⌅ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 49 2|| 4 3 4 3 4dd d d d d d 2||4 34 34 r rrr rr r r r r r r r r ``̀ D DD ⌅ ⌅⌅ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 50 K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 27 Connection types Vertex neighborhood extensions 2V 2V 2V 2V 3 3dd d d d d d 2V 2V 2V 2V 33 r rrr rr r r r r r r r r ``̀ ``̀D DD D DD ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ 51A r rrr rr r r r r r r r r ``̀ ``̀ ⌅ ⌅⌅ ⌅ ⌅⌅ ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 51B 2V 2V 2V 2V 4 4dd d d d d d 2V 2V 2V 2V 44 r rrr rr r r r r r r r r ``̀ ``̀D DD D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 52A r rrr rr r r r r r r r r ``̀ ``̀ ⌅ ⌅⌅ ⌅ ⌅⌅ ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 52B 2V 2V 2V 3 2V 3dd d d d d d 2V 2V 2V 3 2V 3 r rrr rr r r r r r r r r ``̀D DD ⌅ ⌅⌅ D DD ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 53A r rrr rr r r r r r r r r D DD ⌅ ⌅⌅ D DD ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 53B r rrr rr r r r r r r r r ``̀ ⌅ ⌅⌅ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 53C 2V 2V 2V 4 2V 4dd d d d d d 2V 2V 2V 4 2V 4 r rrr rr r r r r r r r r ``̀D DD ⌅ ⌅⌅ D DD ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 54A r rrr rr r r r r r r r r D DD ⌅ ⌅⌅ D DD ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 54B 3 3 3 3 3 3dd d d d d d 33 33 33 r rrr rr r r r r r r r r D DD D DD ⌦ ⌦ ⌦ ⌦ ⌦ ⌦ 55A r rrr rr r r r r r r r r ``̀ ``̀ D DD ⌅ ⌅⌅ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 55B 3 3 3 3 4 4dd d d d d d 33 33 44 r rrr rr r r r r r r r r D DD D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 56A r rrr rr r r r r r r r r ``̀ ``̀ D DD ⌅ ⌅⌅ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 56B 28 Art Discrete Appl. Math. 4 (2021) #P2.04 Connection types Vertex neighborhood extensions 3 3 3 4 3 4dd d d d d d 33 34 34 r rrr rr r r r r r r r r ``̀ ``̀ D DD ⌅ ⌅⌅ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 57 3 3 4 4 4 4dd d d d d d 33 44 44 r rrr rr r r r r r r r r ``̀ ``̀ D DD ⌅ ⌅⌅ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 58 3 4 3 4 4 4dd d d d d d 34 34 44 r rrr rr r r r r r r r r ``̀ ``̀ D DD ⌅ ⌅⌅ ⌅ ⌅⌅ D DD ⌦ ⌦ ⌦ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌦ ⌦ ⌦ 59 A list of 2872 saturated realizations generated using Algorithm 1 and thinned out using Algorithm 2 is given below in Table 5 at the end of the parer. Saturated realizations are defined by triples H,L,X (see above): group H 2 H is given in the fourth column, sub- group L of index 2 in H(0,0,0) is presented in the fifth column, a subset of X of elements of the group H is in the seventh column. In addition, the sixth column contains an element m such that L[mL = H(0,0,0). The realization is given in the third column. Saturated realizations are sorted by vertex neighborhood extensions given in Table 3. ( of vertex neighborhood extension is given in the first column of Table 5). The set of realizations with given vertex neighborhood extension, in turn, are divided into classes defined by sizes of spheres of radii 1, 2, ..., 10 of graph of a realization. We call such a set of sizes of spheres of radii 1, 2, ..., 10 growth and give it in the second column of Table 5. In this subdi- vision, classes are lexicographically sorted by growth increasing. In Table 5, along with saturated realizations, we give non-saturated realizations by of saturated realizations with an asterisk in the third column (see details before Corollary 1 below). Theorem 3.1. The saturated realizations of symmetrical 2-extensions of the grid ⇤3 of class I up to equivalence are exhausted by 2872 pairwise nonequivalent saturated realiza- tions given in Table 5. Note that listing all, up to equivalence, realizations of the symmetrical 2-extensions of the grid ⇤3 is reduced to listing all, up to equivalence, saturated realizations of sym- metrical 2-extensions of ⇤3. Indeed, it is obvious that every non-saturated realization of a symmetrical 2-extension of ⇤3 is obtained from a uniquely defined saturated realization by removing an edge in each block. All maximal non-saturated realizations obtained in this way are given in Table 5 by of saturated maximal realizations taken with an asterisk. K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 29 Corollary 3.2. The non-saturated realizations of symmetrical 2-extensions of the grid ⇤3 of class I up to equivalence are exhausted by 2701 pairwise nonequivalent non-saturated realizations given in Table 5. Proof. Using a computer, it is easily verified that when removing edges inside blocks of 2872 realizations given in Table 5, 171 their graphs become disconnected and the remaining 2701 realizations (see with an asterisk in Table 5) give all, up to equivalence, non- saturated realizations of symmetrical 2-extensions of the grid ⇤3 of class I. According to Theorem 1 and Corollary 1, all realizations given in Table 5 are pair- wise nonequivalent. However, among the graphs of these realizations, isomorphic ones are found. With GAP, we built partition of the set of graphs of realizations from Table 5 into classes of isomorphic graphs (for details, see the proof of Corollary 3.3 below). In the following Table 4, we give all the non-singleton classes of this partition. T a b l e 4 Non-singleton classes of isomorphic graphs of realizations of symmetrical 2-extensions of ⇤3 of class I 1) 1, 45* 2) 7, 8, 76* 3) 9, 10 4) 16, 17 5) 18, 19 6) 26, 64* 7) 30, 97* 8) 36, 198*, 199*, 337* 9) 37, 38, 202*, 203*, 204*, 205*, 206*, 207*, 1119*, 1120*, 1484* 10) 39, 1121* 11) 41, 279 12) 42, 280 13) 43, 44, 214*, 215*, 343* 14) 46, 123* 15) 47, 122* 16) 54, 293 17) 55, 294 18) 57, 348*, 352* 19) 59, 144* 20) 64, 65 21) 110, 250* 22) 122, 123 23) 198, 199 24) 202, 203 25) 206, 207 26) 216, 343 27) 229, 348 28) 233, 352 29) 286, 216* 30) 296, 1123*, 1124* 31) 299, 229* 32) 301, 233* 33) 324, 363* 34) 332, 366* 35) 427, 2114* 36) 428, 503*, 658* 37) 431, 667*, 1489* 38) 439, 1422* 39) 440, 1421* 40) 455, 1440* 41) 456, 1439* 42) 651, 652 43) 653, 654 44) 664, 1489 45) 672, 1421 46) 673, 1422 47) 712, 1439 48) 713, 1440 49) 1091, 1094 50) 1092, 1093 51) 1282, 1412* 52) 1283, 1414* 53) 1288, 664*, 1423* 54) 1289, 1420* 55) 1290, 1424* 56) 1294, 672* 57) 1295, 673* 58) 1307, 1437* 59) 1313, 712* 60) 1314, 713* 61) 1716, 1717 62) 1720, 1721 63) 1722, 1724 64) 1725, 1726 65) 1727, 1728 66) 1756, 1757 67) 1759, 1760 68) 1762, 1763 69) 1857, 1858 70) 1919, 1920 71) 1928, 1929 72) 1941, 1943 73) 1944, 1945 74) 1962, 1963 75) 1990, 1991 76) 1998, 1999 77) 2002, 2006 78) 2004, 2005 79) 2043, 2047 80) 2050, 2051 81) 2053, 2054 82) 2055, 2056 83) 2058, 2059 84) 2071, 2085 85) 2072, 2086 86) 2075, 2084 87) 2080, 2082 88) 2089, 2090 89) 2091, 2092 90) 2093, 2094 91) 2109, 2110 92) 2346, 2349 93) 2347, 2348 94) 2353, 2355 95) 2357, 2358 96) 2362, 2365 97) 2363, 2364 98) 2368, 2369 99) 2371, 2372 100) 2374, 2375 101) 2376, 2378 102) 2492, 2495 103) 2552, 2553 104) 2651, 2659 105) 2652, 2660 106) 2653, 2658 107) 2654, 2655 108) 2656, 2657 109) 2702, 2703 110) 2707, 2708 111) 2713, 2714 112) 2718, 2720 113) 2724, 2726 114) 63*, 64*, 65* 115) 89*, 314* 116) 102*, 105* 117) 116*, 1121* 118) 117*, 339* 119) 121*, 219* 120) 130*, 1127* 121) 141*, 1125* 122) 146*, 353* 123) 167*, 245* 124) 175*, 1139* 125) 209*, 210* 126) 211*, 212* 127) 228*, 350* 128) 234*, 235* 129) 328*, 329* 130) 360*, 361* 131) 651*, 652*, 653*, 654*, 2114* 132) 660*, 2115* 133) 662*, 2116* 134) 817*, 818* 135) 819*, 820* 136) 821*, 2117* 137) 825*, 2118* 138) 1091*, 1094* 139) 1092*, 1093* 140) 1133*, 1134* 141) 1136*, 1137* 142) 1143*, 1144* 143) 1147*, 1148* 144) 1149*, 1150* 145) 1152*, 2521* 146) 1163*, 1622* 147) 1716*, 1717* 148) 1720*, 1721* 149) 1722*, 1724* 150) 1725*, 1726* 151) 1727*, 1728* 152) 1756*, 1757* 153) 1759*, 1760* 154) 1762*, 1763* 155) 1857*, 1858* 156) 1919*, 1920* 157) 1928*, 1929* 158) 1941*, 1943* 159) 1944*, 1945* 160) 1962*, 1963* 161) 1990*, 1991* 162) 1998*, 1999* 163) 2002*, 2006* 164) 2004*, 2005* 165) 2043*, 2047* 166) 2050*, 2051* 167) 2053*, 2054* 168) 2055*, 2056* 169) 2058*, 2059* 170) 2071*, 2085* 171) 2072*, 2086* 172) 2075*, 2084* 173) 2080*, 2082* 174) 2089*, 2090* 175) 2091*, 2092* 176) 2093*, 2094* 177) 2109*, 2110* 178) 2346*, 2349* 179) 2347*, 2348* 180) 2353*, 2355* 181) 2357*, 2358* 182) 2362*, 2365* 183) 2363*, 2364* 184) 2368*, 2369* 185) 2371*, 2372* 186) 2374*, 2375* 187) 2376*, 2378* 188) 2492*, 2495* 189) 2552*, 2553* 190) 2651*, 2659* 191) 2652*, 2660* 192) 2653*, 2658* 193) 2654*, 2655* 194) 2656*, 2657* 195) 2702*, 2703* 196) 2707*, 2708* 197) 2713*, 2714* 198) 2718*, 2720* 199) 2724*, 2726* Corollary 3.3. (i) Up to isomorphism, there are 2792 graphs of saturated realizations of symmetrical 2-extensions of the grid ⇤3 of class I. 30 Art Discrete Appl. Math. 4 (2021) #P2.04 (ii) Up to isomorphism, there are 2594 graphs of non-saturated realizations of symmet- rical 2-extensions of the grid ⇤3 of class I. Corollary 3.4. Up to isomorphism, there are 5350 graphs of realizations of symmetrical 2-extensions of the grid ⇤3 of class I. Proof. (for Corollaries 3.3 and 3.4) Using GAP for each of the graphs of 5573 (2872 sat- urated and 2701 non-saturated) realizations of symmetrical 2-extensions of ⇤3 of class I, a subgraph B was generated by a set of vertices that are at a distance of  4 from some arbitrary vertex (i.e., B is a ball of radius 4). In the obtained set of 5573 finite graphs, balls are isomorphic if and only if they correspond to realizations which are in the same line of Table 4. After that, we continued each isomorphism 'b between the balls B1 and B2 to the isomorphism ' of whole graphs of the corresponding realizations R1 = (1, G1, 1,'1) and R2 = (2, G2,2,'2) as follows. Let a realization R1 satisfies the condition of [px1, py1, pz1] - periodicity and R2 satis- fies the condition of [px2, py2, pz2] - periodicity. Recall that, according to [5], a realization R = (, G,,') of a symmetrical 2-extension of ⇤3 satisfies the condition of [px, py, pz]- periodicity, where px, py, pz are positive integers, if there exist g1, g2, g3 2 G such that [gi, gj ] = 1 for i 6= j and 'g1'1 = tpxx , 'g2'1 = t py y , 'g3'1 = t py z . We iden- tify the set of vertices of 1 with the set {(x, y, z, w) : x, y, z 2 Z, w 2 {0, 1}}, so that 1 = {{(x, y, z, 0), (x, y, z, 1)} : x, y, z 2 Z} and {(x1, y1, z1, w1), (x2, y2, z2, w2)} 2 E(1) , {(x1+px1, y1, z1, w1), (x2+px1, y2, z2, w2)} 2 E(1) , {(x1, y1+py1, z1, w1), (x2, y2 + py1, z2, w2)} 2 E(1) , {(x1, y1, z1 + pz1, w1), (x2, y2, z2 + pz1, w2)} 2 E(1). Similarly, we identify the set of vertices of 2 with the set {(x, y, z, w) : x, y, z 2 Z, w 2 {0, 1}}. We select positive integers px, py, pz so that the realization R1 satisfies the conditions of [px, py, pz]-periodicity, px|px1, py|py1, pz|pz1 and (px, 0, 0)M , (0, py, 0)M , (0, 0, pz)M 2 h(px2, 0, 0), (0, py2, 0), (0, 0, pz2)i, where angle brackets mean generation in the additive group of row-vectors and the 3⇥ 3-matrix M is obtained from the 3⇥ 4 matrix 0 @ (px, 0, 0, 0)'b/px (0, py, 0, 0)'b/py (0, 0, pz, 0)'b/pz 1 A by deleting the last column. We need to ensure that the extension of the fragment [0...px 1]⇥ [0...py 1]⇥ [0...pz 1] is inside of the ball B1. To choose px, py, pz in such a way for some pairs of realizations R1 and R2 we had to take the radii of balls B1 and B2 greater than 4. The image of an arbitrary vertex u of the extension 1 is now defined by u' = = ut1'btM , where the shift t1 = tpxn1x t pyn2 y t pzn3 z maps u into the extension of the fragment [0...px 1] ⇥ [0...py 1] ⇥ [0...pz 1], where n1, n2, n3 are suitable positive integers. K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 31 T a b l e 5 2872 saturated and 2701 non-saturated maximal realizations of symmetrical 2-extensions of the grid ⇤3 of class I Nbr. gr Hi L m X 1A [ 3, 6, 12, 20, 32, 50, 65, 82, 105, 132 ] 20*, [ 4, 9, 19, 35, 52, 72, 100, 131, 163, 201 ] 1 H778 hmzrx, ryrxi r 2 z rx mxt 1 x , r 1 z t 1 y ,mxr 1 y r 1 z t 1 z [ 4, 10, 21, 36, 54, 78, 106, 136, 173, 214 ] 2 H657 hmzrxi r 2 z rx mxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 z [ 4, 10, 21, 37, 57, 81, 109, 142, 180, 222 ] 3 H316 1 i r 2 z t 1 x ,myt 1 y ,mzt 1 z [ 4, 10, 21, 37, 57, 81, 110, 143, 180, 223 ] 4 H413 1 r 2 z rx mxt 1 x , r 2 z t 1 y ,mzt 1 z [ 4, 10, 21, 37, 58, 83, 111, 145, 184, 226 ] 5 H422 1 r 2 z rx r 2 z tx,myt 1 y ,mzt 1 z [ 4, 11, 22, 39, 60, 86, 116, 151, 190, 235 ] 6 H660 hmzrxi r 2 z rx mxt 1 x , r 2 z t 1 y ,mxr 1 x t 1 z [ 4, 11, 24, 41, 62, 90, 122, 157, 200, 247 ] 7 H302 1 i it 1 x , r 2 z t 1 y ,mzt 1 z 8 H659 hmzrxi r 2 z rx r 2 z rxt 1 x ,myt 1 y , r 1 x t 1 z [ 4, 11, 24, 41, 63, 91, 123, 160, 202, 249 ] 9 H317 1 i r 2 z t 1 x ,myt 1 y , r 2 y t 1 z 10 H395 1 r 2 z rx mxt 1 x , it 1 y , r 2 x t 1 z [ 4, 11, 24, 43, 68, 102, 138, 181, 229, 283 ] 11 H315 1 i r 2 z t 1 x , r 2 z t 1 y , r 2 y t 1 z [ 4, 11, 24, 43, 68, 102, 139, 183, 230, 283 ] 12 H423 1 r 2 z rx r 2 z tx,mxrxt 1 y ,mxr 1 x t 1 z [ 4, 11, 24, 43, 69, 102, 141, 184, 233, 287 ] 13 H409 1 r 2 z rx r 2 z tx, r 2 x t 1 y , r 2 x t 1 z [ 4, 11, 24, 43, 69, 104, 148, 195, 245, 304 ] 14 H412 1 r 2 z rx r 2 z rxt 1 x , r 2 z t 1 y ,mzt 1 z [ 4, 11, 24, 45, 76, 118, 162, 213, 280, 361 ] 15 H658 hmzrxi r 2 z rx r 2 z rxt 1 x , r 2 z t 1 y ,mxr 1 x t 1 z [ 4, 12, 24, 42, 64, 92, 124, 162, 204, 252 ] 16 H777 hmzrx, ryrxi r 2 z rx r 2 z rxt 1 x ,mxr 1 y r 1 x t 1 y , r 2 x ryt 1 z 17 H656 hmzrxi r 2 z rx r 2 z rxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 z [ 4, 12, 25, 44, 67, 96, 130, 170, 214, 264 ] 18 H655 hmzrxi r 2 z rx mxt 1 x , it 1 y , r 2 z rxt 1 z 19 H314 1 i it 1 x ,myt 1 y , r 2 x t 1 z [ 4, 12, 26, 44, 72, 104, 138, 178, 228, 282 ] 20 H393 1 r 2 z rx r 1 y r 1 z t 1 x , ryrxty, r 2 y rztz [ 4, 12, 27, 49, 77, 109, 148, 194, 244, 301 ] 21 H734 hr 1 y r 1 z i i r2 z t 1 x , ryr 1 x t 1 y , ryrzt 1 z [ 4, 12, 27, 50, 78, 116, 159, 210, 266, 330 ] 22 H301 1 i it 1 x , it 1 y , r 2 y t 1 z [ 4, 12, 27, 50, 78, 116, 162, 216, 272, 334 ] 23 H394 1 r 2 z rx r 2 z rxt 1 x , it 1 y , r 2 x t 1 z [ 4, 12, 27, 50, 79, 118, 162, 214, 271, 336 ] 24 H406 1 r 2 z rx r 2 z tx, r 2 z rxt 1 y , r 2 z rxt 1 z [ 4, 12, 27, 50, 80, 120, 167, 222, 279, 344 ] 25 H312 1 i it 1 x , r 2 z t 1 y , r 2 x t 1 z 1B [ 4, 10, 24, 50, 86, 130, 182, 242, 310, 386 ] 26 H758 hmz, r 2 x i r2 z rx mxt 1 x , r 2 x ty, r 2 x t 1 y [ 4, 11, 27, 55, 97, 153, 220, 300, 393, 497 ] 27 H413 1 r 2 z rx mxt 1 x ,myty, r 2 z t 1 y [ 4, 11, 27, 55, 97, 153, 221, 302, 395, 501 ] 28 H395 1 r 2 z rx mxt 1 x , r 2 x ty, it 1 y [ 4, 12, 28, 58, 102, 162, 234, 322, 422, 538 ] 29 H760 hmz, r 2 x i r2 z rx mxt 1 x , r 2 z ty,mxt 1 y [ 4, 12, 30, 58, 94, 138, 190, 250, 318, 394 ] 30 H757 hmz, r 2 x i r2 z rx r 2 y rxt 1 x , r 2 x ty, r 2 x t 1 y [ 4, 12, 30, 60, 105, 165, 232, 306, 398, 507 ] 31 H412 1 r 2 z rx r 2 z rxt 1 x ,myty, r 2 z t 1 y [ 4, 12, 30, 62, 108, 166, 231, 308, 397, 498 ] 32 H393 1 r 2 z rx r 2 x ryt 1 x , r 2 y rztz, r 2 x ryt 1 z [ 4, 12, 30, 62, 108, 172, 244, 328, 429, 548 ] 33 H394 1 r 2 z rx r 2 z rxt 1 x , r 2 x ty, it 1 y [ 4, 12, 30, 66, 112, 174, 246, 334, 432, 546 ] 34 H759 hmz, r 2 x i r2 z rx r 2 y rxt 1 x , r 2 z ty,mxt 1 y [ 4, 12, 33, 69, 124, 179, 247, 330, 429, 525 ] 35 H393 1 r 2 z rx ryrxtx, r 2 y rztz, r 1 y r 1 z t 1 z 32 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 2A [ 4, 9, 19, 35, 52, 72, 100, 131, 163, 201 ] 45*, [ 4, 9, 19, 39, 64, 97, 136, 171, 217, 271 ] 48*, [ 4, 9, 19, 39, 76, 140, 229, 329, 437, 556 ] 52*, [ 4, 10, 24, 50, 86, 130, 182, 242, 310, 386 ] 64*, 65*, 63*, [ 4, 10, 24, 52, 91, 135, 183, 239, 307, 380 ] 66*, [ 4, 10, 24, 52, 91, 137, 192, 257, 332, 414 ] 69*, [ 4, 10, 24, 53, 93, 140, 202, 271, 339, 421 ] 68*, [ 4, 10, 24, 54, 96, 150, 216, 292, 377, 473 ] 67*, [ 4, 10, 24, 54, 104, 168, 243, 333, 436, 551 ] 73*, [ 4, 10, 24, 56, 111, 177, 257, 353, 461, 581 ] 74*, [ 4, 10, 24, 56, 112, 187, 263, 344, 448, 573 ] 75*, [ 4, 11, 24, 41, 62, 90, 122, 157, 200, 247 ] 76*, [ 4, 11, 27, 57, 102, 156, 214, 280, 358, 446 ] 84*, [ 4, 11, 27, 57, 102, 156, 214, 281, 363, 456 ] 83*, [ 4, 11, 27, 57, 105, 172, 250, 338, 442, 559 ] 85*, [ 4, 11, 27, 62, 103, 143, 201, 274, 345, 421 ] 82*, [ 4, 11, 30, 63, 100, 142, 194, 254, 322, 398 ] 86*, [ 4, 11, 30, 65, 107, 155, 216, 284, 362, 451 ] 87*, [ 4, 11, 30, 65, 109, 159, 217, 283, 357, 442 ] 89*, [ 4, 11, 30, 66, 110, 159, 222, 293, 370, 458 ] 88*, [ 4, 11, 30, 66, 110, 160, 222, 294, 378, 467 ] 90*, [ 4, 11, 30, 67, 117, 179, 258, 341, 430, 541 ] 93*, [ 4, 11, 30, 68, 111, 159, 219, 286, 366, 453 ] 92*, [ 4, 11, 30, 69, 113, 160, 222, 294, 372, 461 ] 91*, [ 4, 12, 30, 58, 94, 138, 190, 250, 318, 394 ] 97*, [ 4, 12, 30, 64, 105, 151, 210, 279, 358, 447 ] 96*, [ 4, 12, 33, 69, 114, 168, 230, 301, 382, 474 ] 100*, [ 4, 12, 33, 69, 123, 197, 278, 369, 469, 583 ] 101*, [ 4, 12, 33, 70, 121, 182, 251, 331, 422, 523 ] 105*, 102*, [ 4, 12, 33, 70, 121, 184, 260, 350, 453, 570 ] 106*, [ 4, 12, 33, 71, 116, 166, 230, 305, 390, 485 ] 103*, [ 4, 12, 33, 73, 133, 206, 288, 383, 491, 611 ] 108*, [ 4, 12, 33, 74, 128, 191, 266, 352, 450, 560 ] 104*, [ 4, 12, 33, 75, 138, 208, 288, 383, 491, 611 ] 109*, [ 4, 12, 36, 76, 128, 190, 264, 350, 448, 558 ] 107*, [ 5, 13, 26, 45, 69, 98, 133, 173, 218, 269 ] 36 H747 hmzr 1 x , r 2 z rxi i r2zrxtx, r 2 z rxt 1 x ,mxr 1 x t 1 y , r 2 y tz [ 5, 14, 29, 50, 77, 110, 149, 194, 245, 302 ] 37 H745 hmzr 1 x , r 2 z rxi i r2zrxtx, r 2 z rxt 1 x ,mzrxt 1 y , r 2 x tz 38 H615 hmzi r 2 z r 2 z t 1 x ,myt 1 y ,mztz, t 1 z [ 5, 14, 30, 52, 79, 114, 155, 200, 254, 314 ] 39 H483 1 r 2 z mxt 1 x ,myt 1 y , r 2 y tz,mzt 1 z [ 5, 14, 30, 53, 83, 121, 164, 212, 270, 335 ] 40 H529 1 r 2 z mxt 1 x ,myty, r 2 y rztz, r 2 y t 1 z [ 5, 14, 30, 54, 87, 130, 182, 242, 310, 386 ] 41 H326 1 i r 2 z rxt 1 x ,myty,myt 1 y ,mzt 1 z 42 H541 1 r 2 z r 2 z rxtx,myt 1 y ,mztz,mzt 1 z [ 5, 15, 32, 55, 85, 122, 165, 215, 272, 335 ] 43 H618 hmzi r 2 z mxt 1 x ,myt 1 y , r 2 y tz,mxt 1 z 44 H461 1 r 2 z r 2 z t 1 x ,myt 1 y , r 2 x tz,mzt 1 z [ 5, 15, 34, 60, 92, 132, 178, 234, 299, 363 ] 45 H535 1 r 2 z r 1 z t 1 x , rzty, r 2 y tz,mzt 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 33 Nbr. gr Hi L m X [ 5, 15, 34, 61, 94, 135, 185, 242, 306, 378 ] 46 H461 1 r 2 z r 2 z t 1 x ,myt 1 y , r 2 y tz,mzt 1 z [ 5, 15, 34, 61, 95, 138, 189, 247, 314, 389 ] 47 H675 hr 2 y rxi r2zrx r 2 y tx, r 2 z t 1 x ,myt 1 y ,mxr 1 x t 1 z [ 5, 15, 34, 62, 98, 142, 191, 249, 316, 385 ] 48 H512 1 r 2 z r 1 z t 1 x , rzty, r 2 y rztz, r 2 y t 1 z [ 5, 15, 34, 63, 103, 155, 216, 285, 363, 450 ] 49 H460 1 r 2 z r 2 z t 1 x , r 2 z t 1 y , r 2 y tz,mzt 1 z [ 5, 15, 34, 64, 105, 159, 223, 293, 373, 463 ] 50 H503 1 r 2 z r 2 z t 1 x , r 2 z t 1 y , r 2 x rztz, r 2 y t 1 z [ 5, 15, 35, 63, 95, 137, 191, 249, 313, 389 ] 51 H640 hmzr 1 x i r2 x r 2 z rxtx, r 2 y rxt 1 x , r 1 x t 1 y ,mztz [ 5, 15, 38, 74, 126, 201, 289, 386, 497, 630 ] 52 H541 1 r 2 z r 2 z rxtx,myt 1 y ,mztz, it 1 z [ 5, 16, 33, 58, 89, 128, 173, 226, 285, 352 ] 53 H616 hmzi r 2 z r 2 z t 1 x ,myt 1 y , r 2 x tz,myt 1 z [ 5, 16, 34, 60, 95, 138, 190, 250, 318, 394 ] 54 H308 1 i r 2 z t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rzt 1 z 55 H511 1 r 2 z r 2 z rxtx, r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 5, 16, 36, 63, 99, 145, 197, 258, 330, 407 ] 56 H629 hmzr 1 x i r2 x r 2 z rxtx, r 2 y rxt 1 x ,mzrxt 1 y , r 2 x tz [ 5, 16, 37, 66, 103, 150, 205, 268, 341, 422 ] 57 H616 hmzi r 2 z r 2 z t 1 x ,myt 1 y , r 2 y tz,mxt 1 z [ 5, 16, 37, 66, 103, 150, 205, 269, 343, 424 ] 58 H483 1 r 2 z mxt 1 x ,myt 1 y , r 2 y tz, it 1 z [ 5, 16, 37, 66, 104, 151, 206, 271, 344, 426 ] 59 H469 1 r 2 z r 2 z t 1 x ,myt 1 y , r 2 x tz, r 2 y t 1 z [ 5, 16, 37, 66, 104, 152, 208, 274, 349, 431 ] 60 H529 1 r 2 z mxt 1 x ,myt 1 y , r 2 y rztz, r 2 y t 1 z [ 5, 16, 37, 67, 105, 151, 206, 271, 345, 427 ] 61 H468 1 r 2 z r 2 z t 1 x ,myt 1 y ,mztz, it 1 z [ 5, 16, 37, 70, 112, 168, 230, 304, 386, 480 ] 62 H614 hmzi r 2 z r 2 z t 1 x , r 2 z t 1 y , r 2 y tz,mxt 1 z [ 5, 16, 38, 70, 113, 168, 232, 305, 388, 480 ] 63 H531 1 r 2 z mxrzt 1 x ,mxrzt 1 y , r 2 y tz,mzt 1 z [ 5, 16, 38, 70, 114, 170, 237, 314, 401, 498 ] 64 H305 1 i it 1 x , ty, t 1 y , r 2 y rzt 1 z 65 H504 1 r 2 z r 2 z t 1 x , r 2 x ryt 1 y , tz, t 1 z [ 5, 16, 38, 72, 114, 162, 220, 288, 364, 449 ] 66 H479 1 r 2 z r 2 y t 1 x ,myt 1 y , r 2 x tz, r 2 y t 1 z [ 5, 16, 38, 72, 119, 183, 253, 336, 426, 530 ] 67 H533 1 r 2 z mxr 1 z t 1 x ,mxr 1 z ty, r 2 y rztz, r 2 y t 1 z [ 5, 16, 38, 73, 115, 167, 233, 302, 379, 470 ] 68 H571 1 mzrx mxtx, r 2 y rxt 1 x , r 2 z t 1 y ,mztz [ 5, 16, 38, 73, 116, 167, 228, 297, 377, 466 ] 69 H486 1 r 2 z r 2 y t 1 x ,myt 1 y ,mztz, it 1 z [ 5, 16, 38, 76, 124, 191, 271, 367, 473, 595 ] 70 H540 1 r 2 z it 1 x , r 2 x ryt 1 y ,mytz,myt 1 z [ 5, 16, 38, 76, 127, 197, 281, 380, 492, 620 ] 71 H323 1 i r 2 z tx, r 2 z t 1 x , r 2 x ryt 1 y , r 2 x t 1 z [ 5, 16, 39, 72, 119, 178, 244, 322, 410, 508 ] 72 H669 hr 2 y rxi r2zrx r 2 y tx, r 2 z t 1 x , r 2 x t 1 y , r 2 x t 1 z [ 5, 16, 41, 80, 131, 200, 283, 380, 487, 608 ] 73 H308 1 i r 2 z t 1 x ,mzty,mzt 1 y , r 2 y rzt 1 z [ 5, 16, 41, 81, 134, 205, 292, 388, 500, 620 ] 74 H511 1 r 2 z r 2 z rxtx, r 2 x t 1 y , r 2 y tz, r 2 y t 1 z [ 5, 16, 41, 83, 145, 216, 290, 387, 501, 629 ] 75 H511 1 r 2 z r 2 z rxtx, r 2 x t 1 y , r 2 x tz, r 2 y t 1 z [ 5, 17, 38, 67, 103, 148, 203, 263, 332, 411 ] 76 H626 hmzi r 2 z r 1 z t 1 x , rzty, r 2 y tz,mxt 1 z [ 5, 17, 38, 70, 109, 159, 217, 285, 362, 450 ] 77 H461 1 r 2 z r 2 z t 1 x ,myt 1 y , r 2 x tz, it 1 z [ 5, 17, 39, 72, 118, 179, 249, 335, 427, 532 ] 78 H460 1 r 2 z r 2 z t 1 x , r 2 z t 1 y , r 2 y tz, it 1 z [ 5, 17, 39, 72, 119, 179, 253, 338, 434, 539 ] 79 H503 1 r 2 z r 2 z t 1 x , r 2 z t 1 y , r 2 y rztz, r 2 y t 1 z [ 5, 17, 40, 71, 111, 164, 225, 293, 376, 467 ] 80 H747 hmzr 1 x , r 2 z rxi i r2yrxtx,mzrxt 1 x ,mxr 1 x t 1 y , r 2 y tz [ 5, 17, 40, 72, 114, 166, 228, 300, 381, 473 ] 81 H461 1 r 2 z r 2 z t 1 x ,myt 1 y , r 2 y tz, it 1 z [ 5, 17, 40, 80, 116, 166, 236, 302, 370, 468 ] 82 H639 hmzr 1 x i r2 x r 2 z rxtx, r 2 y rxt 1 x ,mxr 1 x t 1 y , r 2 y tz [ 5, 17, 41, 77, 125, 182, 248, 325, 413, 510 ] 83 H475 1 r 2 z it 1 x ,myt 1 y ,mztz, it 1 z [ 5, 17, 41, 77, 126, 182, 247, 322, 409, 505 ] 34 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 84 H485 1 r 2 z it 1 x ,myt 1 y , r 2 x tz, r 2 y t 1 z [ 5, 17, 41, 79, 135, 209, 296, 395, 506, 629 ] 85 H541 1 r 2 z r 2 z rxtx,myty,mztz, it 1 z [ 5, 17, 42, 79, 119, 170, 231, 301, 380, 469 ] 86 H671 hr 2 y rxi r2zrx r 2 y tx, r 2 z t 1 x , r 2 z t 1 y , r 1 x t 1 z [ 5, 17, 43, 77, 119, 171, 234, 303, 386, 475 ] 87 H433 1 r 2 z it 1 x , r 2 z t 1 y ,mztz, it 1 z [ 5, 17, 43, 77, 122, 176, 241, 312, 397, 489 ] 88 H561 1 mzrx mxtx, r 2 y rxt 1 x , it 1 y , r 2 x tz [ 5, 17, 43, 78, 121, 174, 235, 307, 386, 477 ] 89 H512 1 r 2 z rzt 1 x , r 1 z t 1 y , r 2 y rztz, r 2 y t 1 z [ 5, 17, 43, 79, 124, 176, 242, 316, 402, 492 ] 90 H466 1 r 2 z r 2 z t 1 x , r 2 x t 1 y ,mztz, it 1 z [ 5, 17, 43, 80, 121, 175, 240, 313, 392, 487 ] 91 H535 1 r 2 z r 1 z t 1 x , rzty, r 2 y tz, it 1 z [ 5, 17, 43, 81, 124, 177, 243, 317, 400, 494 ] 92 H427 1 r 2 z r 2 z t 1 x , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z [ 5, 17, 43, 82, 135, 199, 281, 362, 462, 575 ] 93 H462 1 r 2 z it 1 x , r 2 z t 1 y , r 2 x tz, r 2 y t 1 z [ 5, 18, 39, 72, 113, 166, 227, 300, 381, 474 ] 94 H745 hmzr 1 x , r 2 z rxi i r2yrxtx,mzrxt 1 x ,mzrxt 1 y , r 2 x tz [ 5, 18, 41, 76, 119, 174, 239, 316, 401, 498 ] 95 H615 hmzi r 2 z r 2 z t 1 x ,myt 1 y , itz, r 2 z t 1 z [ 5, 18, 42, 81, 124, 179, 244, 319, 404, 499 ] 96 H631 hmzr 1 x i r2 x r 2 z rxtx, r 2 y rxt 1 x , r 2 y rxt 1 y , itz [ 5, 18, 43, 76, 123, 178, 245, 319, 406, 500 ] 97 H624 hmzi r 2 z mxrzt 1 x ,mxrzt 1 y , r 2 y tz,mxt 1 z [ 5, 18, 43, 84, 135, 204, 285, 382, 489, 612 ] 98 H305 1 i it 1 x , ity, it 1 y , r 2 y rzt 1 z [ 5, 18, 43, 84, 137, 208, 291, 390, 499, 624 ] 99 H504 1 r 2 z r 2 z t 1 x , r 2 x ryt 1 y , r 2 z tz, r 2 z t 1 z [ 5, 18, 44, 82, 132, 188, 255, 331, 420, 517 ] 100 H474 1 r 2 z it 1 x , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z [ 5, 18, 44, 85, 149, 223, 301, 398, 505, 629 ] 101 H511 1 r 2 z r 2 z rxtx, r 2 x ty, r 2 x tz, r 2 y t 1 z [ 5, 18, 45, 83, 138, 201, 280, 367, 468, 576 ] 102 H533 1 r 2 z mxrzt 1 x ,mxrzt 1 y , r 2 y rztz, r 2 y t 1 z [ 5, 18, 45, 84, 131, 186, 257, 334, 425, 522 ] 103 H473 1 r 2 z it 1 x , r 2 x t 1 y ,mztz, it 1 z [ 5, 18, 45, 85, 140, 206, 289, 377, 484, 596 ] 104 H531 1 r 2 z mxrzt 1 x ,mxrzt 1 y , r 2 y tz, it 1 z [ 5, 18, 46, 85, 139, 206, 285, 376, 481, 597 ] 105 H323 1 i r 2 z t 1 x , r 2 x ryt 1 y ,mxtz,mxt 1 z [ 5, 18, 46, 85, 139, 207, 288, 383, 492, 614 ] 106 H540 1 r 2 z it 1 x , r 2 x ryt 1 y ,mxtz,mxt 1 z [ 5, 18, 47, 88, 147, 209, 293, 379, 485, 595 ] 107 H674 hr 2 y rxi r2zrx r 2 y tx, r 2 z t 1 x , it 1 y ,mzr 1 x t 1 z [ 5, 18, 49, 94, 151, 224, 310, 408, 518, 640 ] 108 H326 1 i r 2 z rxt 1 x ,myt 1 y , r 2 z tz, r 2 z t 1 z [ 5, 18, 49, 95, 152, 223, 311, 407, 519, 639 ] 109 H541 1 r 2 z r 2 z rxtx,myt 1 y , itz, it 1 z 2B [ 4, 11, 30, 73, 147, 243, 346, 462, 602, 756 ] 112*, [ 4, 11, 30, 76, 156, 253, 350, 466, 608, 762 ] 113*, [ 4, 12, 33, 85, 153, 240, 344, 464, 600, 752 ] 115*, [ 5, 14, 32, 64, 112, 176, 256, 352, 464, 592 ] 110 H779 hi, r 2 x , r 2 z i r2 z rx mxtx, t 1 x ,myty, t 1 y [ 5, 16, 42, 88, 152, 232, 328, 440, 568, 712 ] 111 H766 hmy, r 2 x i mzr1x r 2 z rxtx,mxt 1 x ,myty, t 1 y [ 5, 17, 46, 101, 177, 262, 364, 490, 628, 774 ] 112 H571 1 mzrx mxtx, r 2 y rxt 1 x ,myty, r 2 z t 1 y [ 5, 17, 46, 103, 179, 266, 365, 493, 629, 777 ] 113 H561 1 mzrx mxtx, r 2 y rxt 1 x , r 2 x ty, it 1 y [ 5, 18, 48, 96, 160, 240, 336, 448, 576, 720 ] 114 H779 hi, r 2 x , r 2 z i r2 z rx mzr 1 x tx, r 2 z rxt 1 x ,myty, t 1 y [ 5, 18, 48, 106, 174, 266, 366, 490, 622, 778 ] 115 H768 hmy, r 2 x i mzr1x r 2 z rxtx,mxt 1 x , r 2 z ty,mxt 1 y 3A [ 5, 14, 30, 52, 79, 114, 155, 200, 254, 314 ] 116*, [ 5, 14, 31, 57, 90, 131, 181, 238, 303, 377 ] 117*, [ 5, 14, 31, 58, 95, 141, 195, 260, 335, 416 ] 119*, [ 5, 14, 31, 59, 96, 141, 200, 268, 338, 421 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 35 Nbr. gr Hi L m X 118*, [ 5, 14, 31, 61, 100, 143, 201, 274, 345, 421 ] 120*, [ 5, 15, 34, 61, 94, 135, 185, 242, 306, 378 ] 123*, [ 5, 15, 34, 61, 95, 138, 189, 247, 314, 389 ] 122*, [ 5, 15, 34, 61, 96, 141, 193, 253, 324, 401 ] 121*, [ 5, 15, 34, 62, 97, 139, 190, 249, 314, 387 ] 125*, [ 5, 15, 34, 62, 98, 142, 194, 254, 322, 398 ] 128*, [ 5, 15, 34, 62, 99, 145, 199, 262, 334, 415 ] 124*, [ 5, 15, 34, 63, 102, 151, 210, 279, 358, 447 ] 131*, [ 5, 15, 34, 63, 103, 154, 214, 284, 366, 458 ] 129*, [ 5, 15, 35, 64, 100, 146, 201, 264, 337, 418 ] 126*, [ 5, 15, 35, 65, 103, 150, 206, 271, 345, 427 ] 127*, [ 5, 15, 35, 67, 114, 178, 255, 344, 446, 560 ] 132*, 133*, [ 5, 15, 37, 68, 108, 158, 218, 287, 365, 453 ] 134*, [ 5, 15, 37, 68, 108, 159, 222, 293, 370, 458 ] 135*, [ 5, 15, 37, 69, 111, 163, 224, 297, 381, 469 ] 136*, [ 5, 15, 37, 70, 111, 166, 228, 301, 392, 487 ] 137*, [ 5, 15, 37, 70, 113, 166, 230, 305, 390, 485 ] 138*, [ 5, 15, 37, 74, 122, 178, 244, 322, 410, 508 ] 130*, [ 5, 15, 37, 84, 147, 214, 293, 387, 495, 615 ] 139*, [ 5, 16, 36, 62, 96, 140, 189, 246, 314, 386 ] 140*, [ 5, 16, 36, 63, 97, 139, 189, 247, 312, 384 ] 141*, [ 5, 16, 36, 64, 100, 144, 196, 256, 324, 400 ] 147*, 142*, 143*, [ 5, 16, 37, 66, 104, 151, 206, 271, 344, 426 ] 144*, [ 5, 16, 37, 66, 105, 157, 217, 283, 357, 442 ] 146*, [ 5, 16, 37, 67, 106, 155, 213, 280, 357, 443 ] 145*, [ 5, 16, 37, 67, 107, 159, 222, 294, 372, 461 ] 148*, [ 5, 16, 37, 67, 108, 159, 218, 289, 369, 457 ] 154*, [ 5, 16, 39, 72, 113, 164, 223, 292, 373, 461 ] 152*, [ 5, 16, 39, 73, 115, 165, 224, 294, 375, 467 ] 151*, [ 5, 16, 39, 73, 115, 166, 228, 300, 381, 473 ] 153*, [ 5, 16, 39, 75, 124, 186, 259, 342, 436, 541 ] 149*, [ 5, 16, 39, 76, 125, 189, 265, 349, 444, 550 ] 150*, [ 5, 16, 40, 75, 115, 165, 228, 298, 378, 471 ] 155*, [ 5, 16, 42, 77, 118, 172, 234, 305, 389, 480 ] 158*, [ 5, 16, 42, 82, 132, 192, 263, 346, 440, 545 ] 156*, [ 5, 16, 42, 82, 132, 199, 278, 360, 454, 571 ] 159*, [ 5, 16, 42, 83, 133, 194, 266, 349, 444, 550 ] 157*, [ 5, 16, 43, 89, 144, 210, 292, 387, 495, 615 ] 160*, [ 5, 17, 39, 71, 112, 162, 221, 289, 369, 455 ] 161*, [ 5, 17, 39, 72, 115, 168, 231, 303, 384, 476 ] 166*, [ 5, 17, 40, 76, 129, 197, 277, 369, 470, 585 ] 170*, [ 5, 17, 40, 76, 129, 198, 279, 368, 469, 587 ] 171*, [ 5, 17, 41, 73, 114, 168, 229, 298, 381, 471 ] 164*, [ 5, 17, 41, 74, 114, 165, 228, 298, 378, 471 ] 162*, 163*, [ 5, 17, 41, 74, 116, 170, 232, 303, 387, 478 ] 169*, [ 5, 17, 41, 75, 118, 171, 234, 307, 390, 483 ] 165*, 36 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X [ 5, 17, 41, 76, 124, 184, 253, 333, 424, 525 ] 167*, [ 5, 17, 41, 77, 127, 191, 266, 352, 450, 560 ] 168*, [ 5, 17, 42, 78, 124, 179, 244, 320, 406, 503 ] 172*, [ 5, 17, 42, 79, 125, 179, 244, 321, 408, 505 ] 173*, [ 5, 17, 43, 86, 142, 210, 292, 387, 495, 615 ] 180*, [ 5, 17, 44, 82, 131, 194, 268, 352, 448, 556 ] 174*, [ 5, 17, 44, 83, 131, 190, 259, 339, 430, 531 ] 175*, [ 5, 17, 44, 84, 134, 196, 270, 356, 454, 564 ] 177*, 178*, [ 5, 17, 44, 84, 135, 198, 273, 360, 459, 570 ] 179*, [ 5, 17, 44, 84, 139, 206, 282, 370, 470, 582 ] 176*, [ 5, 18, 41, 73, 114, 165, 228, 298, 378, 471 ] 181*, [ 5, 18, 43, 76, 118, 171, 234, 307, 390, 483 ] 182*, 183*, [ 5, 18, 44, 79, 123, 178, 244, 321, 408, 505 ] 184*, 185*, [ 5, 18, 46, 87, 141, 210, 292, 387, 495, 615 ] 186*, [ 5, 19, 43, 75, 118, 171, 234, 307, 390, 483 ] 187*, [ 5, 19, 44, 78, 123, 178, 244, 321, 408, 505 ] 188*, [ 5, 19, 45, 80, 127, 185, 256, 340, 433, 536 ] 189*, [ 6, 18, 37, 63, 99, 142, 189, 249, 317, 384 ] 116 H645 hmzr 1 x i r2 x r 2 z rxtx, r 2 z t 1 x ,mxrxt 1 x , r 1 x t 1 y ,mztz [ 6, 18, 38, 66, 102, 146, 198, 258, 326, 402 ] 117 H640 hmzr 1 x i r2 x r 2 z rxtx, r 2 y rxt 1 x , r 2 z rxt 1 x , r 1 x t 1 y ,mztz [ 6, 18, 40, 74, 116, 168, 233, 302, 379, 470 ] 118 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z t 1 y ,mztz [ 6, 18, 40, 74, 117, 168, 229, 299, 378, 467 ] 119 H486 1 r 2 z r 2 y t 1 x ,myt 1 y ,mztz, it 1 z ,mzt 1 z [ 6, 18, 42, 78, 116, 166, 236, 302, 370, 468 ] 120 H639 hmzr 1 x i r2 x r 2 z rxtx, r 2 y rxt 1 x , r 2 z rxt 1 x ,mxr 1 x t 1 y , r 2 y tz [ 6, 19, 40, 69, 108, 154, 208, 273, 344, 423 ] 121 H629 hmzr 1 x i r2 x r 2 z rxtx, r 2 y rxt 1 x , r 2 z rxt 1 x ,mzrxt 1 y , r 2 x tz [ 6, 19, 41, 71, 110, 158, 214, 279, 353, 435 ] 122 H461 1 r 2 z r 2 z t 1 x ,myt 1 y , r 2 y tz, r 2 x tz,mzt 1 z 123 H482 1 r 2 z mxt 1 x ,myt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 19, 41, 72, 112, 160, 217, 283, 358, 442 ] 124 H468 1 r 2 z r 2 z t 1 x ,myt 1 y ,mztz, it 1 z ,mzt 1 z [ 6, 19, 41, 72, 112, 161, 219, 285, 360, 444 ] 125 H530 1 r 2 z mxt 1 x ,myt 1 y ,mzr 1 z tz,mzrztz, r 2 y t 1 z [ 6, 19, 42, 72, 111, 161, 217, 283, 359, 441 ] 126 H483 1 r 2 z mxt 1 x ,myt 1 y , r 2 y tz, it 1 z ,mzt 1 z [ 6, 19, 42, 73, 113, 164, 222, 289, 366, 450 ] 127 H529 1 r 2 z mxt 1 x ,myt 1 y , r 2 y rztz, r 2 x rztz, r 2 y t 1 z [ 6, 19, 43, 77, 119, 170, 231, 301, 380, 469 ] 128 H479 1 r 2 z r 2 y t 1 x ,myt 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 19, 43, 79, 126, 182, 249, 327, 414, 511 ] 129 H475 1 r 2 z it 1 x ,myt 1 y ,mztz, it 1 z ,mzt 1 z [ 6, 19, 43, 80, 131, 192, 262, 342, 432, 534 ] 130 H638 hmzr 1 x i r2 x r 2 z rxtx, r 2 z t 1 x ,mxrxt 1 x ,mzrxt 1 y , r 2 x tz [ 6, 19, 44, 79, 124, 179, 244, 319, 404, 499 ] 131 H631 hmzr 1 x i r2 x r 2 z rxtx, r 2 y rxt 1 x , r 2 z rxt 1 x , r 2 y rxt 1 y , itz [ 6, 19, 44, 84, 141, 212, 299, 398, 507, 629 ] 132 H541 1 r 2 z r 2 z rxtx,myt 1 y ,mztz, it 1 z ,mzt 1 z 133 H541 1 r 2 z r 2 z rxtx,myty,mztz, it 1 z ,mzt 1 z [ 6, 19, 45, 77, 121, 172, 235, 305, 387, 476 ] 134 H433 1 r 2 z it 1 x , r 2 z t 1 y ,mztz, it 1 z ,mzt 1 z [ 6, 19, 45, 77, 123, 176, 241, 312, 397, 489 ] 135 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , it 1 y , r 2 x tz [ 6, 19, 45, 79, 126, 177, 243, 318, 403, 493 ] 136 H466 1 r 2 z r 2 z t 1 x , r 2 x t 1 y ,mztz, it 1 z ,mzt 1 z [ 6, 19, 47, 82, 130, 189, 263, 343, 445, 545 ] 137 H539 1 r 2 z it 1 x , r 2 x ty,mzr 1 z tz,mzrztz, r 2 y rzt 1 z [ 6, 19, 47, 82, 131, 186, 257, 334, 425, 522 ] 138 H473 1 r 2 z it 1 x , r 2 x t 1 y ,mztz, it 1 z ,mzt 1 z [ 6, 19, 50, 100, 159, 226, 312, 410, 520, 642 ] 139 H511 1 r 2 z r 2 z rxtx, r 2 x t 1 y , r 2 y tz, r 2 x tz, r 2 y t 1 z [ 6, 20, 42, 73, 115, 163, 221, 290, 365, 450 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 37 Nbr. gr Hi L m X 140 H479 1 r 2 z r 2 y t 1 x ,myt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 20, 43, 75, 117, 167, 226, 296, 374, 461 ] 141 H537 1 r 2 z r 1 z t 1 x , rzty,mzr 1 z tz,mzrztz, r 2 y t 1 z [ 6, 20, 43, 76, 119, 170, 231, 301, 380, 469 ] 142 H524 1 r 2 z mzr 1 z t 1 x ,mzrzty, r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 143 H509 1 r 2 z r 1 z t 1 x , rzty, r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 20, 44, 75, 118, 168, 229, 298, 377, 465 ] 144 H469 1 r 2 z r 2 z t 1 x ,myt 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 20, 44, 76, 118, 170, 230, 300, 380, 468 ] 145 H461 1 r 2 z r 2 z t 1 x ,myt 1 y , r 2 x tz, it 1 z ,mzt 1 z [ 6, 20, 44, 76, 120, 174, 235, 307, 386, 477 ] 146 H512 1 r 2 z r 1 z t 1 x , rzty, r 2 y rztz, r 2 x rztz, r 2 y t 1 z [ 6, 20, 44, 77, 119, 170, 231, 301, 380, 469 ] 147 H429 1 r 2 z r 2 y t 1 x , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 20, 44, 77, 121, 175, 240, 313, 392, 487 ] 148 H535 1 r 2 z r 1 z t 1 x , rzty, r 2 y tz, it 1 z ,mzt 1 z [ 6, 20, 45, 83, 134, 197, 275, 359, 456, 563 ] 149 H460 1 r 2 z r 2 z t 1 x , r 2 z t 1 y , r 2 y tz, it 1 z ,mzt 1 z [ 6, 20, 45, 84, 134, 201, 278, 366, 463, 572 ] 150 H503 1 r 2 z r 2 z t 1 x , r 2 z t 1 y , r 2 y rztz, r 2 x rztz, r 2 y t 1 z [ 6, 20, 46, 81, 124, 176, 239, 313, 395, 488 ] 151 H645 hmzr 1 x i r2 x r 2 y rxtx, r 2 z t 1 x ,mxrxt 1 x , r 1 x t 1 y ,mztz [ 6, 20, 46, 81, 125, 179, 243, 317, 401, 495 ] 152 H469 1 r 2 z r 2 z t 1 x ,myt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 20, 46, 82, 125, 179, 245, 318, 402, 497 ] 153 H461 1 r 2 z r 2 z t 1 x ,myt 1 y , r 2 y tz, it 1 z ,mzt 1 z [ 6, 20, 46, 82, 130, 186, 254, 331, 419, 517 ] 154 H485 1 r 2 z it 1 x ,myt 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 20, 47, 82, 123, 178, 243, 313, 398, 493 ] 155 H640 hmzr 1 x i r2 x r 2 y rxtx, r 2 y rxt 1 x , r 2 z rxt 1 x , r 1 x t 1 y ,mztz [ 6, 20, 48, 88, 143, 204, 279, 363, 460, 567 ] 156 H425 1 r 2 z r 2 z t 1 x , r 2 z t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 20, 48, 89, 143, 207, 280, 367, 463, 572 ] 157 H519 1 r 2 z r 2 z t 1 x , r 2 z t 1 y ,mzr 1 z tz,mzrztz, r 2 y t 1 z [ 6, 20, 50, 84, 131, 188, 255, 332, 421, 518 ] 158 H427 1 r 2 z r 2 z t 1 x , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 20, 50, 90, 147, 216, 295, 379, 483, 602 ] 159 H462 1 r 2 z it 1 x , r 2 z t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 20, 54, 101, 155, 226, 312, 410, 520, 642 ] 160 H541 1 r 2 z r 2 z rxtx,myt 1 y , itz,mztz, it 1 z [ 6, 21, 45, 81, 126, 178, 244, 319, 401, 495 ] 161 H427 1 r 2 z r 2 z t 1 x , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 21, 47, 81, 123, 178, 243, 313, 398, 493 ] 162 H486 1 r 2 z r 2 y t 1 x ,myt 1 y , itz, it 1 z ,mzt 1 z 163 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z t 1 y ,mztz [ 6, 21, 47, 81, 128, 184, 249, 325, 412, 507 ] 164 H485 1 r 2 z it 1 x ,myt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 21, 47, 82, 127, 182, 247, 322, 407, 502 ] 165 H629 hmzr 1 x i r2 x r 2 y rxtx, r 2 y rxt 1 x , r 2 z rxt 1 x ,mzrxt 1 y , r 2 x tz [ 6, 21, 47, 83, 131, 187, 255, 331, 420, 517 ] 166 H474 1 r 2 z it 1 x , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 21, 47, 84, 138, 201, 280, 367, 468, 576 ] 167 H533 1 r 2 z mxrzt 1 x ,mxrzt 1 y , r 2 y rztz, r 2 x rztz, r 2 y t 1 z [ 6, 21, 47, 85, 140, 206, 289, 377, 484, 596 ] 168 H531 1 r 2 z mxrzt 1 x ,mxrzt 1 y , r 2 y tz, it 1 z ,mzt 1 z [ 6, 21, 48, 83, 131, 188, 255, 332, 421, 518 ] 169 H474 1 r 2 z it 1 x , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 21, 48, 89, 150, 222, 302, 397, 506, 628 ] 170 H511 1 r 2 z r 2 z rxtx, r 2 x ty, r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 21, 48, 92, 149, 220, 302, 397, 508, 628 ] 171 H511 1 r 2 z r 2 z rxtx, r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 21, 49, 86, 134, 190, 259, 336, 426, 525 ] 172 H461 1 r 2 z r 2 z t 1 x ,myt 1 y , r 2 y tz, r 2 x tz, it 1 z [ 6, 21, 49, 87, 134, 190, 259, 338, 427, 526 ] 173 H468 1 r 2 z r 2 z t 1 x ,myt 1 y , itz, it 1 z ,mzt 1 z [ 6, 21, 49, 87, 143, 205, 282, 368, 468, 576 ] 174 H462 1 r 2 z it 1 x , r 2 z t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 21, 50, 89, 146, 208, 284, 370, 469, 577 ] 175 H526 1 r 2 z mxrzt 1 x ,mxrzt 1 y ,mzr 1 z tz,mzrztz, r 2 y t 1 z [ 6, 21, 50, 89, 147, 212, 296, 384, 488, 600 ] 176 H638 hmzr 1 x i r2 x r 2 y rxtx, r 2 z t 1 x ,mxrxt 1 x ,mzrxt 1 y , r 2 x tz [ 6, 21, 50, 90, 148, 212, 292, 380, 484, 596 ] 177 H507 1 r 2 z r 2 y rzt 1 x , r 2 y rzt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 178 H522 1 r 2 z mxrzt 1 x ,mxrzt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 21, 51, 91, 149, 213, 294, 382, 486, 598 ] 38 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 179 H471 1 r 2 z it 1 x , it 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 21, 53, 99, 155, 226, 312, 410, 520, 642 ] 180 H511 1 r 2 z r 2 z rxtx, r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 6, 22, 46, 81, 123, 178, 243, 313, 398, 493 ] 181 H639 hmzr 1 x i r2 x r 2 y rxtx, r 2 y rxt 1 x , r 2 z rxt 1 x ,mxr 1 x t 1 y , r 2 y tz [ 6, 22, 48, 82, 127, 182, 247, 322, 407, 502 ] 182 H433 1 r 2 z it 1 x , r 2 z t 1 y , itz, it 1 z ,mzt 1 z 183 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , it 1 y , r 2 x tz [ 6, 22, 50, 86, 133, 190, 259, 338, 427, 526 ] 184 H475 1 r 2 z it 1 x ,myt 1 y , itz, it 1 z ,mzt 1 z 185 H466 1 r 2 z r 2 z t 1 x , r 2 x t 1 y , itz, it 1 z ,mzt 1 z [ 6, 22, 55, 98, 155, 226, 312, 410, 520, 642 ] 186 H541 1 r 2 z r 2 z rxtx,myt 1 y , itz, it 1 z ,mzt 1 z [ 6, 23, 47, 82, 127, 182, 247, 322, 407, 502 ] 187 H631 hmzr 1 x i r2 x r 2 y rxtx, r 2 y rxt 1 x , r 2 z rxt 1 x , r 2 y rxt 1 y , itz [ 6, 23, 49, 86, 133, 190, 259, 338, 427, 526 ] 188 H473 1 r 2 z it 1 x , r 2 x t 1 y , itz, it 1 z ,mzt 1 z [ 6, 23, 50, 87, 141, 202, 277, 360, 459, 566 ] 189 H539 1 r 2 z it 1 x , r 2 x t 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z 3B [ 5, 15, 37, 78, 142, 228, 332, 452, 588, 740 ] 190*, [ 5, 15, 37, 80, 151, 242, 342, 460, 602, 756 ] 191*, [ 5, 15, 37, 83, 156, 248, 348, 466, 608, 762 ] 192*, [ 5, 15, 37, 84, 150, 240, 344, 464, 600, 752 ] 193*, [ 5, 17, 47, 100, 172, 260, 364, 484, 620, 772 ] 194*, [ 5, 18, 49, 101, 172, 260, 364, 484, 620, 772 ] 195*, 196*, [ 5, 19, 49, 100, 172, 260, 364, 484, 620, 772 ] 197*, [ 6, 19, 46, 94, 164, 252, 356, 476, 612, 764 ] 190 H766 hmy, r 2 x i mzr1x mxtx,mxt 1 x , r 2 z rxt 1 x ,myty, t 1 y [ 6, 19, 48, 103, 178, 262, 364, 490, 628, 774 ] 191 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, r 2 z t 1 y [ 6, 19, 48, 105, 179, 265, 365, 493, 629, 777 ] 192 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty, it 1 y [ 6, 19, 50, 104, 174, 266, 366, 490, 622, 778 ] 193 H768 hmy, r 2 x i mzr1x mxtx,mxt 1 x , r 2 z rxt 1 x , r 2 z ty,mxt 1 y [ 6, 21, 56, 112, 184, 272, 376, 496, 632, 784 ] 194 H766 hmy, r 2 x i mzr1x r 2 z rxtx,mxt 1 x , r 2 z rxt 1 x ,myty, t 1 y [ 6, 22, 57, 112, 184, 272, 376, 496, 632, 784 ] 195 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty, it 1 y 196 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, r 2 z t 1 y [ 6, 23, 56, 112, 184, 272, 376, 496, 632, 784 ] 197 H768 hmy, r 2 x i mzr1x r 2 z rxtx,mxt 1 x , r 2 z rxt 1 x , r 2 z ty,mxt 1 y 4A [ 5, 13, 26, 45, 69, 98, 133, 173, 218, 269 ] 198*, 199*, [ 5, 13, 26, 46, 75, 112, 157, 213, 275, 343 ] 200*, [ 5, 13, 26, 47, 81, 126, 176, 236, 309, 396 ] 201*, [ 5, 14, 29, 50, 77, 110, 149, 194, 245, 302 ] 202*, 206*, 203*, 207*, 204*, 205*, [ 5, 14, 31, 57, 88, 125, 173, 228, 287, 355 ] 208*, [ 5, 14, 32, 59, 94, 139, 193, 256, 327, 406 ] 209*, 210*, [ 5, 14, 32, 59, 94, 139, 195, 262, 335, 414 ] 212*, 211*, [ 5, 14, 32, 68, 117, 178, 256, 346, 449, 564 ] 213*, [ 5, 15, 32, 55, 85, 122, 165, 215, 272, 335 ] 214*, 215*, [ 5, 15, 33, 58, 89, 127, 173, 226, 285, 351 ] 216*, [ 5, 15, 33, 59, 94, 138, 190, 250, 318, 394 ] 217*, [ 5, 15, 34, 61, 96, 141, 193, 253, 324, 401 ] 219*, [ 5, 15, 35, 64, 99, 141, 192, 251, 318, 393 ] 218*, [ 5, 15, 35, 65, 103, 152, 212, 279, 357, 447 ] 220*, [ 5, 15, 37, 70, 112, 166, 228, 299, 383, 474 ] 221*, [ 5, 15, 37, 71, 113, 165, 227, 299, 381, 473 ] 222*, [ 5, 15, 37, 76, 130, 193, 264, 353, 458, 566 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 39 Nbr. gr Hi L m X 223*, [ 5, 15, 38, 79, 135, 206, 290, 385, 493, 613 ] 225*, [ 5, 15, 39, 75, 120, 177, 243, 320, 409, 507 ] 224*, [ 5, 16, 36, 66, 106, 156, 216, 286, 366, 456 ] 226*, [ 5, 16, 36, 67, 105, 149, 205, 268, 338, 419 ] 227*, [ 5, 16, 36, 67, 109, 162, 225, 296, 378, 471 ] 228*, [ 5, 16, 37, 69, 111, 163, 226, 300, 384, 478 ] 232*, [ 5, 16, 37, 70, 115, 170, 235, 310, 395, 490 ] 234*, 235*, [ 5, 16, 38, 68, 106, 156, 213, 278, 356, 440 ] 229*, [ 5, 16, 38, 69, 109, 160, 218, 285, 365, 452 ] 233*, [ 5, 16, 38, 71, 112, 163, 226, 296, 376, 469 ] 230*, [ 5, 16, 38, 71, 113, 165, 228, 298, 378, 471 ] 231*, [ 5, 16, 40, 80, 135, 204, 286, 381, 489, 609 ] 239*, [ 5, 16, 41, 78, 125, 183, 252, 332, 423, 525 ] 236*, [ 5, 16, 41, 79, 128, 187, 258, 342, 436, 540 ] 237*, [ 5, 16, 43, 84, 133, 194, 266, 349, 444, 550 ] 238*, [ 5, 17, 40, 73, 114, 165, 228, 298, 378, 471 ] 240*, [ 5, 17, 40, 73, 116, 169, 232, 305, 388, 481 ] 241*, [ 5, 17, 40, 74, 118, 171, 234, 307, 390, 483 ] 242*, [ 5, 17, 41, 76, 121, 176, 242, 319, 406, 503 ] 243*, 244*, [ 5, 17, 41, 76, 124, 184, 253, 333, 424, 525 ] 245*, [ 5, 17, 43, 84, 139, 208, 290, 385, 493, 613 ] 247*, [ 5, 17, 43, 86, 148, 224, 311, 415, 535, 664 ] 246*, [ 5, 18, 42, 75, 118, 171, 234, 307, 390, 483 ] 248*, [ 5, 18, 43, 78, 123, 178, 244, 321, 408, 505 ] 249*, [ 6, 17, 36, 63, 96, 138, 188, 243, 308, 381 ] 198 H302 1 i tx, it 1 x , t 1 x , r 2 z t 1 y ,mzt 1 z 199 H659 hmzrxi r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,myt 1 y , r 1 x t 1 z [ 6, 17, 36, 65, 101, 146, 203, 264, 333, 416 ] 200 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 z t 1 y ,mzt 1 z [ 6, 17, 36, 67, 106, 154, 213, 281, 363, 459 ] 201 H658 hmzrxi r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 z t 1 y ,mxr 1 x t 1 z [ 6, 18, 38, 66, 102, 146, 198, 258, 326, 402 ] 202 H654 hmzrxi r 2 z rx tx, r 2 z rxt 1 x , t 1 x , it 1 y , r 2 z rxt 1 z 203 H656 hmzrxi r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 x t 1 y ,mzr 1 x t 1 z 204 H304 1 i mztx, r 2 z t 1 x ,mzt 1 x , r 2 z t 1 y ,mzt 1 z [ 6, 18, 38, 67, 104, 148, 201, 262, 331, 409 ] 205 H316 1 i mztx, r 2 z t 1 x ,mzt 1 x ,myt 1 y ,mzt 1 z [ 6, 18, 39, 68, 105, 150, 204, 266, 336, 414 ] 206 H300 1 i tx, it 1 x , t 1 x , it 1 y ,mzt 1 z 207 H314 1 i tx, it 1 x , t 1 x ,myt 1 y , r 2 x t 1 z [ 6, 18, 40, 72, 111, 160, 219, 284, 360, 447 ] 208 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,myt 1 y ,mzt 1 z [ 6, 18, 41, 72, 113, 162, 222, 292, 370, 458 ] 209 H301 1 i tx, it 1 x , t 1 x , it 1 y , r 2 y t 1 z 210 H301 1 i it 1 x , ty, it 1 y , t 1 y , r 2 y t 1 z [ 6, 18, 41, 72, 113, 162, 223, 292, 370, 456 ] 211 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , it 1 y , r 2 x t 1 z [ 6, 18, 41, 72, 114, 166, 230, 304, 384, 474 ] 212 H312 1 i tx, it 1 x , t 1 x , r 2 z t 1 y , r 2 x t 1 z [ 6, 18, 45, 87, 138, 208, 292, 388, 496, 616 ] 213 H308 1 i mztx, r 2 z t 1 x ,mzt 1 x , r 2 z ty, r 2 y rzt 1 z [ 6, 19, 41, 71, 110, 158, 214, 279, 353, 435 ] 214 H302 1 i it 1 x ,mzty, r 2 z t 1 y ,mzt 1 y ,mzt 1 z [ 6, 19, 41, 71, 111, 158, 216, 280, 355, 437 ] 215 H317 1 i r 2 z t 1 x ,myt 1 y ,mytz, r 2 y t 1 z ,myt 1 z [ 6, 19, 41, 72, 111, 159, 217, 282, 356, 440 ] 216 H317 1 i mztx, r 2 z t 1 x ,mzt 1 x ,myt 1 y , r 2 y t 1 z [ 6, 19, 41, 73, 114, 164, 225, 295, 374, 463 ] 217 H315 1 i mztx, r 2 z t 1 x ,mzt 1 x , r 2 z t 1 y , r 2 y t 1 z 40 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X [ 6, 19, 42, 75, 116, 166, 226, 293, 372, 458 ] 218 H423 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,mxrxt 1 y ,mxr 1 x t 1 z [ 6, 19, 43, 75, 118, 170, 231, 302, 383, 472 ] 219 H318 1 i r 2 z t 1 x ,myt 1 y ,mxtz,mxt 1 z , r 2 x t 1 z [ 6, 19, 43, 76, 117, 171, 234, 303, 387, 480 ] 220 H661 hmzrxi r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,myt 1 y , r 1 x t 1 z [ 6, 19, 45, 79, 126, 183, 250, 327, 416, 513 ] 221 H315 1 i r 2 z t 1 x ,mzty, r 2 z t 1 y ,mzt 1 y , r 2 y t 1 z [ 6, 19, 45, 82, 134, 193, 264, 345, 438, 541 ] 222 H315 1 i r 2 z t 1 x , r 2 z t 1 y ,mytz, r 2 y t 1 z ,myt 1 z [ 6, 19, 45, 86, 143, 206, 283, 375, 480, 590 ] 223 H311 1 i it 1 x , r 2 z t 1 y ,mytz, r 2 y t 1 z ,myt 1 z [ 6, 19, 47, 84, 140, 201, 275, 360, 458, 564 ] 224 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 x t 1 y , r 2 x t 1 z [ 6, 19, 50, 95, 152, 225, 311, 409, 519, 641 ] 225 H326 1 i r 2 z rxt 1 x ,myt 1 y , r 2 z tz,mztz, r 2 z t 1 z [ 6, 20, 43, 77, 120, 174, 237, 311, 394, 488 ] 226 H657 hmzrxi r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 x t 1 y ,mzr 1 x t 1 z [ 6, 20, 44, 79, 123, 174, 239, 311, 392, 485 ] 227 H319 1 i r 2 z t 1 x ,mxty,mxt 1 y , r 2 x t 1 y , r 2 y t 1 z [ 6, 20, 44, 80, 126, 180, 248, 326, 414, 512 ] 228 H312 1 i it 1 x ,mzty, r 2 z t 1 y ,mzt 1 y , r 2 x t 1 z [ 6, 20, 45, 77, 120, 173, 234, 305, 387, 476 ] 229 H303 1 i it 1 x ,mxty,mxt 1 y , r 2 x t 1 y , r 2 x t 1 z [ 6, 20, 45, 80, 122, 177, 242, 312, 397, 492 ] 230 H316 1 i r 2 z t 1 x , r 2 y ty, r 2 y t 1 y ,myt 1 y ,mzt 1 z [ 6, 20, 45, 80, 123, 178, 243, 313, 398, 493 ] 231 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 z t 1 y ,mzt 1 z [ 6, 20, 45, 81, 126, 182, 249, 327, 414, 512 ] 232 H316 1 i r 2 z t 1 x ,myt 1 y , r 2 z tz, r 2 z t 1 z ,mzt 1 z [ 6, 20, 46, 80, 125, 179, 243, 317, 402, 495 ] 233 H314 1 i it 1 x ,myt 1 y ,mxtz,mxt 1 z , r 2 x t 1 z [ 6, 20, 46, 83, 134, 197, 272, 358, 456, 565 ] 234 H305 1 i tx, it 1 x , t 1 x , it 1 y , r 2 y rzt 1 z [ 6, 20, 47, 85, 137, 201, 278, 366, 466, 577 ] 235 H305 1 i tx, it 1 x , t 1 x , ity, r 2 y rzt 1 z [ 6, 20, 48, 85, 138, 197, 270, 351, 446, 549 ] 236 H301 1 i it 1 x , it 1 y ,mytz, r 2 y t 1 z ,myt 1 z [ 6, 20, 48, 87, 143, 203, 280, 365, 463, 569 ] 237 H312 1 i it 1 x , r 2 z t 1 y ,mxtz,mxt 1 z , r 2 x t 1 z [ 6, 20, 49, 89, 143, 207, 280, 367, 463, 572 ] 238 H406 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 z rxt 1 y , r 2 z rxt 1 z [ 6, 20, 50, 93, 150, 221, 307, 405, 515, 637 ] 239 H308 1 i mztx, r 2 z t 1 x ,mzt 1 x , r 2 z t 1 y , r 2 y rzt 1 z [ 6, 21, 46, 81, 123, 178, 243, 313, 398, 493 ] 240 H660 hmzrxi r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 z t 1 y ,mxr 1 x t 1 z [ 6, 21, 46, 81, 126, 181, 246, 321, 406, 501 ] 241 H302 1 i it 1 x , r 2 z t 1 y , r 2 z tz, r 2 z t 1 z ,mzt 1 z [ 6, 21, 46, 82, 127, 182, 247, 322, 407, 502 ] 242 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , it 1 y , r 2 x t 1 z [ 6, 21, 48, 85, 132, 189, 258, 337, 426, 525 ] 243 H313 1 i it 1 x ,myt 1 y , r 2 z tz, r 2 z t 1 z ,mzt 1 z 244 H317 1 i r 2 z t 1 x , r 2 y ty, r 2 y t 1 y ,myt 1 y , r 2 y t 1 z [ 6, 21, 50, 90, 146, 213, 292, 384, 488, 603 ] 245 H323 1 i r 2 z t 1 x , r 2 x ryt 1 y ,mxtz, r 2 x tz,mxt 1 z [ 6, 21, 51, 95, 157, 232, 323, 428, 549, 680 ] 246 H323 1 i r 2 z t 1 x , r 2 x ryt 1 y ,mxtz,mxt 1 z , r 2 x t 1 z [ 6, 21, 53, 97, 154, 225, 311, 409, 519, 641 ] 247 H326 1 i r 2 z rxt 1 x ,myt 1 y , r 2 z tz, r 2 z t 1 z ,mzt 1 z [ 6, 22, 47, 82, 127, 182, 247, 322, 407, 502 ] 248 H655 hmzrxi r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , it 1 y , r 2 z rxt 1 z [ 6, 22, 49, 86, 133, 190, 259, 338, 427, 526 ] 249 H314 1 i it 1 x , r 2 y ty, r 2 y t 1 y ,myt 1 y , r 2 x t 1 z 4B [ 5, 14, 32, 64, 112, 176, 256, 352, 464, 592 ] 250*, [ 5, 14, 32, 65, 120, 201, 300, 409, 537, 692 ] 251*, [ 5, 14, 32, 68, 124, 208, 312, 424, 555, 711 ] 252*, [ 5, 14, 32, 68, 130, 210, 310, 430, 566, 718 ] 253*, [ 5, 16, 42, 88, 152, 232, 328, 440, 568, 712 ] 254*, [ 5, 17, 46, 98, 171, 260, 364, 484, 620, 772 ] 255*, [ 5, 17, 46, 98, 172, 261, 364, 484, 620, 772 ] 256*, K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 41 Nbr. gr Hi L m X [ 5, 18, 48, 100, 172, 260, 364, 484, 620, 772 ] 257*, [ 6, 18, 44, 88, 148, 224, 316, 424, 548, 688 ] 250 H757 hmz, r 2 x i r2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , r 2 x ty, r 2 x t 1 y [ 6, 18, 44, 90, 157, 242, 340, 455, 592, 746 ] 251 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,myty, r 2 z t 1 y [ 6, 18, 44, 92, 158, 248, 346, 464, 601, 756 ] 252 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 x ty, it 1 y [ 6, 18, 44, 96, 160, 248, 348, 472, 604, 760 ] 253 H759 hmz, r 2 x i r2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , r 2 z ty,mxt 1 y [ 6, 20, 52, 104, 172, 256, 356, 472, 604, 752 ] 254 H758 hmz, r 2 x i r2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 x ty, r 2 x t 1 y [ 6, 21, 55, 111, 184, 272, 376, 496, 632, 784 ] 255 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,myty, r 2 z t 1 y [ 6, 21, 55, 111, 185, 272, 376, 496, 632, 784 ] 256 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 x ty, it 1 y [ 6, 22, 56, 112, 184, 272, 376, 496, 632, 784 ] 257 H760 hmz, r 2 x i r2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 z ty,mxt 1 y 5A [ 6, 18, 38, 66, 102, 146, 198, 258, 326, 402 ] 258*, [ 6, 18, 39, 71, 114, 165, 224, 294, 375, 467 ] 259*, [ 6, 18, 40, 73, 114, 165, 228, 298, 378, 471 ] 260*, [ 6, 19, 42, 75, 118, 171, 234, 307, 390, 483 ] 261*, [ 6, 19, 43, 76, 118, 172, 234, 305, 389, 480 ] 262*, [ 6, 19, 43, 77, 122, 178, 244, 320, 406, 503 ] 263*, [ 6, 19, 43, 78, 123, 178, 244, 321, 408, 505 ] 264*, [ 6, 19, 44, 80, 127, 185, 256, 340, 433, 536 ] 265*, [ 6, 19, 45, 84, 139, 206, 282, 370, 470, 582 ] 266*, [ 6, 19, 45, 86, 141, 210, 292, 387, 495, 615 ] 268*, 269*, 270*, 271*, [ 6, 19, 48, 90, 142, 206, 282, 370, 470, 582 ] 267*, [ 6, 21, 51, 95, 153, 224, 308, 406, 516, 638 ] 273*, [ 6, 21, 51, 95, 154, 227, 313, 413, 526, 652 ] 272*, [ 6, 21, 51, 95, 155, 230, 318, 420, 535, 663 ] 274*, [ 6, 21, 51, 95, 156, 233, 323, 427, 545, 677 ] 275*, [ 7, 21, 44, 77, 119, 170, 231, 301, 380, 469 ] 258 H748 hmzr 1 x , r 2 z rxi i r2ztx,mxrxtx,myt 1 x , rxt 1 x ,mxr 1 x t 1 y , r 2 y tz [ 7, 21, 45, 80, 124, 176, 239, 313, 395, 488 ] 259 H645 hmzr 1 x i r2 x r 2 y rxtx, r 2 z rxtx, r 2 z t 1 x ,mxrxt 1 x , r 1 x t 1 y ,mztz [ 7, 21, 46, 81, 123, 178, 243, 313, 398, 493 ] 260 H747 hmzr 1 x , r 2 z rxi i r2yrxtx, r 2 z rxtx,mzrxt 1 x , r 2 z rxt 1 x ,mxr 1 x t 1 y , r 2 y tz [ 7, 22, 47, 82, 127, 182, 247, 322, 407, 502 ] 261 H745 hmzr 1 x , r 2 z rxi i r2yrxtx, r 2 z rxtx,mzrxt 1 x , r 2 z rxt 1 x ,mzrxt 1 y , r 2 x tz [ 7, 22, 49, 84, 131, 188, 255, 332, 421, 518 ] 262 H616 hmzi r 2 z r 2 z t 1 x ,myt 1 y , r 2 y tz, r 2 x tz,mxt 1 z ,myt 1 z [ 7, 22, 49, 85, 133, 190, 259, 336, 426, 525 ] 263 H461 1 r 2 z r 2 z t 1 x ,myt 1 y , r 2 y tz, r 2 x tz, it 1 z ,mzt 1 z [ 7, 22, 49, 86, 133, 190, 259, 338, 427, 526 ] 264 H615 hmzi r 2 z r 2 z t 1 x ,myt 1 y , itz,mztz, r 2 z t 1 z , t 1 z [ 7, 22, 50, 87, 141, 202, 277, 360, 459, 566 ] 265 H539 1 r 2 z it 1 x , r 2 x t 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z , r 2 x rzt 1 z [ 7, 22, 50, 89, 147, 212, 296, 384, 488, 600 ] 266 H638 hmzr 1 x i r2 x r 2 y rxtx, r 2 z rxtx, r 2 z t 1 x ,mxrxt 1 x ,mzrxt 1 y , r 2 x tz [ 7, 22, 53, 93, 152, 216, 296, 384, 488, 600 ] 267 H746 hmzr 1 x , r 2 z rxi i r2ztx,mxrxtx,myt 1 x , rxt 1 x ,mzrxt 1 y , r 2 x tz [ 7, 22, 54, 98, 155, 226, 312, 410, 520, 642 ] 268 H308 1 i r 2 z t 1 x , r 2 z ty,mzty, r 2 z t 1 y ,mzt 1 y , r 2 y rzt 1 z 269 H326 1 i r 2 z rxt 1 x ,myt 1 y , r 2 z tz,mztz, r 2 z t 1 z ,mzt 1 z 270 H511 1 r 2 z r 2 z rxtx, r 2 x t 1 y , r 2 y tz, r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 271 H541 1 r 2 z r 2 z rxtx,myt 1 y , itz,mztz, it 1 z ,mzt 1 z [ 7, 24, 57, 102, 163, 236, 325, 426, 541, 668 ] 272 H305 1 i it 1 x , ity, ty, it 1 y , t 1 y , r 2 y rzt 1 z 273 H540 1 r 2 z it 1 x , r 2 x ryt 1 y ,mxtz,mytz,mxt 1 z ,myt 1 z [ 7, 24, 57, 102, 165, 240, 331, 434, 551, 680 ] 274 H504 1 r 2 z r 2 z t 1 x , r 2 x ryt 1 y , r 2 z tz, tz, r 2 z t 1 z , t 1 z [ 7, 24, 57, 102, 165, 240, 331, 434, 554, 686 ] 42 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 275 H323 1 i r 2 z tx,mztx, r 2 z t 1 x ,mzt 1 x , r 2 x ryt 1 y , r 2 x t 1 z 5B [ 6, 19, 48, 100, 172, 260, 364, 484, 620, 772 ] 276*, [ 7, 22, 56, 112, 184, 272, 376, 496, 632, 784 ] 276 H779 hi, r 2 x , r 2 z i r2 z rx mxtx,mzr 1 x tx, t 1 x , r 2 z rxt 1 x ,myty, t 1 y 6 [ 4, 10, 24, 52, 104, 180, 268, 354, 452, 570 ] 308*, [ 4, 11, 27, 64, 127, 209, 291, 377, 477, 591 ] 317*, [ 4, 11, 30, 65, 109, 159, 217, 283, 357, 442 ] 314*, [ 4, 11, 30, 65, 112, 164, 229, 302, 383, 486 ] 316*, [ 4, 11, 30, 65, 116, 181, 258, 347, 449, 565 ] 315*, [ 4, 11, 30, 66, 119, 190, 262, 342, 438, 547 ] 318*, [ 4, 11, 30, 67, 135, 213, 285, 375, 478, 600 ] 319*, [ 4, 12, 30, 66, 107, 154, 211, 278, 353, 438 ] 328*, 329*, [ 4, 12, 30, 66, 114, 174, 243, 326, 419, 526 ] 333*, [ 4, 12, 30, 66, 124, 195, 272, 368, 472, 589 ] 335*, [ 4, 12, 33, 73, 129, 189, 257, 343, 441, 549 ] 334*, [ 4, 12, 33, 78, 140, 210, 292, 389, 499, 617 ] 331*, [ 4, 12, 36, 76, 123, 176, 241, 318, 401, 500 ] 336*, [ 5, 13, 27, 51, 86, 130, 182, 242, 310, 386 ] 277 H422 1 r 2 z rx r 2 z tx,myty,myt 1 y ,mzt 1 z 278 H419 1 r 2 z rx mxt 1 x ,myty,myt 1 y ,mzt 1 z [ 5, 14, 30, 54, 87, 130, 182, 242, 310, 386 ] 279 H415 1 r 2 z rx it 1 x ,myty,myt 1 y ,mzt 1 z 280 H422 1 r 2 z rx r 2 z tx,myt 1 y ,mztz,mzt 1 z [ 5, 14, 31, 56, 85, 120, 165, 216, 271, 334 ] 281 H413 1 r 2 z rx mxt 1 x ,myty, r 1 x t 1 y , rxtz [ 5, 14, 31, 58, 94, 138, 190, 250, 318, 394 ] 282 H402 1 r 2 z rx r 2 z rxt 1 x ,myty,myt 1 y ,mzt 1 z 283 H404 1 r 2 z rx r 2 y rxt 1 x ,myty,myt 1 y ,mzt 1 z [ 5, 14, 31, 59, 100, 155, 223, 303, 395, 500 ] 284 H413 1 r 2 z rx mxt 1 x ,myty, r 2 z t 1 y ,mzt 1 z [ 5, 14, 31, 60, 105, 165, 232, 306, 398, 507 ] 285 H412 1 r 2 z rx r 2 z rxt 1 x ,myty, r 2 z t 1 y ,mzt 1 z [ 5, 15, 33, 58, 89, 127, 173, 226, 285, 351 ] 286 H395 1 r 2 z rx mxt 1 x ,mzrxt 1 y ,mzrxtz, r 2 x t 1 z [ 5, 15, 33, 59, 94, 138, 190, 250, 318, 394 ] 287 H409 1 r 2 z rx r 2 z tx, r 2 x ty, r 2 x t 1 y , r 2 x t 1 z 288 H401 1 r 2 z rx mxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 5, 15, 33, 60, 101, 156, 222, 303, 396, 499 ] 289 H413 1 r 2 z rx mxt 1 x ,myty, r 2 z t 1 y , r 2 y tz [ 5, 15, 34, 59, 89, 130, 177, 227, 290, 359 ] 290 H412 1 r 2 z rx r 2 z rxt 1 x ,myty, r 1 x t 1 y , rxtz [ 5, 15, 36, 68, 106, 152, 208, 271, 344, 427 ] 291 H418 1 r 2 z rx mxt 1 x , r 1 x ty, r 2 z t 1 y , rxt 1 z [ 5, 15, 36, 71, 115, 175, 248, 322, 415, 519 ] 292 H407 1 r 2 z rx r 2 z rxt 1 x , r 1 x ty, r 2 z t 1 y , rxt 1 z [ 5, 16, 34, 60, 95, 138, 190, 250, 318, 394 ] 293 H396 1 r 2 z rx it 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z 294 H409 1 r 2 z rx r 2 z tx, r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 5, 16, 34, 64, 107, 166, 238, 326, 426, 542 ] 295 H418 1 r 2 z rx mxt 1 x , r 2 z ty, r 2 z t 1 y , r 2 y t 1 z [ 5, 16, 35, 60, 93, 134, 181, 236, 299, 368 ] 296 H394 1 r 2 z rx r 2 z rxt 1 x ,mzrxt 1 y ,mzrxtz, r 2 x t 1 z [ 5, 16, 35, 62, 98, 142, 194, 254, 322, 398 ] 297 H391 1 r 2 z rx r 2 z rxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z 298 H392 1 r 2 z rx r 2 y rxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 5, 16, 38, 68, 106, 156, 213, 278, 356, 440 ] 299 H388 1 r 2 z rx r 2 z rxt 1 x ,mzrxty, it 1 y ,mzrxt 1 z [ 5, 16, 38, 69, 108, 155, 209, 274, 347, 427 ] 300 H419 1 r 2 z rx mxt 1 x ,myty,mxr 1 x t 1 y ,mxrxtz [ 5, 16, 38, 69, 109, 160, 218, 285, 365, 452 ] 301 H400 1 r 2 z rx mxt 1 x ,mzrxty, it 1 y ,mzrxt 1 z [ 5, 16, 38, 70, 114, 174, 246, 329, 427, 537 ] 302 H395 1 r 2 z rx mxt 1 x , r 2 x ty, it 1 y , r 2 x t 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 43 Nbr. gr Hi L m X [ 5, 16, 38, 72, 120, 186, 261, 344, 447, 566 ] 303 H394 1 r 2 z rx r 2 z rxt 1 x , r 2 x ty, it 1 y , r 2 x t 1 z [ 5, 16, 38, 74, 125, 189, 265, 354, 455, 568 ] 304 H412 1 r 2 z rx r 2 z rxt 1 x ,myty, r 2 z t 1 y , r 2 y tz [ 5, 16, 38, 78, 133, 204, 288, 388, 500, 628 ] 305 H407 1 r 2 z rx r 2 z rxt 1 x , r 2 z ty, r 2 z t 1 y , r 2 y t 1 z [ 5, 16, 39, 78, 126, 190, 266, 358, 462, 582 ] 306 H414 1 r 2 z rx it 1 x , r 2 z ty, r 2 z t 1 y , r 2 y t 1 z [ 5, 16, 39, 80, 136, 208, 292, 392, 504, 632 ] 307 H410 1 r 2 z rx r 2 y rxt 1 x , r 2 z ty, r 2 z t 1 y , r 2 y t 1 z [ 5, 16, 43, 81, 137, 201, 286, 377, 482, 597 ] 308 H422 1 r 2 z rx r 2 z tx,mxr 1 x t 1 y ,mxrxtz,mzt 1 z [ 5, 17, 39, 74, 116, 168, 228, 298, 376, 466 ] 309 H413 1 r 2 z rx mxt 1 x ,mxr 1 x ty, r 2 z t 1 y ,mxrxt 1 z [ 5, 17, 40, 78, 126, 192, 269, 352, 449, 562 ] 310 H412 1 r 2 z rx r 2 z rxt 1 x ,mxr 1 x ty, r 2 z t 1 y ,mxrxt 1 z [ 5, 17, 41, 74, 117, 171, 232, 304, 388, 478 ] 311 H401 1 r 2 z rx mxt 1 x , r 2 y rxt 1 y , r 2 y rxtz, r 2 x t 1 z [ 5, 17, 41, 75, 122, 184, 256, 343, 445, 555 ] 312 H395 1 r 2 z rx mxt 1 x , it 1 y , itz, r 2 x t 1 z [ 5, 17, 41, 76, 123, 182, 248, 324, 417, 516 ] 313 H402 1 r 2 z rx r 2 z rxt 1 x ,myty,mxr 1 x t 1 y ,mxrxtz [ 5, 17, 44, 81, 130, 188, 256, 334, 419, 524 ] 314 H410 1 r 2 z rx r 2 y rxt 1 x , r 1 x ty, r 2 z t 1 y , rxt 1 z [ 5, 17, 44, 81, 136, 205, 286, 381, 493, 616 ] 315 H414 1 r 2 z rx it 1 x , r 1 x ty, r 2 z t 1 y , rxt 1 z [ 5, 17, 44, 83, 129, 185, 261, 329, 423, 533 ] 316 H422 1 r 2 z rx r 2 z tx,myty,mxr 1 x t 1 y ,mxrxtz [ 5, 17, 46, 88, 149, 218, 299, 396, 498, 618 ] 317 H409 1 r 2 z rx r 2 z tx, r 2 y rxt 1 y , r 2 y rxtz, r 2 x t 1 z [ 5, 17, 47, 88, 140, 209, 286, 378, 469, 586 ] 318 H404 1 r 2 z rx r 2 y rxt 1 x ,myty,mxr 1 x t 1 y ,mxrxtz [ 5, 17, 47, 91, 150, 223, 308, 402, 503, 631 ] 319 H415 1 r 2 z rx it 1 x ,myty,mxr 1 x t 1 y ,mxrxtz [ 5, 18, 41, 78, 124, 182, 249, 328, 417, 518 ] 320 H393 1 r 2 z rx r 2 y rztx, r 2 y rzty, r 2 x ryt 1 y , r 2 y rztz [ 5, 18, 41, 78, 128, 194, 272, 366, 472, 594 ] 321 H389 1 r 2 z rx it 1 x , it 1 y , itz, it 1 z [ 5, 18, 42, 79, 124, 181, 247, 325, 412, 511 ] 322 H395 1 r 2 z rx mxt 1 x , r 2 y rxty, it 1 y , r 2 y rxt 1 z [ 5, 18, 42, 80, 129, 194, 272, 366, 472, 594 ] 323 H400 1 r 2 z rx mxt 1 x , it 1 y , itz, it 1 z [ 5, 18, 43, 81, 131, 191, 263, 347, 443, 549 ] 324 H391 1 r 2 z rx r 2 z rxt 1 x , r 2 y rxt 1 y , r 2 y rxtz, r 2 x t 1 z [ 5, 18, 43, 82, 135, 202, 280, 372, 478, 594 ] 325 H394 1 r 2 z rx r 2 z rxt 1 x , it 1 y , itz, r 2 x t 1 z [ 5, 18, 43, 83, 134, 199, 275, 367, 467, 581 ] 326 H394 1 r 2 z rx r 2 z rxt 1 x , r 2 y rxty, it 1 y , r 2 y rxt 1 z [ 5, 18, 43, 84, 137, 206, 286, 382, 489, 612 ] 327 H388 1 r 2 z rx r 2 z rxt 1 x , it 1 y , itz, it 1 z [ 5, 18, 44, 82, 126, 183, 248, 325, 411, 508 ] 328 H389 1 r 2 z rx it 1 x ,mzrxty, it 1 y ,mzrxt 1 z [ 5, 18, 44, 83, 129, 185, 254, 332, 420, 518 ] 329 H398 1 r 2 z rx r 2 y rxt 1 x ,mzrxty, it 1 y ,mzrxt 1 z [ 5, 18, 44, 86, 140, 210, 292, 390, 500, 626 ] 330 H398 1 r 2 z rx r 2 y rxt 1 x , it 1 y , itz, it 1 z [ 5, 18, 46, 94, 160, 233, 318, 430, 547, 665 ] 331 H393 1 r 2 z rx r 1 y r 1 z t 1 x , ryrxty, r 2 x ryt 1 y , r 2 y rztz [ 5, 18, 47, 87, 140, 207, 284, 376, 480, 596 ] 332 H393 1 r 2 z rx r 1 y r 1 z t 1 x , ryrxty, r 1 y r 1 z t 1 y , ryrxtz [ 5, 18, 47, 88, 141, 204, 282, 369, 471, 582 ] 333 H392 1 r 2 z rx r 2 y rxt 1 x , r 2 y rxt 1 y , r 2 y rxtz, r 2 x t 1 z [ 5, 18, 47, 88, 142, 203, 279, 368, 467, 576 ] 334 H409 1 r 2 z rx r 2 z tx, r 2 x ty, r 2 y rxt 1 y , r 2 y rxtz [ 5, 18, 47, 91, 146, 217, 305, 401, 508, 633 ] 335 H396 1 r 2 z rx it 1 x , r 2 y rxt 1 y , r 2 y rxtz, r 2 x t 1 z [ 5, 18, 49, 90, 137, 199, 270, 351, 446, 548 ] 336 H393 1 r 2 z rx r 2 y rztx, r 2 y rzty, r 1 y r 1 z t 1 y , ryrxtz 7A [ 5, 13, 26, 45, 69, 98, 133, 173, 218, 269 ] 337*, [ 5, 14, 31, 57, 90, 131, 181, 238, 303, 377 ] 339*, [ 5, 14, 31, 57, 91, 134, 185, 244, 313, 390 ] 338*, [ 5, 14, 31, 57, 91, 136, 192, 253, 320, 399 ] 340*, 44 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X [ 5, 14, 34, 71, 120, 185, 258, 341, 438, 547 ] 342*, [ 5, 14, 34, 72, 127, 193, 269, 359, 461, 573 ] 341*, [ 5, 15, 32, 55, 85, 122, 165, 215, 272, 335 ] 343*, [ 5, 15, 33, 61, 99, 146, 202, 267, 341, 425 ] 344*, [ 5, 15, 34, 67, 111, 162, 229, 302, 383, 486 ] 345*, [ 5, 15, 38, 74, 127, 194, 270, 356, 457, 571 ] 346*, [ 5, 16, 36, 67, 109, 162, 225, 296, 378, 471 ] 350*, [ 5, 16, 37, 66, 103, 150, 205, 268, 341, 422 ] 348*, 352*, [ 5, 16, 37, 66, 105, 157, 217, 283, 357, 442 ] 353*, [ 5, 16, 37, 67, 107, 158, 217, 286, 367, 455 ] 349*, [ 5, 16, 37, 71, 117, 170, 236, 314, 399, 498 ] 351*, [ 5, 16, 37, 72, 117, 176, 245, 328, 421, 528 ] 355*, [ 5, 16, 37, 72, 122, 188, 271, 355, 445, 559 ] 347*, [ 5, 16, 39, 78, 134, 202, 280, 374, 485, 604 ] 354*, [ 5, 16, 39, 79, 135, 204, 281, 373, 481, 602 ] 356*, [ 5, 17, 40, 73, 116, 170, 235, 310, 395, 491 ] 357*, [ 5, 17, 41, 77, 127, 187, 258, 345, 443, 551 ] 358*, [ 5, 17, 41, 81, 137, 204, 283, 373, 478, 597 ] 359*, [ 5, 18, 39, 70, 109, 158, 215, 282, 357, 442 ] 360*, 361*, [ 5, 18, 43, 76, 121, 176, 241, 318, 401, 500 ] 362*, [ 5, 18, 43, 81, 131, 191, 263, 347, 443, 549 ] 363*, [ 5, 18, 43, 83, 136, 203, 282, 377, 481, 599 ] 365*, [ 5, 18, 46, 88, 140, 204, 284, 376, 478, 594 ] 364*, [ 5, 18, 47, 87, 140, 207, 284, 376, 480, 596 ] 366*, [ 6, 17, 36, 64, 98, 140, 191, 248, 313, 387 ] 337 H413 1 r 2 z rx mxt 1 x , r 2 z t 1 y , r 1 x t 1 y , rxtz,mzt 1 z [ 6, 18, 40, 71, 109, 156, 211, 275, 348, 429 ] 338 H419 1 r 2 z rx mxt 1 x ,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mzt 1 z [ 6, 18, 41, 73, 114, 163, 222, 289, 366, 452 ] 339 H418 1 r 2 z rx mxt 1 x , r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z [ 6, 18, 41, 74, 115, 171, 236, 304, 387, 479 ] 340 H412 1 r 2 z rx r 2 z rxt 1 x , r 2 z t 1 y , r 1 x t 1 y , rxtz,mzt 1 z [ 6, 18, 43, 83, 141, 208, 292, 382, 487, 607 ] 341 H407 1 r 2 z rx r 2 z rxt 1 x , r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z [ 6, 18, 47, 88, 140, 209, 286, 378, 469, 586 ] 342 H404 1 r 2 z rx r 2 y rxt 1 x ,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mzt 1 z [ 6, 19, 41, 72, 111, 159, 217, 282, 356, 440 ] 343 H395 1 r 2 z rx mxt 1 x , it 1 y ,mzrxt 1 y ,mzrxtz, r 2 x t 1 z [ 6, 19, 42, 76, 118, 170, 230, 300, 378, 468 ] 344 H413 1 r 2 z rx mxt 1 x ,mxr 1 x ty, r 2 z t 1 y ,mxrxt 1 z ,mzt 1 z [ 6, 19, 43, 83, 129, 185, 261, 329, 423, 533 ] 345 H422 1 r 2 z rx r 2 z tx,myty,mxr 1 x t 1 y ,mxrxtz,mztz [ 6, 19, 48, 87, 145, 211, 290, 379, 483, 600 ] 346 H422 1 r 2 z rx r 2 z tx,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mzt 1 z [ 6, 20, 44, 85, 137, 208, 293, 372, 471, 591 ] 347 H412 1 r 2 z rx r 2 z rxt 1 x ,mxr 1 x ty, r 2 z t 1 y ,mxrxt 1 z ,mzt 1 z [ 6, 20, 45, 77, 120, 173, 234, 305, 387, 476 ] 348 H388 1 r 2 z rx r 2 z rxt 1 x , it 1 y ,mzrxt 1 y ,mzrxtz, it 1 z [ 6, 20, 45, 79, 124, 177, 240, 315, 397, 489 ] 349 H401 1 r 2 z rx mxt 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z [ 6, 20, 45, 81, 128, 188, 259, 339, 427, 529 ] 350 H394 1 r 2 z rx r 2 z rxt 1 x , it 1 y ,mzrxt 1 y ,mzrxtz, r 2 x t 1 z [ 6, 20, 45, 85, 132, 188, 262, 339, 428, 535 ] 351 H402 1 r 2 z rx r 2 z rxt 1 x ,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mzt 1 z [ 6, 20, 46, 80, 125, 179, 243, 317, 402, 495 ] 352 H400 1 r 2 z rx mxt 1 x , it 1 y ,mzrxt 1 y ,mzrxtz, it 1 z [ 6, 20, 47, 81, 131, 191, 259, 338, 423, 526 ] 353 H410 1 r 2 z rx r 2 y rxt 1 x , r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z [ 6, 20, 48, 88, 145, 214, 297, 395, 508, 628 ] 354 H414 1 r 2 z rx it 1 x , r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z [ 6, 20, 49, 88, 142, 205, 283, 370, 472, 583 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 45 Nbr. gr Hi L m X 355 H392 1 r 2 z rx r 2 y rxt 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z [ 6, 20, 50, 94, 152, 220, 305, 402, 508, 629 ] 356 H415 1 r 2 z rx it 1 x ,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mzt 1 z [ 6, 21, 48, 85, 132, 190, 258, 337, 426, 526 ] 357 H395 1 r 2 z rx mxt 1 x , r 2 y rxty, it 1 y , r 2 y rxt 1 z , r 2 x t 1 z [ 6, 21, 49, 88, 142, 204, 280, 369, 468, 577 ] 358 H409 1 r 2 z rx r 2 z tx, r 2 x ty, r 2 y rxt 1 y , r 2 y rxtz, r 2 x tz [ 6, 21, 50, 93, 150, 219, 300, 393, 502, 618 ] 359 H409 1 r 2 z rx r 2 z tx, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z [ 6, 22, 47, 83, 128, 185, 250, 327, 413, 510 ] 360 H389 1 r 2 z rx it 1 x , it 1 y ,mzrxt 1 y ,mzrxtz, it 1 z [ 6, 22, 48, 85, 132, 189, 258, 336, 425, 524 ] 361 H398 1 r 2 z rx r 2 y rxt 1 x , it 1 y ,mzrxt 1 y ,mzrxtz, it 1 z [ 6, 22, 50, 89, 137, 199, 270, 351, 446, 548 ] 362 H393 1 r 2 z rx r 2 y rztx, r 2 y rzty, r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz [ 6, 22, 50, 92, 144, 206, 282, 370, 466, 576 ] 363 H391 1 r 2 z rx r 2 z rxt 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z [ 6, 22, 52, 96, 151, 219, 301, 394, 498, 617 ] 364 H394 1 r 2 z rx r 2 z rxt 1 x , r 2 y rxty, it 1 y , r 2 y rxt 1 z , r 2 x t 1 z [ 6, 22, 53, 96, 155, 224, 309, 405, 515, 637 ] 365 H396 1 r 2 z rx it 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z [ 6, 22, 56, 96, 156, 222, 306, 398, 508, 624 ] 366 H393 1 r 2 z rx r 1 y r 1 z t 1 x , ryrxty, r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz 7B [ 5, 14, 34, 68, 112, 170, 244, 326, 420, 532 ] 367*, [ 5, 15, 37, 77, 129, 189, 267, 358, 454, 569 ] 369*, [ 5, 15, 38, 77, 126, 183, 253, 336, 429, 535 ] 368*, [ 5, 15, 39, 77, 128, 193, 266, 350, 453, 570 ] 370*, [ 5, 15, 39, 81, 135, 202, 284, 380, 490, 614 ] 371*, [ 5, 15, 41, 84, 134, 200, 278, 367, 471, 591 ] 372*, [ 5, 15, 41, 86, 147, 226, 320, 418, 530, 661 ] 374*, [ 5, 15, 41, 91, 152, 230, 320, 421, 537, 670 ] 375*, [ 5, 15, 42, 90, 151, 228, 315, 412, 525, 654 ] 373*, [ 5, 16, 41, 84, 142, 211, 294, 390, 499, 624 ] 376*, [ 5, 16, 41, 84, 142, 212, 289, 382, 493, 613 ] 377*, [ 5, 16, 42, 89, 149, 214, 289, 380, 485, 599 ] 378*, [ 5, 16, 42, 89, 152, 227, 313, 413, 530, 661 ] 379*, [ 5, 16, 42, 95, 157, 229, 317, 421, 534, 663 ] 382*, [ 5, 16, 45, 95, 155, 225, 308, 403, 513, 637 ] 380*, [ 5, 16, 45, 96, 154, 217, 297, 392, 499, 618 ] 381*, [ 5, 16, 45, 98, 161, 234, 319, 418, 530, 655 ] 383*, [ 5, 17, 45, 87, 141, 207, 285, 378, 482, 597 ] 385*, [ 5, 17, 45, 88, 153, 236, 323, 420, 531, 661 ] 384*, [ 5, 17, 45, 96, 165, 242, 330, 430, 544, 674 ] 386*, [ 5, 17, 47, 93, 155, 232, 317, 419, 541, 672 ] 389*, [ 5, 17, 47, 94, 155, 229, 318, 419, 536, 665 ] 387*, [ 5, 17, 49, 96, 155, 227, 316, 415, 528, 655 ] 388*, [ 5, 18, 48, 102, 161, 233, 320, 423, 538, 669 ] 392*, [ 5, 18, 50, 100, 155, 223, 305, 399, 509, 629 ] 390*, [ 5, 18, 50, 101, 160, 225, 309, 407, 509, 632 ] 395*, [ 5, 18, 50, 102, 161, 233, 321, 423, 539, 669 ] 396*, [ 5, 18, 50, 104, 162, 231, 318, 419, 532, 661 ] 393*, [ 5, 18, 51, 103, 166, 239, 327, 429, 542, 672 ] 391*, [ 5, 18, 52, 102, 156, 222, 303, 396, 503, 622 ] 394*, [ 6, 18, 45, 87, 138, 205, 287, 374, 477, 599 ] 367 H413 1 r 2 z rx mxt 1 x ,myty, r 2 z t 1 y , r 1 x t 1 y , rxtz [ 6, 19, 48, 92, 143, 204, 281, 367, 464, 577 ] 46 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 368 H419 1 r 2 z rx mxt 1 x ,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz [ 6, 19, 48, 95, 148, 215, 305, 394, 496, 629 ] 369 H395 1 r 2 z rx mxt 1 x , r 2 x ty, it 1 y ,mzrxt 1 y ,mzrxtz [ 6, 19, 50, 90, 145, 213, 290, 377, 481, 599 ] 370 H412 1 r 2 z rx r 2 z rxt 1 x ,myty, r 2 z t 1 y , r 1 x t 1 y , rxtz [ 6, 19, 50, 97, 157, 230, 319, 420, 537, 666 ] 371 H418 1 r 2 z rx mxt 1 x , r 2 z ty, r 2 z t 1 y , r 1 x t 1 y , rxtz [ 6, 19, 52, 98, 150, 226, 302, 400, 506, 636 ] 372 H422 1 r 2 z rx r 2 z tx,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz [ 6, 19, 52, 101, 162, 241, 323, 428, 541, 673 ] 373 H407 1 r 2 z rx r 2 z rxt 1 x , r 2 z ty, r 2 z t 1 y , r 1 x t 1 y , rxtz [ 6, 19, 53, 101, 164, 242, 332, 434, 549, 680 ] 374 H422 1 r 2 z rx r 2 z tx,mxr 1 x t 1 y ,mxrxtz,mztz,mzt 1 z [ 6, 19, 55, 106, 165, 247, 341, 443, 557, 691 ] 375 H415 1 r 2 z rx it 1 x ,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz [ 6, 20, 51, 99, 160, 232, 321, 419, 534, 663 ] 376 H413 1 r 2 z rx mxt 1 x ,mxr 1 x ty,myty, r 2 z t 1 y ,mxrxt 1 z [ 6, 20, 52, 97, 154, 223, 306, 400, 508, 628 ] 377 H394 1 r 2 z rx r 2 z rxt 1 x , r 2 x ty, it 1 y ,mzrxt 1 y ,mzrxtz [ 6, 20, 52, 102, 159, 224, 305, 400, 504, 622 ] 378 H401 1 r 2 z rx mxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz [ 6, 20, 53, 104, 169, 244, 334, 439, 560, 691 ] 379 H400 1 r 2 z rx mxt 1 x , ity, it 1 y ,mzrxt 1 y ,mzrxtz [ 6, 20, 55, 104, 163, 234, 319, 417, 530, 654 ] 380 H388 1 r 2 z rx r 2 z rxt 1 x , ity, it 1 y ,mzrxt 1 y ,mzrxtz [ 6, 20, 56, 106, 162, 231, 315, 412, 521, 640 ] 381 H409 1 r 2 z rx r 2 z tx, r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz [ 6, 20, 56, 109, 169, 248, 340, 444, 559, 696 ] 382 H396 1 r 2 z rx it 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz [ 6, 20, 57, 108, 168, 243, 330, 434, 544, 671 ] 383 H409 1 r 2 z rx r 2 z tx, r 2 y rxt 1 y , r 2 y rxtz, r 2 x tz, r 2 x t 1 z [ 6, 21, 53, 97, 168, 244, 331, 432, 545, 677 ] 384 H412 1 r 2 z rx r 2 z rxt 1 x ,mxr 1 x ty,myty, r 2 z t 1 y ,mxrxt 1 z [ 6, 21, 54, 97, 154, 220, 302, 397, 500, 619 ] 385 H402 1 r 2 z rx r 2 z rxt 1 x ,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz [ 6, 21, 55, 109, 175, 249, 340, 441, 559, 690 ] 386 H395 1 r 2 z rx mxt 1 x , r 2 y rxty, r 2 x ty, it 1 y , r 2 y rxt 1 z [ 6, 21, 58, 104, 172, 243, 342, 437, 568, 687 ] 387 H414 1 r 2 z rx it 1 x , r 2 z ty, r 2 z t 1 y , r 1 x t 1 y , rxtz [ 6, 21, 59, 104, 168, 238, 332, 427, 550, 670 ] 388 H410 1 r 2 z rx r 2 y rxt 1 x , r 2 z ty, r 2 z t 1 y , r 1 x t 1 y , rxtz [ 6, 21, 59, 105, 170, 245, 338, 444, 561, 690 ] 389 H404 1 r 2 z rx r 2 y rxt 1 x ,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz [ 6, 22, 59, 106, 162, 232, 316, 414, 522, 644 ] 390 H391 1 r 2 z rx r 2 z rxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz [ 6, 22, 59, 108, 173, 244, 336, 437, 554, 684 ] 391 H394 1 r 2 z rx r 2 z rxt 1 x , r 2 y rxty, r 2 x ty, it 1 y , r 2 y rxt 1 z [ 6, 22, 59, 111, 171, 248, 339, 444, 563, 696 ] 392 H389 1 r 2 z rx it 1 x , ity, it 1 y ,mzrxt 1 y ,mzrxtz [ 6, 22, 60, 111, 168, 242, 330, 433, 548, 678 ] 393 H398 1 r 2 z rx r 2 y rxt 1 x , ity, it 1 y ,mzrxt 1 y ,mzrxtz [ 6, 22, 61, 112, 172, 245, 339, 438, 560, 687 ] 394 H393 1 r 2 z rx ryrxtx, r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz, r 1 y r 1 z t 1 z [ 6, 22, 61, 113, 172, 250, 343, 443, 561, 700 ] 395 H393 1 r 2 z rx r 2 x ryt 1 x , r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz, r 2 x ryt 1 z [ 6, 22, 62, 111, 173, 249, 341, 445, 565, 697 ] 396 H392 1 r 2 z rx r 2 y rxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz 8 [ 6, 17, 37, 70, 115, 173, 246, 329, 423, 534 ] 397*, [ 6, 17, 38, 75, 125, 183, 253, 336, 429, 535 ] 398*, [ 6, 17, 40, 82, 134, 200, 278, 367, 471, 591 ] 399*, [ 6, 18, 42, 82, 136, 204, 286, 382, 492, 616 ] 400*, [ 6, 18, 42, 83, 140, 210, 294, 390, 499, 624 ] 401*, [ 6, 18, 44, 85, 140, 208, 285, 375, 481, 600 ] 403*, [ 6, 18, 44, 86, 141, 207, 285, 378, 482, 597 ] 404*, [ 6, 18, 44, 87, 153, 236, 323, 420, 531, 661 ] 402*, [ 6, 18, 45, 90, 153, 229, 317, 416, 532, 663 ] 405*, [ 6, 18, 45, 92, 154, 232, 317, 419, 540, 672 ] 407*, [ 6, 18, 46, 92, 155, 232, 317, 419, 541, 672 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 47 Nbr. gr Hi L m X 408*, [ 6, 18, 47, 96, 160, 238, 327, 431, 549, 681 ] 406*, [ 6, 20, 47, 89, 145, 212, 289, 380, 485, 599 ] 409*, [ 6, 20, 47, 89, 145, 213, 294, 389, 496, 615 ] 410*, [ 6, 20, 48, 93, 150, 216, 297, 392, 499, 618 ] 411*, [ 6, 20, 52, 99, 164, 241, 332, 433, 553, 685 ] 412*, [ 6, 20, 52, 101, 165, 241, 332, 435, 554, 685 ] 413*, [ 6, 21, 50, 99, 158, 233, 321, 423, 538, 669 ] 421*, [ 6, 21, 51, 96, 153, 223, 304, 399, 510, 629 ] 414*, [ 6, 21, 51, 97, 153, 223, 305, 399, 509, 629 ] 415*, [ 6, 21, 51, 99, 159, 233, 321, 423, 539, 669 ] 422*, [ 6, 21, 51, 100, 159, 231, 318, 417, 530, 657 ] 416*, [ 6, 21, 52, 99, 161, 237, 326, 428, 544, 674 ] 417*, [ 6, 21, 52, 99, 161, 237, 326, 429, 547, 679 ] 418*, [ 6, 21, 54, 101, 159, 231, 317, 416, 528, 654 ] 419*, [ 6, 21, 54, 103, 162, 236, 327, 429, 542, 672 ] 420*, [ 6, 23, 54, 100, 159, 225, 309, 407, 509, 632 ] 424*, [ 6, 23, 54, 103, 163, 239, 327, 431, 547, 679 ] 423*, [ 6, 23, 56, 105, 166, 241, 330, 435, 552, 683 ] 425*, [ 6, 23, 59, 105, 160, 230, 312, 407, 515, 635 ] 426*, [ 7, 20, 46, 88, 139, 204, 287, 375, 476, 599 ] 397 H413 1 r 2 z rx mxt 1 x ,myty, r 2 z t 1 y , r 1 x t 1 y , r 2 y tz, rxtz [ 7, 20, 47, 91, 143, 204, 281, 367, 464, 577 ] 398 H419 1 r 2 z rx mxt 1 x ,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mztz [ 7, 20, 50, 98, 150, 226, 302, 400, 506, 636 ] 399 H422 1 r 2 z rx r 2 z tx,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mztz [ 7, 21, 51, 97, 157, 230, 319, 420, 537, 666 ] 400 H418 1 r 2 z rx mxt 1 x , r 2 z ty, r 1 x ty, r 2 z t 1 y , r 2 y t 1 z , rxt 1 z [ 7, 21, 51, 98, 159, 232, 321, 419, 534, 663 ] 401 H413 1 r 2 z rx mxt 1 x ,mxr 1 x ty,myty, r 2 z t 1 y ,mxrxt 1 z ,mzt 1 z [ 7, 21, 52, 97, 168, 244, 331, 432, 545, 677 ] 402 H412 1 r 2 z rx r 2 z rxt 1 x ,mxr 1 x ty,myty, r 2 z t 1 y ,mxrxt 1 z ,mzt 1 z [ 7, 21, 53, 96, 153, 221, 302, 395, 500, 620 ] 403 H412 1 r 2 z rx r 2 z rxt 1 x ,myty, r 2 z t 1 y , r 1 x t 1 y , r 2 y tz, rxtz [ 7, 21, 53, 97, 154, 220, 302, 397, 500, 619 ] 404 H402 1 r 2 z rx r 2 z rxt 1 x ,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mztz [ 7, 21, 55, 103, 166, 244, 330, 433, 549, 683 ] 405 H422 1 r 2 z rx r 2 z tx,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mztz,mzt 1 z [ 7, 21, 55, 104, 168, 246, 334, 442, 558, 694 ] 406 H407 1 r 2 z rx r 2 z rxt 1 x , r 2 z ty, r 1 x ty, r 2 z t 1 y , r 2 y t 1 z , rxt 1 z [ 7, 21, 57, 105, 170, 245, 338, 443, 562, 689 ] 407 H415 1 r 2 z rx it 1 x ,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mztz [ 7, 21, 58, 105, 170, 245, 338, 444, 561, 690 ] 408 H404 1 r 2 z rx r 2 y rxt 1 x ,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mztz [ 7, 23, 54, 100, 157, 224, 305, 400, 504, 622 ] 409 H401 1 r 2 z rx mxt 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 x t 1 z [ 7, 23, 54, 100, 157, 226, 312, 409, 515, 641 ] 410 H395 1 r 2 z rx mxt 1 x , it 1 y ,mzrxt 1 y , itz,mzrxtz, r 2 x t 1 z [ 7, 23, 56, 103, 161, 231, 315, 412, 521, 640 ] 411 H409 1 r 2 z rx r 2 z tx, r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz [ 7, 23, 61, 107, 176, 247, 346, 440, 572, 691 ] 412 H410 1 r 2 z rx r 2 y rxt 1 x , r 2 z ty, r 1 x ty, r 2 z t 1 y , r 2 y t 1 z , rxt 1 z [ 7, 23, 61, 108, 176, 247, 346, 441, 572, 691 ] 413 H414 1 r 2 z rx it 1 x , r 2 z ty, r 1 x ty, r 2 z t 1 y , r 2 y t 1 z , rxt 1 z [ 7, 24, 58, 103, 162, 232, 316, 414, 522, 643 ] 414 H394 1 r 2 z rx r 2 z rxt 1 x , it 1 y ,mzrxt 1 y , itz,mzrxtz, r 2 x t 1 z [ 7, 24, 58, 104, 162, 232, 316, 414, 522, 644 ] 415 H391 1 r 2 z rx r 2 z rxt 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 x t 1 z [ 7, 24, 59, 108, 166, 244, 329, 432, 545, 672 ] 416 H409 1 r 2 z rx r 2 z tx, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 x t 1 z [ 7, 24, 59, 108, 172, 247, 338, 441, 559, 690 ] 417 H395 1 r 2 z rx mxt 1 x , r 2 y rxty, r 2 x ty, it 1 y , r 2 y rxt 1 z , r 2 x t 1 z [ 7, 24, 59, 108, 172, 247, 338, 443, 564, 695 ] 418 H400 1 r 2 z rx mxt 1 x , it 1 y ,mzrxt 1 y , itz,mzrxtz, it 1 z 48 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X [ 7, 24, 60, 105, 166, 238, 326, 425, 540, 666 ] 419 H388 1 r 2 z rx r 2 z rxt 1 x , it 1 y ,mzrxt 1 y , itz,mzrxtz, it 1 z [ 7, 24, 60, 107, 170, 244, 336, 437, 554, 684 ] 420 H394 1 r 2 z rx r 2 z rxt 1 x , r 2 y rxty, r 2 x ty, it 1 y , r 2 y rxt 1 z , r 2 x t 1 z [ 7, 24, 60, 109, 172, 250, 340, 445, 564, 698 ] 421 H396 1 r 2 z rx it 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 x t 1 z [ 7, 24, 61, 109, 173, 249, 341, 445, 565, 697 ] 422 H392 1 r 2 z rx r 2 y rxt 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 x t 1 z [ 7, 26, 61, 110, 173, 250, 341, 446, 565, 698 ] 423 H389 1 r 2 z rx it 1 x , it 1 y ,mzrxt 1 y , itz,mzrxtz, it 1 z [ 7, 26, 61, 112, 172, 250, 343, 443, 561, 700 ] 424 H393 1 r 2 z rx r 2 y rztx, r 2 y rzty, r 2 x ryt 1 y , r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz [ 7, 26, 63, 111, 174, 249, 342, 446, 566, 697 ] 425 H398 1 r 2 z rx r 2 y rxt 1 x , it 1 y ,mzrxt 1 y , itz,mzrxtz, it 1 z [ 7, 26, 66, 110, 176, 249, 344, 444, 566, 694 ] 426 H393 1 r 2 z rx r 1 y r 1 z t 1 x , ryrxty, r 2 x ryt 1 y , r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz 9 [ 5, 14, 32, 66, 117, 179, 251, 335, 431, 539 ] 435*, [ 5, 14, 33, 68, 119, 182, 254, 336, 431, 539 ] 437*, [ 5, 14, 33, 68, 119, 183, 259, 347, 448, 563 ] 438*, [ 5, 15, 38, 77, 126, 186, 260, 342, 434, 541 ] 442*, [ 5, 15, 39, 85, 152, 236, 336, 452, 584, 732 ] 445*, [ 5, 15, 39, 87, 158, 247, 353, 474, 610, 762 ] 446*, [ 5, 15, 40, 89, 154, 226, 310, 408, 520, 646 ] 448*, [ 5, 15, 40, 89, 158, 231, 313, 413, 526, 653 ] 449*, [ 5, 15, 40, 89, 158, 236, 322, 421, 536, 667 ] 450*, [ 5, 15, 40, 91, 155, 228, 316, 415, 530, 660 ] 451*, [ 5, 15, 41, 85, 137, 198, 271, 356, 455, 566 ] 447*, [ 5, 16, 42, 86, 141, 204, 279, 367, 467, 579 ] 457*, [ 5, 16, 42, 86, 143, 211, 292, 387, 495, 615 ] 461*, [ 5, 16, 42, 86, 145, 216, 297, 391, 499, 619 ] 458*, [ 5, 16, 42, 96, 160, 229, 317, 419, 535, 665 ] 464*, [ 5, 16, 42, 96, 165, 235, 319, 425, 547, 674 ] 465*, [ 5, 16, 43, 90, 150, 220, 302, 396, 505, 629 ] 459*, [ 5, 16, 45, 97, 168, 256, 360, 480, 616, 768 ] 462*, [ 5, 16, 45, 98, 163, 237, 327, 431, 549, 681 ] 466*, [ 5, 16, 45, 99, 170, 256, 360, 480, 616, 768 ] 463*, [ 5, 16, 45, 102, 168, 236, 323, 427, 543, 671 ] 467*, [ 5, 16, 45, 102, 171, 243, 331, 435, 553, 685 ] 468*, [ 5, 16, 46, 90, 144, 216, 299, 389, 491, 607 ] 460*, [ 5, 17, 44, 83, 132, 196, 268, 350, 450, 556 ] 469*, [ 5, 17, 44, 85, 138, 203, 276, 362, 464, 575 ] 470*, [ 5, 17, 45, 93, 158, 235, 322, 422, 538, 666 ] 472*, [ 5, 17, 47, 96, 155, 225, 309, 405, 516, 641 ] 471*, [ 5, 17, 47, 99, 170, 259, 364, 484, 620, 772 ] 473*, [ 5, 17, 47, 101, 174, 264, 371, 492, 628, 780 ] 474*, [ 5, 17, 48, 101, 167, 240, 327, 430, 545, 675 ] 475*, [ 5, 17, 50, 103, 162, 232, 318, 416, 528, 654 ] 476*, [ 5, 17, 50, 112, 188, 276, 383, 505, 640, 790 ] 477*, [ 5, 17, 51, 112, 188, 278, 383, 503, 639, 790 ] 478*, [ 5, 18, 47, 90, 142, 205, 281, 369, 469, 581 ] 479*, [ 5, 18, 48, 89, 142, 213, 294, 386, 492, 609 ] 481*, [ 5, 18, 48, 90, 143, 210, 289, 379, 480, 597 ] 482*, K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 49 Nbr. gr Hi L m X [ 5, 18, 48, 90, 143, 211, 288, 377, 483, 599 ] 480*, [ 5, 18, 49, 96, 153, 224, 308, 402, 513, 638 ] 483*, [ 5, 18, 49, 100, 168, 252, 352, 468, 600, 748 ] 484*, [ 5, 18, 50, 105, 179, 268, 372, 492, 628, 780 ] 486*, [ 5, 18, 50, 107, 181, 268, 372, 492, 628, 780 ] 487*, [ 5, 18, 52, 106, 170, 245, 335, 439, 557, 689 ] 488*, [ 5, 18, 52, 110, 184, 272, 376, 496, 632, 784 ] 485*, [ 5, 18, 55, 109, 171, 244, 329, 431, 547, 675 ] 489*, [ 5, 18, 55, 121, 198, 285, 390, 509, 643, 795 ] 490*, [ 5, 18, 56, 121, 198, 285, 389, 509, 643, 795 ] 491*, [ 5, 19, 52, 101, 159, 230, 316, 414, 526, 652 ] 492*, [ 5, 19, 52, 102, 163, 237, 326, 427, 543, 674 ] 494*, [ 5, 19, 52, 102, 165, 239, 328, 432, 547, 678 ] 495*, [ 5, 19, 53, 102, 160, 236, 328, 426, 538, 670 ] 493*, [ 5, 20, 54, 104, 165, 239, 328, 431, 548, 679 ] 496*, [ 5, 20, 54, 106, 168, 241, 330, 433, 550, 681 ] 497*, [ 5, 20, 56, 111, 183, 272, 376, 496, 632, 784 ] 498*, [ 5, 20, 58, 110, 170, 244, 332, 434, 550, 680 ] 499*, [ 5, 20, 58, 111, 173, 247, 335, 439, 556, 687 ] 500*, [ 5, 20, 58, 112, 172, 245, 335, 439, 556, 687 ] 501*, [ 5, 20, 59, 118, 191, 280, 384, 504, 640, 792 ] 502*, [ 6, 18, 38, 66, 102, 146, 198, 258, 326, 402 ] 427 H779 hmz,mzrx, r 2 x i r2 z rx mxt 1 x , r 2 x ty,mzr 1 x t 1 y ,mzr 1 x tz, r 2 x t 1 z [ 6, 19, 43, 78, 123, 179, 246, 323, 411, 510 ] 428 H680 hmxi r 2 z mxtx,mxt 1 x ,myt 1 y , r 2 y tz, r 2 x t 1 z [ 6, 19, 44, 84, 141, 217, 312, 424, 552, 696 ] 429 H527 1 r 2 z mxtx,mxt 1 x ,myt 1 y , r 2 y rztz,mzt 1 z 430 H529 1 r 2 z mxtx,mxt 1 x ,myt 1 y , r 2 y rztz, r 2 y t 1 z [ 6, 20, 46, 84, 134, 196, 270, 356, 454, 564 ] 431 H749 hmy, r 2 z i i r2 y tx,mzt 1 x , r 2 z ty, r 2 z t 1 y ,mzt 1 z [ 6, 20, 47, 86, 137, 202, 278, 366, 469, 582 ] 432 H650 hmzrxi mx mxt 1 x ,myty, r 2 z t 1 y , r 1 x tz,mxr 1 x t 1 z [ 6, 20, 47, 88, 145, 220, 313, 424, 552, 696 ] 433 H527 1 r 2 z mxtx,mxt 1 x ,myt 1 y , r 2 y rztz, it 1 z 434 H529 1 r 2 z mxt 1 x ,myty,myt 1 y , r 2 y rztz, r 2 y t 1 z [ 6, 20, 47, 89, 144, 210, 288, 378, 480, 594 ] 435 H682 hmyi mx r 2 y t 1 x ,myty, t 1 y ,mztz, r 2 y t 1 z [ 6, 20, 48, 88, 140, 208, 286, 376, 484, 600 ] 436 H648 hmzrxi mx mxt 1 x , r 2 x ty, it 1 y ,mzr 1 x tz, r 2 z rxt 1 z [ 6, 20, 48, 91, 145, 208, 283, 371, 472, 585 ] 437 H378 1 mz mxtx, r 2 y t 1 x ,myty,myt 1 y , r 2 y rzt 1 z [ 6, 20, 48, 91, 149, 220, 306, 404, 514, 636 ] 438 H541 1 r 2 z r 2 z rxtx,myty,myt 1 y ,mztz, it 1 z [ 6, 20, 48, 92, 153, 232, 328, 440, 568, 712 ] 439 H604 hr 2 y i r2 z mxtx,mxt 1 x ,myt 1 y , r 2 x rztz, rzt 1 z 440 H622 hmzi r 2 z mxtx,mxt 1 x ,myt 1 y , r 2 y rztz,mxrzt 1 z [ 6, 21, 49, 90, 145, 213, 294, 389, 497, 618 ] 441 H680 hmxi r 2 z r 2 z tx,myt 1 x ,myt 1 y , r 2 y tz, r 2 x t 1 z [ 6, 21, 50, 90, 140, 203, 277, 361, 458, 566 ] 442 H359 1 mz r 2 z tx,mxt 1 x ,myty, r 2 x t 1 y ,mzt 1 z [ 6, 21, 50, 94, 155, 233, 328, 440, 568, 712 ] 443 H452 1 r 2 z r 2 z t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rztz,mzt 1 z 444 H503 1 r 2 z r 2 z t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rztz, r 2 x t 1 z [ 6, 21, 51, 97, 162, 247, 349, 467, 601, 751 ] 445 H452 1 r 2 z r 2 z t 1 x , ty, t 1 y , r 2 y rztz,mzt 1 z [ 6, 21, 51, 98, 165, 254, 359, 479, 615, 767 ] 446 H503 1 r 2 z r 2 z t 1 x , ty, t 1 y , r 2 y rztz, r 2 x t 1 z [ 6, 21, 52, 94, 147, 212, 286, 374, 475, 587 ] 447 H377 1 mz r 2 y tx, r 1 z t 1 x , rzty,myt 1 y ,mzt 1 z [ 6, 21, 54, 104, 164, 237, 324, 423, 537, 664 ] 448 H368 1 mz mxtx, r 2 y t 1 x ,myty, r 2 x t 1 y , r 2 y t 1 z [ 6, 21, 54, 104, 168, 239, 328, 429, 544, 670 ] 50 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 449 H409 1 r 2 z rx r 2 z tx, r 2 x ty, r 2 y rxt 1 y , r 2 y rxtz, r 2 x t 1 z [ 6, 21, 54, 104, 169, 245, 333, 434, 553, 685 ] 450 H422 1 r 2 z rx r 2 z tx,myty,mxr 1 x t 1 y ,mxrxtz,mzt 1 z [ 6, 21, 54, 105, 167, 243, 332, 434, 553, 683 ] 451 H361 1 mz r 2 z tx, it 1 x , r 2 z ty, it 1 y , r 2 y t 1 z [ 6, 22, 50, 94, 150, 222, 306, 406, 518, 646 ] 452 H779 hmz,mzrx, r 2 x i r2 z rx mxt 1 x , r 2 z ty,mxt 1 y ,mxr 1 x tz, r 2 y rxt 1 z [ 6, 22, 51, 96, 157, 234, 329, 440, 568, 712 ] 453 H452 1 r 2 z r 2 z t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rztz, it 1 z 454 H503 1 r 2 z r 2 z t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rztz, r 2 y t 1 z [ 6, 22, 52, 98, 161, 240, 336, 448, 576, 720 ] 455 H598 hr 2 y i r2 z r 2 z tx, r 2 x t 1 x , r 2 x t 1 y , r 2 x rztz, rzt 1 z 456 H611 hmzi r 2 z r 2 z t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rztz,mxrzt 1 z [ 6, 22, 53, 98, 153, 218, 296, 386, 488, 602 ] 457 H370 1 mz r 2 z tx,mxt 1 x ,myty, r 2 x t 1 y , r 2 x t 1 z [ 6, 22, 54, 100, 158, 228, 312, 407, 516, 639 ] 458 H359 1 mz r 2 z tx, r 2 y t 1 x ,myty, r 2 x t 1 y ,mzt 1 z [ 6, 22, 55, 101, 158, 230, 315, 411, 522, 647 ] 459 H359 1 mz itx,mxt 1 x ,myty, r 2 x t 1 y ,mzt 1 z [ 6, 22, 56, 100, 157, 226, 308, 401, 505, 624 ] 460 H382 1 mz r 2 y tx, r 1 z t 1 x , rzty,myt 1 y , r 2 y rzt 1 z [ 6, 22, 56, 102, 160, 230, 316, 414, 524, 646 ] 461 H750 hmy, r 2 z i i r2 y tx,mzt 1 x , r 2 z ty, r 2 z t 1 y , r 2 x rzt 1 z [ 6, 22, 56, 106, 174, 259, 362, 482, 618, 770 ] 462 H503 1 r 2 z r 2 z t 1 x , ty, t 1 y , r 2 y rztz, r 2 y t 1 z [ 6, 22, 56, 107, 174, 259, 362, 482, 618, 770 ] 463 H452 1 r 2 z r 2 z t 1 x , ty, t 1 y , r 2 y rztz, it 1 z [ 6, 22, 56, 110, 171, 247, 339, 443, 563, 695 ] 464 H649 hmzrxi mx r 2 z rxt 1 x , r 2 x ty, it 1 y ,mzr 1 x tz, r 2 z rxt 1 z [ 6, 22, 56, 111, 172, 245, 340, 448, 563, 690 ] 465 H652 hmzrxi mx r 2 z rxt 1 x ,myty, r 2 z t 1 y , r 1 x tz,mxr 1 x t 1 z [ 6, 22, 59, 109, 172, 248, 340, 444, 564, 696 ] 466 H682 hmyi mx r 2 y t 1 x , r 2 z ty,mxt 1 y ,mztz, r 2 y t 1 z [ 6, 22, 59, 112, 171, 245, 337, 441, 557, 686 ] 467 H541 1 r 2 z r 2 z rxtx,mxty,mxt 1 y ,mztz, it 1 z [ 6, 22, 59, 112, 175, 250, 342, 446, 566, 698 ] 468 H378 1 mz r 2 y tx, r 2 y t 1 x ,myty, r 2 x t 1 y , r 2 y rzt 1 z [ 6, 23, 53, 92, 145, 209, 282, 371, 469, 576 ] 469 H682 hmxi r 2 z r 2 y tx, r 2 y t 1 x ,myt 1 y , r 2 y tz, r 2 x t 1 z [ 6, 23, 53, 94, 150, 215, 290, 381, 482, 595 ] 470 H343 1 mz r 2 z tx,mxrzt 1 x , ity,mxrzt 1 y ,mzt 1 z [ 6, 23, 57, 105, 165, 239, 325, 424, 539, 665 ] 471 H370 1 mz itx,mxt 1 x ,myty, r 2 x t 1 y , r 2 x t 1 z [ 6, 23, 57, 105, 168, 243, 332, 435, 552, 681 ] 472 H359 1 mz itx, r 2 y t 1 x ,myty, r 2 x t 1 y ,mzt 1 z [ 6, 23, 58, 110, 180, 268, 372, 492, 628, 780 ] 473 H527 1 r 2 z mxt 1 x ,mxty,mxt 1 y , r 2 y rztz,mzt 1 z [ 6, 23, 58, 111, 182, 271, 376, 496, 632, 784 ] 474 H529 1 r 2 z mytx,myt 1 x ,myt 1 y , r 2 y rztz, r 2 y t 1 z [ 6, 23, 59, 109, 172, 246, 338, 441, 558, 689 ] 475 H377 1 mz r 2 y tx,mzr 1 z t 1 x ,mzrzty,myt 1 y ,mzt 1 z [ 6, 23, 60, 109, 167, 241, 328, 427, 541, 668 ] 476 H370 1 mz r 2 z tx, r 2 y t 1 x ,myty, r 2 x t 1 y , r 2 x t 1 z [ 6, 23, 61, 119, 190, 279, 385, 506, 639, 792 ] 477 H541 1 r 2 z rxtx, r 1 x t 1 x ,myt 1 y ,mztz, it 1 z [ 6, 23, 61, 119, 191, 280, 384, 505, 639, 792 ] 478 H541 1 r 2 z r 2 z rxtx, r 2 y rxt 1 x ,myt 1 y ,mztz, it 1 z [ 6, 24, 55, 99, 152, 218, 296, 386, 488, 602 ] 479 H683 hmxi r 2 z r 2 y tx, r 2 y t 1 x , r 2 x t 1 y , r 2 y tz, r 2 x t 1 z [ 6, 24, 56, 100, 158, 225, 304, 400, 506, 622 ] 480 H344 1 mz r 2 z tx, it 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rzt 1 z [ 6, 24, 56, 100, 159, 231, 316, 412, 519, 644 ] 481 H347 1 mz r 2 z tx,mxrzt 1 x , ity,mxrzt 1 y , r 2 x rzt 1 z [ 6, 24, 56, 100, 160, 230, 312, 407, 516, 638 ] 482 H511 1 r 2 z r 2 z rxtx, r 2 x ty, r 2 x t 1 y , r 2 x tz, r 2 y t 1 z [ 6, 24, 58, 103, 161, 234, 319, 416, 529, 653 ] 483 H350 1 mz r 2 z tx, it 1 x ,myty, r 2 x t 1 y ,mzt 1 z [ 6, 24, 58, 108, 177, 263, 365, 483, 617, 767 ] 484 H611 hmzi r 2 z r 2 z t 1 x , ty, t 1 y , r 2 y rztz,mxrzt 1 z [ 6, 24, 60, 114, 186, 274, 378, 498, 634, 786 ] 485 H598 hr 2 y i r2 z r 2 y tx, r 2 y t 1 x , r 2 x t 1 y , r 2 x rztz, rzt 1 z [ 6, 24, 61, 115, 186, 273, 376, 496, 632, 784 ] 486 H529 1 r 2 z mxt 1 x ,mxty,mxt 1 y , r 2 y rztz, r 2 y t 1 z [ 6, 24, 61, 116, 186, 273, 376, 496, 632, 784 ] 487 H527 1 r 2 z mxt 1 x ,mxty,mxt 1 y , r 2 y rztz, it 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 51 Nbr. gr Hi L m X [ 6, 24, 62, 112, 176, 252, 344, 448, 568, 700 ] 488 H370 1 mz itx, r 2 y t 1 x ,myty, r 2 x t 1 y , r 2 x t 1 z [ 6, 24, 64, 113, 177, 250, 339, 443, 560, 689 ] 489 H382 1 mz r 2 y tx,mzr 1 z t 1 x ,mzrzty,myt 1 y , r 2 y rzt 1 z [ 6, 24, 65, 126, 196, 285, 389, 508, 643, 796 ] 490 H511 1 r 2 z rxtx, r 1 x t 1 x , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z [ 6, 24, 65, 126, 197, 284, 389, 508, 643, 796 ] 491 H511 1 r 2 z r 2 z rxtx, r 2 y rxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z [ 6, 25, 59, 108, 167, 241, 328, 427, 541, 668 ] 492 H365 1 mz r 2 z tx, it 1 x ,myty, r 2 x t 1 y , r 2 x t 1 z [ 6, 25, 60, 108, 170, 248, 335, 434, 554, 686 ] 493 H372 1 mz r 2 z tx, it 1 x ,myty, r 2 x t 1 y , r 2 y t 1 z [ 6, 25, 61, 108, 170, 246, 336, 438, 557, 687 ] 494 H682 hmxi r 2 z itx, r 2 x t 1 x ,myt 1 y , r 2 y tz, r 2 x t 1 z [ 6, 25, 61, 108, 172, 247, 339, 442, 560, 692 ] 495 H343 1 mz r 2 z tx, r 2 y rzt 1 x , ity, r 2 y rzt 1 y ,mzt 1 z [ 6, 26, 61, 111, 174, 250, 342, 446, 566, 698 ] 496 H683 hmxi r 2 z itx, r 2 x t 1 x , r 2 x t 1 y , r 2 y tz, r 2 x t 1 z [ 6, 26, 62, 114, 176, 252, 344, 448, 568, 700 ] 497 H364 1 mz r 2 z tx, it 1 x ,myty, r 2 x t 1 y , it 1 z [ 6, 26, 64, 118, 190, 278, 382, 502, 638, 790 ] 498 H622 hmzi r 2 z mxt 1 x ,mxty,mxt 1 y , r 2 y rztz,mxrzt 1 z [ 6, 26, 65, 113, 176, 251, 341, 444, 563, 693 ] 499 H347 1 mz r 2 z tx, r 2 y rzt 1 x , ity, r 2 y rzt 1 y , r 2 x rzt 1 z [ 6, 26, 65, 114, 177, 251, 343, 447, 566, 697 ] 500 H511 1 r 2 z r 2 z rxtx, r 2 y ty, r 2 y t 1 y , r 2 x tz, r 2 y t 1 z [ 6, 26, 65, 115, 176, 252, 344, 448, 568, 700 ] 501 H344 1 mz r 2 z tx, it 1 x , ity, it 1 y , r 2 y rzt 1 z [ 6, 26, 66, 122, 194, 282, 386, 506, 642, 794 ] 502 H604 hr 2 y i r2 z itx,myt 1 x ,myt 1 y , r 2 x rztz, rzt 1 z 10 [ 6, 19, 43, 78, 123, 179, 246, 323, 411, 510 ] 503*, [ 6, 19, 43, 79, 127, 187, 259, 343, 439, 547 ] 504*, [ 6, 19, 43, 79, 128, 190, 262, 344, 439, 547 ] 505*, [ 6, 19, 43, 79, 128, 191, 267, 355, 456, 571 ] 506*, [ 6, 20, 47, 85, 132, 192, 265, 347, 440, 546 ] 507*, [ 6, 20, 47, 87, 139, 203, 279, 367, 467, 579 ] 510*, [ 6, 20, 48, 89, 139, 200, 274, 360, 459, 569 ] 508*, [ 6, 20, 48, 90, 143, 207, 283, 371, 471, 583 ] 509*, [ 6, 20, 48, 90, 144, 211, 290, 381, 486, 602 ] 511*, [ 6, 20, 48, 91, 147, 216, 297, 388, 491, 607 ] 512*, [ 6, 20, 49, 94, 151, 220, 303, 399, 508, 631 ] 513*, [ 6, 20, 49, 96, 161, 245, 348, 468, 604, 756 ] 514*, 515*, [ 6, 20, 50, 101, 172, 260, 364, 484, 620, 772 ] 516*, [ 6, 20, 51, 100, 160, 232, 318, 416, 528, 654 ] 517*, [ 6, 21, 49, 86, 135, 199, 270, 353, 453, 558 ] 518*, [ 6, 21, 49, 89, 141, 205, 281, 369, 469, 581 ] 520*, [ 6, 21, 49, 89, 143, 210, 289, 379, 480, 597 ] 523*, [ 6, 21, 49, 89, 143, 211, 288, 377, 483, 599 ] 522*, [ 6, 21, 50, 90, 141, 205, 279, 366, 468, 578 ] 519*, [ 6, 21, 50, 91, 143, 207, 283, 371, 471, 583 ] 521*, [ 6, 21, 50, 91, 145, 213, 292, 385, 492, 609 ] 526*, [ 6, 21, 51, 93, 144, 207, 283, 371, 471, 583 ] 524*, [ 6, 21, 51, 94, 149, 217, 295, 386, 492, 607 ] 525*, [ 6, 21, 51, 95, 151, 219, 300, 395, 503, 623 ] 536*, 537*, [ 6, 21, 51, 96, 155, 227, 312, 411, 523, 648 ] 530*, [ 6, 21, 51, 97, 157, 228, 311, 408, 519, 643 ] 531*, [ 6, 21, 52, 97, 152, 218, 295, 386, 492, 607 ] 527*, [ 6, 21, 52, 97, 153, 221, 302, 397, 504, 624 ] 52 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 529*, [ 6, 21, 52, 97, 153, 223, 306, 401, 511, 634 ] 528*, [ 6, 21, 52, 97, 153, 224, 309, 405, 516, 641 ] 532*, [ 6, 21, 52, 98, 157, 230, 315, 413, 526, 651 ] 533*, [ 6, 21, 52, 101, 164, 236, 320, 419, 531, 656 ] 534*, [ 6, 21, 52, 104, 170, 242, 327, 433, 555, 682 ] 541*, [ 6, 21, 53, 100, 161, 237, 324, 425, 542, 670 ] 538*, [ 6, 21, 53, 104, 169, 242, 327, 433, 555, 682 ] 543*, [ 6, 21, 53, 104, 173, 260, 364, 484, 620, 772 ] 535*, [ 6, 21, 54, 108, 180, 268, 372, 492, 628, 780 ] 539*, 540*, [ 6, 21, 55, 103, 160, 232, 318, 416, 528, 654 ] 542*, [ 6, 21, 56, 107, 169, 245, 335, 439, 557, 689 ] 544*, [ 6, 21, 57, 110, 174, 250, 339, 443, 561, 693 ] 545*, [ 6, 22, 52, 92, 143, 207, 283, 371, 471, 583 ] 546*, 547*, [ 6, 22, 53, 96, 152, 221, 302, 397, 504, 624 ] 548*, [ 6, 22, 54, 97, 150, 217, 295, 386, 492, 607 ] 549*, [ 6, 22, 54, 97, 152, 224, 305, 398, 510, 629 ] 550*, [ 6, 22, 54, 98, 154, 226, 310, 405, 516, 640 ] 551*, 552*, [ 6, 22, 54, 100, 158, 230, 316, 414, 526, 652 ] 556*, [ 6, 22, 54, 102, 165, 241, 331, 435, 553, 685 ] 562*, [ 6, 22, 54, 102, 169, 256, 360, 480, 616, 768 ] 553*, 554*, [ 6, 22, 55, 101, 158, 228, 307, 401, 513, 631 ] 555*, [ 6, 22, 55, 101, 159, 236, 328, 426, 538, 670 ] 557*, [ 6, 22, 55, 102, 160, 232, 318, 416, 528, 654 ] 558*, [ 6, 22, 55, 102, 160, 234, 323, 423, 538, 668 ] 559*, [ 6, 22, 55, 103, 162, 233, 318, 416, 528, 654 ] 560*, [ 6, 22, 55, 103, 164, 238, 325, 427, 543, 671 ] 561*, [ 6, 22, 55, 106, 175, 261, 364, 484, 620, 772 ] 564*, 565*, [ 6, 22, 56, 104, 161, 232, 318, 416, 528, 654 ] 563*, [ 6, 22, 56, 109, 180, 268, 372, 492, 628, 780 ] 566*, [ 6, 22, 57, 106, 168, 245, 335, 439, 557, 689 ] 568*, [ 6, 22, 57, 109, 173, 248, 335, 436, 552, 682 ] 567*, [ 6, 22, 58, 110, 173, 249, 339, 443, 561, 693 ] 569*, [ 6, 22, 59, 110, 170, 243, 329, 431, 547, 675 ] 570*, [ 6, 22, 59, 112, 174, 249, 339, 443, 561, 693 ] 571*, [ 6, 23, 52, 91, 143, 207, 283, 371, 471, 583 ] 572*, [ 6, 23, 53, 95, 151, 219, 300, 395, 503, 623 ] 574*, 575*, [ 6, 23, 55, 102, 162, 233, 318, 417, 529, 654 ] 576*, [ 6, 23, 55, 103, 164, 237, 325, 427, 543, 673 ] 586*, [ 6, 23, 56, 99, 155, 227, 307, 401, 513, 631 ] 573*, [ 6, 23, 56, 103, 163, 237, 325, 427, 543, 673 ] 590*, [ 6, 23, 56, 103, 163, 239, 330, 429, 541, 672 ] 578*, [ 6, 23, 56, 103, 164, 239, 328, 431, 548, 679 ] 585*, [ 6, 23, 56, 104, 164, 235, 320, 419, 531, 656 ] 579*, [ 6, 23, 56, 104, 164, 237, 325, 426, 541, 670 ] 580*, [ 6, 23, 56, 105, 167, 241, 330, 433, 550, 681 ] 591*, 592*, [ 6, 23, 57, 101, 156, 227, 307, 401, 513, 631 ] 577*, K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 53 Nbr. gr Hi L m X [ 6, 23, 57, 103, 160, 232, 318, 416, 528, 654 ] 581*, 582*, 583*, [ 6, 23, 57, 104, 164, 239, 328, 430, 546, 676 ] 584*, [ 6, 23, 57, 110, 181, 268, 372, 492, 628, 780 ] 597*, [ 6, 23, 58, 105, 164, 239, 327, 428, 544, 673 ] 589*, [ 6, 23, 58, 107, 169, 245, 335, 439, 557, 689 ] 600*, [ 6, 23, 58, 108, 171, 246, 335, 439, 557, 689 ] 601*, [ 6, 23, 58, 112, 177, 250, 339, 443, 561, 693 ] 602*, 603*, [ 6, 23, 58, 112, 184, 272, 376, 496, 632, 784 ] 604*, 605*, [ 6, 23, 58, 114, 186, 274, 378, 498, 634, 786 ] 587*, 588*, [ 6, 23, 59, 107, 165, 240, 330, 431, 547, 678 ] 598*, 599*, [ 6, 23, 59, 109, 169, 242, 331, 435, 551, 679 ] 606*, [ 6, 23, 59, 109, 172, 249, 339, 443, 561, 693 ] 608*, 609*, [ 6, 23, 59, 111, 174, 249, 339, 443, 561, 693 ] 610*, [ 6, 23, 59, 112, 178, 253, 341, 444, 561, 693 ] 611*, [ 6, 23, 59, 114, 186, 274, 378, 498, 634, 786 ] 593*, 594*, 595*, 596*, [ 6, 23, 60, 111, 172, 247, 336, 438, 554, 685 ] 607*, [ 6, 23, 61, 111, 170, 245, 335, 439, 556, 687 ] 612*, [ 6, 23, 62, 112, 169, 243, 332, 434, 550, 680 ] 613*, [ 6, 23, 63, 125, 197, 282, 385, 505, 642, 793 ] 614*, 615*, [ 6, 24, 57, 102, 160, 232, 318, 416, 528, 654 ] 616*, [ 6, 24, 59, 106, 165, 240, 330, 431, 547, 678 ] 617*, 618*, [ 6, 24, 59, 108, 169, 243, 332, 435, 552, 683 ] 619*, 620*, [ 6, 24, 60, 108, 169, 245, 335, 439, 557, 689 ] 621*, 622*, [ 6, 24, 60, 110, 173, 249, 339, 443, 561, 693 ] 623*, [ 6, 24, 61, 112, 174, 249, 339, 443, 561, 693 ] 624*, [ 6, 24, 61, 112, 176, 252, 341, 444, 561, 693 ] 625*, [ 6, 24, 61, 114, 178, 251, 339, 443, 561, 693 ] 626*, [ 6, 24, 61, 114, 184, 272, 376, 496, 632, 784 ] 627*, 628*, [ 6, 25, 59, 107, 169, 243, 332, 435, 552, 683 ] 629*, 630*, [ 6, 25, 60, 107, 169, 245, 335, 439, 557, 689 ] 631*, [ 6, 25, 61, 109, 170, 245, 335, 439, 556, 687 ] 633*, 634*, [ 6, 25, 61, 109, 171, 246, 335, 439, 556, 687 ] 632*, [ 6, 25, 61, 111, 174, 249, 339, 443, 561, 693 ] 635*, [ 6, 25, 62, 112, 175, 250, 339, 443, 561, 693 ] 636*, [ 6, 25, 62, 113, 175, 249, 339, 443, 561, 693 ] 637*, [ 6, 25, 63, 114, 176, 250, 339, 443, 561, 693 ] 638*, [ 6, 25, 63, 120, 192, 280, 384, 504, 640, 792 ] 639*, 640*, [ 6, 25, 64, 120, 192, 280, 384, 504, 640, 792 ] 641*, 642*, 643*, 644*, [ 6, 25, 67, 126, 197, 284, 388, 508, 644, 796 ] 645*, [ 6, 25, 67, 128, 199, 284, 388, 508, 644, 796 ] 646*, [ 6, 25, 68, 126, 196, 284, 388, 508, 644, 796 ] 647*, [ 6, 25, 68, 128, 198, 284, 388, 508, 644, 796 ] 648*, [ 6, 26, 70, 127, 196, 284, 388, 508, 644, 796 ] 649*, 650*, [ 7, 23, 50, 87, 135, 194, 263, 343, 434, 535 ] 503 H680 hmxi r 2 z mxtx,mxt 1 x ,myt 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z [ 7, 23, 52, 94, 148, 214, 292, 382, 484, 598 ] 504 H682 hmyi mx r 2 y t 1 x ,myty, t 1 y ,mztz, r 2 y t 1 z ,mzt 1 z [ 7, 23, 52, 95, 149, 212, 287, 375, 476, 589 ] 54 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 505 H378 1 mz mxtx, r 2 y t 1 x ,mxt 1 x ,myty,myt 1 y , r 2 y rzt 1 z [ 7, 23, 52, 95, 153, 224, 310, 408, 518, 640 ] 506 H541 1 r 2 z r 2 z rxtx,myty,myt 1 y ,mztz, it 1 z ,mzt 1 z [ 7, 24, 53, 92, 143, 206, 279, 364, 461, 568 ] 507 H359 1 mz r 2 z tx,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y ,mzt 1 z [ 7, 24, 54, 95, 148, 213, 288, 376, 477, 588 ] 508 H377 1 mz r 2 y tx,mxtx, r 1 z t 1 x , rzty,myt 1 y ,mzt 1 z [ 7, 24, 55, 98, 152, 218, 296, 386, 488, 602 ] 509 H680 hmxi r 2 z mxtx,mxt 1 x ,myt 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z 510 H370 1 mz r 2 z tx,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z [ 7, 24, 55, 99, 155, 223, 304, 398, 503, 621 ] 511 H353 1 mz mxtx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,mzt 1 z [ 7, 24, 56, 102, 158, 225, 307, 401, 505, 624 ] 512 H382 1 mz r 2 y tx,mxtx, r 1 z t 1 x , rzty,myt 1 y , r 2 y rzt 1 z [ 7, 24, 56, 102, 160, 231, 316, 413, 523, 648 ] 513 H359 1 mz itx, r 2 z tx,mxt 1 x ,myty, r 2 x t 1 y ,mzt 1 z [ 7, 24, 56, 104, 169, 253, 356, 476, 612, 764 ] 514 H527 1 r 2 z mxtx,mxt 1 x ,myt 1 y , r 2 y rztz, it 1 z ,mzt 1 z 515 H529 1 r 2 z mxtx,mxt 1 x ,myt 1 y , r 2 y rztz, r 2 y t 1 z , r 2 x t 1 z [ 7, 24, 56, 105, 173, 260, 364, 484, 620, 772 ] 516 H690 hmxi r 2 z mxtx,mxt 1 x ,myt 1 y , r 2 y rztz,mzr 1 z tz, r 2 y t 1 z [ 7, 24, 59, 108, 167, 241, 328, 427, 541, 668 ] 517 H368 1 mz mxtx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , r 2 y t 1 z [ 7, 25, 54, 93, 147, 210, 283, 373, 470, 577 ] 518 H682 hmxi r 2 z r 2 y tx, r 2 y t 1 x ,myt 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z [ 7, 25, 55, 96, 151, 216, 292, 383, 484, 596 ] 519 H343 1 mz r 2 z tx,mxrzt 1 x , ity, r 2 z ty,mxrzt 1 y ,mzt 1 z [ 7, 25, 56, 98, 152, 218, 296, 386, 488, 602 ] 520 H683 hmxi r 2 z r 2 y tx, r 2 y t 1 x , r 2 x t 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z 521 H682 hmyi mx r 2 y t 1 x ,myty, t 1 y , r 2 y tz, r 2 y t 1 z ,mzt 1 z [ 7, 25, 56, 100, 158, 225, 304, 400, 506, 622 ] 522 H344 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rzt 1 z [ 7, 25, 56, 100, 160, 230, 312, 407, 516, 638 ] 523 H511 1 r 2 z r 2 z rxtx, r 2 x ty, r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 7, 25, 57, 99, 152, 218, 296, 386, 488, 602 ] 524 H359 1 mz r 2 z tx,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y ,mzt 1 z [ 7, 25, 57, 101, 158, 226, 306, 401, 506, 623 ] 525 H378 1 mz mxtx, r 2 y t 1 x ,mxt 1 x , r 2 x ty,myt 1 y , r 2 y rzt 1 z [ 7, 25, 57, 102, 160, 230, 315, 412, 519, 644 ] 526 H347 1 mz r 2 z tx,mxrzt 1 x , ity, r 2 z ty,mxrzt 1 y , r 2 x rzt 1 z [ 7, 25, 58, 103, 159, 226, 306, 401, 506, 623 ] 527 H377 1 mz r 2 y tx,mxtx, r 1 z t 1 x , rzty, r 2 x t 1 y ,mzt 1 z [ 7, 25, 58, 103, 161, 233, 317, 414, 525, 649 ] 528 H359 1 mz itx,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y ,mzt 1 z [ 7, 25, 58, 104, 161, 230, 315, 410, 518, 642 ] 529 H359 1 mz r 2 z tx, r 2 y t 1 x ,myty, r 2 x t 1 y ,myt 1 y ,mzt 1 z [ 7, 25, 58, 105, 165, 238, 325, 425, 538, 665 ] 530 H680 hmxi r 2 z r 2 z tx,myt 1 x ,myt 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z [ 7, 25, 58, 106, 167, 240, 326, 426, 540, 666 ] 531 H370 1 mz itx, r 2 z tx,mxt 1 x ,myty, r 2 x t 1 y , r 2 x t 1 z [ 7, 25, 59, 105, 164, 239, 325, 424, 539, 665 ] 532 H370 1 mz itx,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z [ 7, 25, 59, 106, 166, 240, 326, 426, 540, 666 ] 533 H353 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,mzt 1 z [ 7, 25, 59, 109, 170, 242, 329, 429, 542, 669 ] 534 H368 1 mz mxtx, r 2 y t 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 y t 1 z [ 7, 25, 59, 109, 176, 261, 364, 484, 620, 772 ] 535 H690 hmxi r 2 z mxtx,mxt 1 x ,myt 1 y , r 2 y rztz,mzr 1 z tz, r 2 x t 1 z [ 7, 25, 60, 106, 164, 234, 320, 418, 528, 650 ] 536 H378 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x ,myty,myt 1 y , r 2 y rzt 1 z 537 H541 1 r 2 z r 2 z rxtx,myty,myt 1 y , itz, it 1 z ,mzt 1 z [ 7, 25, 60, 107, 170, 245, 334, 437, 554, 683 ] 538 H359 1 mz itx, r 2 z tx, r 2 y t 1 x ,myty, r 2 x t 1 y ,mzt 1 z [ 7, 25, 60, 113, 184, 272, 376, 496, 632, 784 ] 539 H496 1 r 2 z mxtx,mxt 1 x ,myt 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 540 H502 1 r 2 z mxtx,mxt 1 x ,myt 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z [ 7, 25, 61, 114, 176, 249, 344, 452, 567, 694 ] 541 H373 1 mz mxtx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , it 1 z [ 7, 25, 62, 108, 167, 241, 328, 427, 541, 668 ] 542 H370 1 mz r 2 z tx, r 2 y t 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z [ 7, 25, 62, 113, 176, 249, 344, 452, 567, 694 ] 543 H378 1 mz mxtx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , r 2 y rzt 1 z [ 7, 25, 64, 112, 176, 252, 344, 448, 568, 700 ] 544 H682 hmyi mx r 2 y t 1 x , r 2 z ty,mxt 1 y ,mztz, r 2 y t 1 z ,mzt 1 z [ 7, 25, 65, 114, 179, 254, 346, 450, 570, 702 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 55 Nbr. gr Hi L m X 545 H378 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , r 2 y rzt 1 z [ 7, 26, 57, 98, 152, 218, 296, 386, 488, 602 ] 546 H682 hmxi r 2 z r 2 y tx, r 2 y t 1 x ,myt 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z 547 H370 1 mz r 2 z tx,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z [ 7, 26, 58, 103, 161, 230, 315, 410, 518, 642 ] 548 H359 1 mz r 2 z tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,mzt 1 z [ 7, 26, 59, 102, 158, 226, 306, 401, 506, 623 ] 549 H382 1 mz r 2 y tx,mxtx, r 1 z t 1 x , rzty, r 2 x t 1 y , r 2 y rzt 1 z [ 7, 26, 59, 103, 162, 233, 315, 414, 523, 643 ] 550 H330 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 z ty, it 1 y ,mzt 1 z [ 7, 26, 59, 104, 163, 235, 320, 418, 530, 654 ] 551 H350 1 mz r 2 z tx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,mzt 1 z 552 H350 1 mz r 2 z tx, it 1 x ,myty, r 2 x t 1 y ,myt 1 y ,mzt 1 z [ 7, 26, 59, 107, 174, 259, 362, 482, 618, 770 ] 553 H452 1 r 2 z r 2 z t 1 x , ty, t 1 y , r 2 y rztz, it 1 z ,mzt 1 z 554 H503 1 r 2 z r 2 z t 1 x , ty, t 1 y , r 2 y rztz, r 2 y t 1 z , r 2 x t 1 z [ 7, 26, 60, 106, 165, 234, 316, 416, 524, 644 ] 555 H343 1 mz itx,mxrzt 1 x , ity, r 2 z ty,mxrzt 1 y ,mzt 1 z [ 7, 26, 60, 107, 167, 241, 328, 427, 541, 668 ] 556 H365 1 mz r 2 z tx, it 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z [ 7, 26, 61, 107, 170, 248, 335, 434, 554, 686 ] 557 H372 1 mz r 2 z tx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y , r 2 y t 1 z [ 7, 26, 61, 108, 167, 241, 328, 427, 541, 668 ] 558 H368 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , r 2 y t 1 z [ 7, 26, 61, 108, 170, 246, 336, 438, 557, 687 ] 559 H361 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 z ty, it 1 y , r 2 y t 1 z [ 7, 26, 61, 109, 168, 241, 328, 427, 541, 668 ] 560 H359 1 mz r 2 z tx, r 2 y t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y ,mzt 1 z [ 7, 26, 61, 109, 171, 245, 335, 438, 554, 684 ] 561 H359 1 mz itx, r 2 y t 1 x ,myty, r 2 x t 1 y ,myt 1 y ,mzt 1 z [ 7, 26, 61, 110, 174, 250, 342, 446, 566, 698 ] 562 H680 hmxi r 2 z r 2 z tx,myt 1 x ,myt 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z [ 7, 26, 62, 109, 167, 241, 328, 427, 541, 668 ] 563 H359 1 mz itx,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y ,mzt 1 z [ 7, 26, 62, 114, 183, 269, 372, 492, 628, 780 ] 564 H452 1 r 2 z r 2 z t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rztz, it 1 z ,mzt 1 z 565 H503 1 r 2 z r 2 z t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rztz, r 2 y t 1 z , r 2 x t 1 z [ 7, 26, 62, 115, 185, 272, 376, 496, 632, 784 ] 566 H690 hmxi r 2 z r 2 z tx,myt 1 x ,myt 1 y , r 2 y rztz,mzr 1 z tz, r 2 y t 1 z [ 7, 26, 63, 113, 176, 251, 341, 444, 562, 692 ] 567 H377 1 mz r 2 y tx,mxtx,mzr 1 z t 1 x ,mzrzty,myt 1 y ,mzt 1 z [ 7, 26, 64, 111, 176, 252, 344, 448, 568, 700 ] 568 H370 1 mz itx, r 2 y t 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z [ 7, 26, 65, 114, 178, 254, 346, 450, 570, 702 ] 569 H370 1 mz itx, r 2 z tx, r 2 y t 1 x ,myty, r 2 x t 1 y , r 2 x t 1 z [ 7, 26, 66, 113, 176, 250, 339, 443, 560, 689 ] 570 H382 1 mz r 2 y tx,mxtx,mzr 1 z t 1 x ,mzrzty,myt 1 y , r 2 y rzt 1 z [ 7, 26, 66, 115, 178, 254, 346, 450, 570, 702 ] 571 H368 1 mz mxtx, r 2 y t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 y t 1 z [ 7, 27, 56, 98, 152, 218, 296, 386, 488, 602 ] 572 H683 hmxi r 2 z r 2 y tx, r 2 y t 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z [ 7, 27, 60, 104, 164, 234, 316, 416, 524, 644 ] 573 H344 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , ity, r 2 z t 1 y , r 2 y rzt 1 z [ 7, 27, 60, 106, 164, 234, 320, 418, 528, 650 ] 574 H344 1 mz itx, it 1 x , r 2 z t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rzt 1 z 575 H511 1 r 2 z r 2 z rxtx, r 2 x ty, r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 7, 27, 60, 109, 169, 242, 329, 429, 542, 669 ] 576 H365 1 mz r 2 z tx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y , r 2 x t 1 z [ 7, 27, 61, 105, 164, 234, 316, 416, 524, 644 ] 577 H347 1 mz itx,mxrzt 1 x , ity, r 2 z ty,mxrzt 1 y , r 2 x rzt 1 z [ 7, 27, 61, 109, 172, 249, 336, 436, 555, 687 ] 578 H372 1 mz r 2 z tx, it 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 y t 1 z [ 7, 27, 61, 110, 169, 242, 329, 429, 542, 669 ] 579 H370 1 mz r 2 z tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , r 2 x t 1 z [ 7, 27, 61, 110, 172, 247, 337, 440, 558, 688 ] 580 H361 1 mz r 2 z tx, it 1 x , r 2 z ty, it 1 y , r 2 z t 1 y , r 2 y t 1 z [ 7, 27, 62, 108, 167, 241, 328, 427, 541, 668 ] 581 H350 1 mz r 2 z tx, it 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y ,mzt 1 z 582 H370 1 mz itx,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z 583 H370 1 mz r 2 z tx, r 2 y t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z [ 7, 27, 62, 109, 172, 247, 337, 440, 558, 688 ] 584 H682 hmxi r 2 z itx, r 2 x t 1 x ,myt 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z [ 7, 27, 62, 110, 174, 250, 342, 446, 566, 698 ] 585 H683 hmxi r 2 z itx, r 2 x t 1 x , r 2 x t 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z 56 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X [ 7, 27, 62, 112, 175, 251, 343, 447, 567, 699 ] 586 H332 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 z ty, it 1 y , it 1 z [ 7, 27, 62, 117, 186, 276, 378, 500, 634, 788 ] 587 H453 1 r 2 z it 1 x ,mzty,mzt 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 588 H539 1 r 2 z mztx,mzt 1 x , r 2 x t 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z [ 7, 27, 63, 109, 171, 246, 336, 438, 555, 685 ] 589 H359 1 mz itx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,mzt 1 z [ 7, 27, 63, 111, 175, 251, 343, 447, 567, 699 ] 590 H344 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 z ty, it 1 y , r 2 y rzt 1 z [ 7, 27, 63, 113, 176, 252, 344, 448, 568, 700 ] 591 H364 1 mz r 2 z tx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y , it 1 z 592 H364 1 mz r 2 z tx, it 1 x ,myty, r 2 x t 1 y ,myt 1 y , it 1 z [ 7, 27, 63, 116, 187, 275, 379, 499, 635, 787 ] 593 H539 1 r 2 z it 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z 594 H438 1 r 2 z r 2 z t 1 x , ty, t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 595 H495 1 r 2 z r 2 z t 1 x , ty, t 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z 596 H505 1 r 2 z r 2 y tx, r 2 y t 1 x , r 2 x t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z [ 7, 27, 63, 117, 186, 273, 376, 496, 632, 784 ] 597 H690 hmxi r 2 z r 2 z tx,myt 1 x ,myt 1 y , r 2 y rztz,mzr 1 z tz, r 2 x t 1 z [ 7, 27, 64, 110, 172, 248, 338, 440, 559, 689 ] 598 H330 1 mz r 2 z tx, it 1 x , ity, it 1 y , r 2 z t 1 y ,mzt 1 z 599 H350 1 mz itx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,mzt 1 z [ 7, 27, 64, 112, 176, 252, 344, 448, 568, 700 ] 600 H682 hmyi mx r 2 y t 1 x , r 2 z ty,mxt 1 y , r 2 y tz, r 2 y t 1 z ,mzt 1 z [ 7, 27, 64, 113, 177, 252, 344, 448, 568, 700 ] 601 H359 1 mz itx, r 2 y t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y ,mzt 1 z [ 7, 27, 64, 117, 179, 254, 346, 450, 570, 702 ] 602 H378 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y , r 2 y rzt 1 z 603 H541 1 r 2 z r 2 z rxtx,mxty,mxt 1 y , itz, it 1 z ,mzt 1 z [ 7, 27, 64, 119, 190, 278, 382, 502, 638, 790 ] 604 H438 1 r 2 z r 2 z t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 605 H495 1 r 2 z r 2 z t 1 x , r 2 z ty, r 2 z t 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z [ 7, 27, 65, 113, 174, 249, 341, 445, 561, 690 ] 606 H541 1 r 2 z r 2 z rxtx,mxty,mxt 1 y ,mztz, it 1 z ,mzt 1 z [ 7, 27, 65, 113, 176, 252, 342, 445, 564, 695 ] 607 H343 1 mz r 2 z tx, r 2 y rzt 1 x , ity, r 2 z ty, r 2 y rzt 1 y ,mzt 1 z [ 7, 27, 65, 113, 178, 254, 346, 450, 570, 702 ] 608 H378 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x , r 2 x ty,myt 1 y , r 2 y rzt 1 z 609 H378 1 mz r 2 y tx, r 2 y t 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 y rzt 1 z [ 7, 27, 65, 115, 178, 254, 346, 450, 570, 702 ] 610 H373 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , it 1 z [ 7, 27, 65, 116, 181, 255, 347, 450, 570, 702 ] 611 H377 1 mz r 2 y tx,mxtx,mzr 1 z t 1 x ,mzrzty, r 2 x t 1 y ,mzt 1 z [ 7, 27, 67, 113, 176, 252, 344, 448, 568, 700 ] 612 H344 1 mz r 2 z tx, it 1 x , ity, it 1 y , r 2 z t 1 y , r 2 y rzt 1 z [ 7, 27, 68, 113, 175, 251, 341, 444, 563, 693 ] 613 H347 1 mz r 2 z tx, r 2 y rzt 1 x , ity, r 2 z ty, r 2 y rzt 1 y , r 2 x rzt 1 z [ 7, 27, 68, 126, 194, 281, 385, 506, 641, 793 ] 614 H541 1 r 2 z r 2 z rxtx, r 2 y rxt 1 x ,myt 1 y ,mztz, it 1 z ,mzt 1 z 615 H541 1 r 2 z rxtx, r 1 x t 1 x ,myt 1 y ,mztz, it 1 z ,mzt 1 z [ 7, 28, 61, 108, 167, 241, 328, 427, 541, 668 ] 616 H365 1 mz r 2 z tx, it 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z [ 7, 28, 63, 110, 172, 248, 338, 440, 559, 689 ] 617 H361 1 mz itx, it 1 x , r 2 z t 1 x , r 2 z ty, it 1 y , r 2 y t 1 z 618 H372 1 mz itx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y , r 2 y t 1 z [ 7, 28, 64, 113, 176, 252, 344, 448, 568, 700 ] 619 H361 1 mz r 2 z tx, it 1 x , ity, it 1 y , r 2 z t 1 y , r 2 y t 1 z 620 H365 1 mz itx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y , r 2 x t 1 z [ 7, 28, 65, 112, 176, 252, 344, 448, 568, 700 ] 621 H682 hmxi r 2 z itx, r 2 x t 1 x ,myt 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z 622 H370 1 mz itx, r 2 y t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z [ 7, 28, 65, 114, 178, 254, 346, 450, 570, 702 ] 623 H370 1 mz itx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , r 2 x t 1 z [ 7, 28, 66, 115, 178, 254, 346, 450, 570, 702 ] 624 H372 1 mz r 2 z tx, it 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 y t 1 z [ 7, 28, 66, 115, 180, 255, 347, 450, 570, 702 ] 625 H343 1 mz itx, r 2 y rzt 1 x , ity, r 2 z ty, r 2 y rzt 1 y ,mzt 1 z [ 7, 28, 66, 117, 180, 254, 346, 450, 570, 702 ] 626 H382 1 mz r 2 y tx,mxtx,mzr 1 z t 1 x ,mzrzty, r 2 x t 1 y , r 2 y rzt 1 z [ 7, 28, 66, 118, 188, 275, 378, 498, 634, 786 ] 627 H527 1 r 2 z mxt 1 x ,mxty,mxt 1 y , r 2 y rztz, it 1 z ,mzt 1 z 628 H529 1 r 2 z mxt 1 x ,mxty,mxt 1 y , r 2 y rztz, r 2 y t 1 z , r 2 x t 1 z [ 7, 29, 63, 113, 176, 252, 344, 448, 568, 700 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 57 Nbr. gr Hi L m X 629 H332 1 mz r 2 z tx, it 1 x , ity, it 1 y , r 2 z t 1 y , it 1 z 630 H364 1 mz itx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y , it 1 z [ 7, 29, 64, 112, 176, 252, 344, 448, 568, 700 ] 631 H683 hmxi r 2 z itx, r 2 x t 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z [ 7, 29, 65, 113, 176, 251, 343, 447, 566, 697 ] 632 H511 1 r 2 z r 2 z rxtx, r 2 y ty, r 2 y t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 7, 29, 65, 113, 176, 252, 344, 448, 568, 700 ] 633 H344 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , ity, it 1 y , r 2 y rzt 1 z 634 H344 1 mz r 2 z tx, it 1 x , ity, r 2 z ty, it 1 y , r 2 y rzt 1 z [ 7, 29, 65, 115, 178, 254, 346, 450, 570, 702 ] 635 H364 1 mz r 2 z tx, it 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , it 1 z [ 7, 29, 66, 115, 179, 254, 346, 450, 570, 702 ] 636 H511 1 r 2 z r 2 z rxtx, r 2 y ty, r 2 y t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 7, 29, 66, 116, 178, 254, 346, 450, 570, 702 ] 637 H344 1 mz itx, it 1 x , r 2 z t 1 x , ity, it 1 y , r 2 y rzt 1 z [ 7, 29, 67, 116, 179, 254, 346, 450, 570, 702 ] 638 H347 1 mz itx, r 2 y rzt 1 x , ity, r 2 z ty, r 2 y rzt 1 y , r 2 x rzt 1 z [ 7, 29, 67, 124, 193, 283, 385, 507, 641, 795 ] 639 H539 1 r 2 z it 1 x , r 2 y ty, r 2 y t 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z 640 H505 1 r 2 z r 2 y t 1 x , r 2 y ty, r 2 y t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z [ 7, 29, 68, 123, 194, 282, 386, 506, 642, 794 ] 641 H453 1 r 2 z it 1 x , ity, it 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 642 H539 1 r 2 z itx, it 1 x , r 2 x t 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z 643 H496 1 r 2 z mxt 1 x ,mxty,mxt 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 644 H502 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z [ 7, 29, 71, 126, 196, 284, 388, 508, 644, 796 ] 645 H511 1 r 2 z rxtx, r 1 x t 1 x , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 7, 29, 71, 128, 196, 284, 388, 508, 644, 796 ] 646 H511 1 r 2 z rxtx, r 1 x t 1 x , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 7, 29, 72, 125, 196, 284, 388, 508, 644, 796 ] 647 H511 1 r 2 z r 2 z rxtx, r 2 y rxt 1 x , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 7, 29, 72, 127, 196, 284, 388, 508, 644, 796 ] 648 H511 1 r 2 z r 2 z rxtx, r 2 y rxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 7, 30, 73, 125, 196, 284, 388, 508, 644, 796 ] 649 H541 1 r 2 z r 2 z rxtx, r 2 y rxt 1 x ,myt 1 y , itz, it 1 z ,mzt 1 z 650 H541 1 r 2 z rxtx, r 1 x t 1 x ,myt 1 y , itz, it 1 z ,mzt 1 z 11 [ 6, 18, 38, 66, 102, 146, 198, 258, 326, 402 ] 651*, 652*, 653*, 654*, [ 6, 19, 43, 78, 122, 175, 239, 314, 398, 491 ] 655*, [ 6, 19, 43, 78, 123, 179, 246, 323, 411, 510 ] 658*, [ 6, 19, 43, 79, 127, 187, 259, 343, 439, 547 ] 656*, [ 6, 19, 43, 79, 128, 191, 267, 355, 455, 567 ] 657*, [ 6, 19, 46, 90, 148, 220, 306, 404, 516, 642 ] 659*, [ 6, 20, 46, 82, 128, 186, 254, 332, 422, 522 ] 660*, [ 6, 20, 46, 84, 134, 194, 263, 343, 434, 535 ] 662*, [ 6, 20, 46, 84, 134, 196, 270, 356, 454, 564 ] 667*, [ 6, 20, 46, 84, 134, 196, 272, 362, 465, 582 ] 663*, [ 6, 20, 46, 84, 135, 199, 275, 363, 463, 575 ] 661*, [ 6, 20, 47, 87, 139, 203, 279, 367, 467, 579 ] 664*, 665*, [ 6, 20, 47, 87, 140, 207, 287, 380, 487, 607 ] 668*, [ 6, 20, 48, 89, 140, 204, 281, 368, 468, 581 ] 666*, [ 6, 20, 48, 90, 143, 207, 283, 370, 469, 581 ] 670*, [ 6, 20, 48, 90, 144, 211, 290, 380, 484, 602 ] 669*, [ 6, 20, 48, 91, 147, 215, 296, 391, 499, 619 ] 674*, 675*, [ 6, 20, 48, 94, 160, 244, 344, 460, 592, 740 ] 671*, [ 6, 20, 49, 96, 160, 240, 336, 448, 576, 720 ] 672*, 673*, [ 6, 20, 51, 98, 156, 228, 314, 412, 524, 650 ] 676*, [ 6, 20, 51, 99, 161, 237, 327, 431, 549, 681 ] 677*, [ 6, 21, 49, 89, 141, 205, 281, 369, 469, 581 ] 678*, [ 6, 21, 49, 91, 147, 217, 301, 399, 511, 637 ] 681*, 58 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X [ 6, 21, 50, 91, 143, 207, 283, 371, 471, 583 ] 679*, [ 6, 21, 50, 94, 150, 217, 298, 392, 499, 619 ] 687*, [ 6, 21, 50, 95, 153, 221, 303, 399, 507, 627 ] 688*, [ 6, 21, 51, 93, 145, 211, 289, 376, 477, 591 ] 683*, [ 6, 21, 51, 94, 148, 216, 298, 391, 497, 617 ] 684*, [ 6, 21, 51, 95, 151, 221, 306, 404, 516, 642 ] 680*, [ 6, 21, 51, 95, 151, 222, 306, 401, 512, 637 ] 686*, [ 6, 21, 51, 96, 155, 227, 312, 410, 521, 645 ] 692*, [ 6, 21, 51, 97, 156, 228, 314, 412, 524, 650 ] 689*, [ 6, 21, 52, 95, 148, 217, 298, 389, 496, 614 ] 682*, [ 6, 21, 52, 97, 155, 230, 319, 418, 532, 661 ] 685*, [ 6, 21, 52, 98, 153, 218, 296, 386, 488, 602 ] 694*, [ 6, 21, 52, 98, 156, 228, 314, 412, 524, 650 ] 690*, 691*, [ 6, 21, 52, 100, 162, 239, 329, 428, 543, 679 ] 695*, [ 6, 21, 53, 100, 161, 237, 327, 431, 549, 681 ] 693*, [ 6, 21, 53, 104, 174, 263, 368, 488, 624, 776 ] 696*, [ 6, 21, 53, 106, 179, 268, 373, 493, 629, 781 ] 697*, [ 6, 21, 56, 110, 181, 269, 373, 493, 629, 781 ] 698*, [ 6, 21, 56, 115, 188, 275, 383, 506, 640, 789 ] 699*, [ 6, 21, 57, 110, 171, 243, 331, 435, 551, 679 ] 700*, [ 6, 22, 51, 91, 143, 207, 283, 371, 471, 583 ] 701*, [ 6, 22, 52, 95, 151, 219, 300, 395, 503, 623 ] 705*, 706*, [ 6, 22, 52, 96, 151, 216, 294, 384, 486, 600 ] 704*, [ 6, 22, 52, 97, 154, 222, 304, 399, 507, 628 ] 703*, [ 6, 22, 53, 97, 154, 224, 307, 403, 512, 634 ] 702*, [ 6, 22, 53, 98, 157, 230, 317, 418, 533, 662 ] 708*, [ 6, 22, 53, 100, 159, 229, 314, 413, 525, 650 ] 709*, [ 6, 22, 53, 100, 163, 240, 331, 435, 553, 685 ] 711*, [ 6, 22, 53, 100, 164, 244, 340, 452, 580, 724 ] 712*, 713*, [ 6, 22, 53, 101, 163, 237, 326, 429, 546, 677 ] 710*, [ 6, 22, 54, 100, 158, 228, 312, 410, 520, 642 ] 717*, [ 6, 22, 54, 100, 158, 230, 316, 414, 526, 652 ] 707*, [ 6, 22, 54, 100, 159, 231, 317, 417, 530, 656 ] 716*, [ 6, 22, 54, 100, 161, 237, 327, 431, 549, 681 ] 718*, [ 6, 22, 54, 102, 165, 241, 331, 435, 553, 685 ] 719*, [ 6, 22, 54, 104, 172, 256, 356, 472, 604, 752 ] 720*, [ 6, 22, 55, 102, 160, 232, 318, 416, 528, 654 ] 714*, 715*, [ 6, 22, 55, 103, 165, 241, 331, 435, 553, 685 ] 725*, [ 6, 22, 56, 104, 163, 238, 328, 429, 545, 676 ] 723*, 724*, [ 6, 22, 56, 109, 179, 265, 367, 485, 619, 769 ] 722*, [ 6, 22, 56, 109, 180, 268, 372, 492, 628, 780 ] 721*, [ 6, 22, 57, 109, 172, 247, 337, 441, 559, 691 ] 733*, 734*, [ 6, 22, 57, 111, 183, 272, 376, 496, 632, 784 ] 735*, [ 6, 22, 57, 111, 183, 273, 378, 498, 634, 786 ] 731*, [ 6, 22, 57, 113, 186, 274, 378, 498, 634, 786 ] 726*, [ 6, 22, 58, 110, 173, 247, 335, 439, 556, 687 ] 737*, [ 6, 22, 58, 110, 177, 264, 368, 488, 624, 776 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 59 Nbr. gr Hi L m X 732*, [ 6, 22, 58, 113, 186, 274, 378, 498, 634, 786 ] 727*, 728*, [ 6, 22, 58, 114, 186, 274, 378, 498, 634, 786 ] 729*, 730*, [ 6, 22, 58, 116, 190, 278, 382, 502, 638, 790 ] 736*, [ 6, 22, 59, 118, 192, 280, 384, 504, 640, 792 ] 738*, [ 6, 22, 59, 122, 198, 284, 389, 510, 644, 794 ] 739*, [ 6, 23, 56, 102, 160, 232, 318, 416, 528, 654 ] 740*, [ 6, 23, 56, 105, 167, 241, 330, 433, 550, 681 ] 741*, 742*, [ 6, 23, 57, 105, 167, 243, 333, 437, 555, 687 ] 745*, [ 6, 23, 58, 106, 165, 240, 330, 431, 547, 678 ] 743*, 744*, [ 6, 23, 58, 107, 169, 245, 335, 439, 557, 689 ] 747*, [ 6, 23, 59, 109, 169, 242, 331, 435, 551, 679 ] 749*, [ 6, 23, 59, 111, 174, 249, 339, 443, 561, 693 ] 750*, 751*, [ 6, 23, 59, 113, 184, 272, 376, 496, 632, 784 ] 746*, 748*, [ 6, 23, 59, 113, 185, 274, 378, 498, 634, 786 ] 752*, [ 6, 23, 59, 116, 190, 278, 382, 502, 638, 790 ] 754*, 755*, [ 6, 23, 60, 117, 190, 279, 382, 502, 640, 791 ] 753*, [ 6, 23, 61, 116, 185, 272, 376, 496, 632, 784 ] 756*, [ 6, 23, 62, 117, 186, 274, 378, 498, 634, 786 ] 757*, [ 6, 24, 58, 104, 163, 235, 321, 421, 534, 660 ] 758*, [ 6, 24, 58, 104, 164, 238, 326, 428, 543, 671 ] 759*, [ 6, 24, 58, 107, 169, 243, 332, 435, 552, 683 ] 760*, 761*, [ 6, 24, 59, 107, 169, 245, 335, 439, 557, 689 ] 762*, 763*, [ 6, 24, 59, 108, 171, 248, 339, 443, 561, 693 ] 764*, [ 6, 24, 60, 109, 171, 246, 335, 439, 556, 687 ] 765*, [ 6, 24, 61, 113, 181, 268, 372, 492, 628, 780 ] 766*, [ 6, 24, 61, 115, 185, 272, 376, 496, 632, 784 ] 768*, [ 6, 24, 61, 115, 186, 274, 378, 498, 634, 786 ] 767*, [ 6, 24, 61, 116, 188, 276, 380, 500, 636, 788 ] 770*, [ 6, 24, 62, 118, 190, 278, 382, 502, 638, 790 ] 769*, [ 6, 24, 62, 119, 192, 280, 384, 504, 640, 792 ] 772*, [ 6, 24, 63, 119, 192, 280, 384, 504, 640, 792 ] 773*, 774*, [ 6, 24, 63, 120, 192, 280, 384, 504, 640, 792 ] 771*, 775*, 776*, [ 6, 24, 63, 124, 198, 284, 387, 508, 645, 796 ] 777*, [ 6, 25, 64, 118, 187, 274, 378, 498, 634, 786 ] 778*, [ 6, 25, 65, 122, 194, 282, 386, 506, 642, 794 ] 779*, 780*, [ 6, 26, 65, 120, 192, 280, 384, 504, 640, 792 ] 781*, [ 6, 27, 67, 122, 194, 282, 386, 506, 642, 794 ] 782*, [ 7, 22, 47, 82, 127, 182, 247, 322, 407, 502 ] 651 H610 hmzi r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y ,mztz, t 1 z 652 H615 hmzi r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y ,mztz, t 1 z [ 7, 22, 48, 84, 130, 186, 253, 330, 417, 514 ] 653 H305 1 i tx, it 1 x , t 1 x , ty, t 1 y , r 2 y rzt 1 z 654 H504 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 x ryt 1 y , tz, t 1 z [ 7, 23, 51, 90, 139, 199, 271, 354, 447, 551 ] 655 H461 1 r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y , r 2 y tz,mzt 1 z [ 7, 23, 51, 91, 143, 207, 283, 371, 471, 583 ] 656 H617 hmzi r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y ,mztz, t 1 z [ 7, 23, 51, 91, 143, 207, 284, 374, 476, 590 ] 657 H460 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y tz,mzt 1 z [ 7, 23, 52, 93, 145, 210, 287, 375, 476, 589 ] 658 H461 1 r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y , r 2 x tz,mzt 1 z 60 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X [ 7, 23, 54, 99, 157, 231, 318, 417, 531, 658 ] 659 H460 1 r 2 z r 2 z t 1 x , ty, r 2 z t 1 y , t 1 y , r 2 y tz,mzt 1 z [ 7, 24, 53, 92, 143, 206, 279, 364, 461, 568 ] 660 H616 hmzi r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y , r 2 y tz,mxt 1 z [ 7, 24, 53, 94, 147, 212, 290, 380, 482, 596 ] 661 H614 hmzi r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y tz,mxt 1 z [ 7, 24, 54, 95, 149, 216, 295, 386, 491, 607 ] 662 H323 1 i mztx, r 2 z t 1 x ,mzt 1 x , r 2 x ryt 1 y ,mxtz,mxt 1 z [ 7, 24, 54, 95, 149, 217, 298, 393, 502, 624 ] 663 H540 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 x ryt 1 y ,mxtz,mxt 1 z [ 7, 24, 54, 96, 150, 216, 294, 384, 486, 600 ] 664 H615 hmzi r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y ,mztz, t 1 z 665 H483 1 r 2 z mytx,mxt 1 x ,myt 1 x ,myt 1 y , r 2 y tz,mzt 1 z [ 7, 24, 55, 97, 150, 217, 295, 384, 487, 601 ] 666 H431 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y ,mztz, it 1 z [ 7, 24, 55, 98, 155, 224, 307, 402, 511, 632 ] 667 H616 hmzi r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y , r 2 x tz,myt 1 z [ 7, 24, 55, 99, 156, 227, 311, 408, 519, 643 ] 668 H483 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz,mzt 1 z [ 7, 24, 56, 101, 157, 226, 307, 400, 508, 628 ] 669 H468 1 r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y ,mztz, it 1 z [ 7, 24, 56, 101, 157, 226, 308, 401, 508, 628 ] 670 H469 1 r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y , r 2 x tz, r 2 y t 1 z [ 7, 24, 56, 104, 170, 255, 357, 475, 609, 759 ] 671 H452 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 x rztz,mzt 1 z [ 7, 24, 56, 105, 172, 256, 356, 472, 604, 752 ] 672 H326 1 i mzrxtx, r 2 z rxt 1 x ,mzrxt 1 x ,myty,myt 1 y ,mzt 1 z 673 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,myt 1 y ,mztz,mzt 1 z [ 7, 24, 58, 104, 162, 232, 318, 416, 526, 648 ] 674 H326 1 i r 2 z rxt 1 x ,myty,myt 1 y , r 2 z tz, r 2 z t 1 z ,mzt 1 z 675 H541 1 r 2 z r 2 z rxtx,mxty,mxt 1 y ,myt 1 y ,mztz,mzt 1 z [ 7, 24, 58, 104, 163, 237, 324, 423, 537, 664 ] 676 H425 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 x tz, r 2 y t 1 z [ 7, 24, 59, 106, 170, 246, 338, 442, 562, 694 ] 677 H614 hmzi r 2 z r 2 z t 1 x , ty, r 2 z t 1 y , t 1 y , r 2 y tz,mxt 1 z [ 7, 25, 55, 97, 151, 217, 295, 385, 487, 601 ] 678 H618 hmzi r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz,myt 1 z [ 7, 25, 56, 98, 152, 218, 296, 386, 488, 602 ] 679 H461 1 r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz,mzt 1 z [ 7, 25, 57, 102, 159, 231, 317, 415, 529, 656 ] 680 H460 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y tz, it 1 z [ 7, 25, 57, 103, 163, 237, 325, 427, 543, 673 ] 681 H618 hmzi r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz,mxt 1 z [ 7, 25, 58, 101, 158, 228, 309, 404, 512, 630 ] 682 H433 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 z t 1 y ,mztz, it 1 z [ 7, 25, 58, 102, 160, 231, 314, 409, 519, 640 ] 683 H485 1 r 2 z mztx, it 1 x ,mzt 1 x ,myt 1 y , r 2 x tz, r 2 y t 1 z [ 7, 25, 58, 103, 159, 230, 314, 408, 517, 639 ] 684 H461 1 r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y , r 2 y tz, it 1 z [ 7, 25, 58, 103, 163, 238, 327, 428, 545, 673 ] 685 H462 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 z t 1 y , r 2 x tz, r 2 y t 1 z [ 7, 25, 58, 104, 164, 237, 324, 423, 537, 662 ] 686 H475 1 r 2 z mztx, it 1 x ,mzt 1 x ,myt 1 y ,mztz, it 1 z [ 7, 25, 58, 105, 163, 234, 319, 416, 527, 650 ] 687 H433 1 r 2 z it 1 x , ty, r 2 z t 1 y , t 1 y ,mztz, it 1 z [ 7, 25, 58, 106, 165, 235, 319, 417, 529, 651 ] 688 H466 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 x t 1 y ,mztz, it 1 z [ 7, 25, 58, 106, 165, 239, 326, 425, 539, 666 ] 689 H482 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz, r 2 y t 1 z [ 7, 25, 59, 106, 165, 239, 326, 425, 539, 666 ] 690 H461 1 r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz,mzt 1 z 691 H481 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y ,mztz, it 1 z [ 7, 25, 59, 106, 167, 240, 327, 426, 540, 666 ] 692 H461 1 r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y , r 2 x tz, it 1 z [ 7, 25, 60, 106, 169, 244, 336, 440, 560, 692 ] 693 H460 1 r 2 z r 2 z t 1 x , ty, r 2 z t 1 y , t 1 y , r 2 y tz, it 1 z [ 7, 25, 60, 107, 165, 236, 321, 418, 529, 652 ] 694 H427 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z [ 7, 25, 60, 108, 171, 249, 340, 441, 563, 699 ] 695 H462 1 r 2 z it 1 x , ty, r 2 z t 1 y , t 1 y , r 2 x tz, r 2 y t 1 z [ 7, 25, 60, 112, 182, 270, 374, 494, 630, 782 ] 696 H527 1 r 2 z mxt 1 x ,mxty,myty,mxt 1 y , r 2 y rztz,mzt 1 z [ 7, 25, 60, 112, 182, 270, 375, 494, 631, 782 ] 697 H503 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 x rztz, r 2 y t 1 z [ 7, 25, 63, 114, 185, 271, 375, 494, 631, 782 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 61 Nbr. gr Hi L m X 698 H503 1 r 2 z r 2 z t 1 x , ty, r 2 z t 1 y , t 1 y , r 2 x rztz, r 2 y t 1 z [ 7, 25, 63, 121, 190, 278, 386, 506, 639, 791 ] 699 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,myty,mztz, it 1 z [ 7, 25, 65, 114, 175, 249, 341, 445, 561, 690 ] 700 H541 1 r 2 z r 2 z rxtx,mxty,mxt 1 y ,myt 1 y ,mztz, it 1 z [ 7, 26, 56, 98, 152, 218, 296, 386, 488, 602 ] 701 H616 hmzi r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz,myt 1 z [ 7, 26, 59, 104, 163, 234, 319, 416, 527, 650 ] 702 H610 hmzi r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , itz, r 2 z t 1 z [ 7, 26, 59, 106, 164, 235, 320, 417, 528, 651 ] 703 H434 1 r 2 z mztx, it 1 x ,mzt 1 x , it 1 y ,mztz, it 1 z [ 7, 26, 59, 106, 165, 236, 321, 418, 529, 652 ] 704 H474 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z [ 7, 26, 60, 106, 164, 234, 320, 418, 528, 650 ] 705 H308 1 i mztx, r 2 z t 1 x ,mzt 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rzt 1 z 706 H511 1 r 2 z r 2 z rxtx, r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 7, 26, 60, 107, 166, 240, 327, 426, 540, 667 ] 707 H483 1 r 2 z mytx,mxt 1 x ,myt 1 x ,myt 1 y , r 2 y tz, it 1 z [ 7, 26, 60, 107, 169, 244, 334, 437, 555, 686 ] 708 H308 1 i mztx, r 2 z t 1 x ,mzt 1 x ,mzty,mzt 1 y , r 2 y rzt 1 z [ 7, 26, 60, 109, 170, 243, 332, 433, 548, 675 ] 709 H473 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 x t 1 y ,mztz, it 1 z [ 7, 26, 60, 109, 171, 247, 340, 444, 564, 696 ] 710 H471 1 r 2 z mztx, it 1 x ,mzt 1 x , it 1 y , r 2 x tz, r 2 y t 1 z [ 7, 26, 60, 109, 173, 250, 342, 446, 566, 698 ] 711 H511 1 r 2 z r 2 z rxtx, r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z [ 7, 26, 60, 111, 178, 262, 362, 478, 610, 758 ] 712 H308 1 i r 2 z t 1 x , r 2 z ty, r 2 z t 1 y ,mxr 1 z tz, r 2 y rzt 1 z ,mxr 1 z t 1 z 713 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 7, 26, 61, 108, 167, 241, 328, 427, 541, 668 ] 714 H468 1 r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y ,mztz, it 1 z 715 H469 1 r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz, r 2 y t 1 z [ 7, 26, 61, 108, 169, 242, 331, 432, 547, 674 ] 716 H615 hmzi r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y , itz, r 2 z t 1 z 717 H540 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 x ryt 1 y ,mytz,myt 1 z [ 7, 26, 61, 108, 171, 246, 337, 440, 560, 692 ] 718 H323 1 i r 2 z tx, r 2 z t 1 x , r 2 x ryt 1 y ,mxtz,mxt 1 z , r 2 x t 1 z [ 7, 26, 61, 110, 174, 250, 342, 446, 566, 698 ] 719 H483 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, it 1 z [ 7, 26, 61, 112, 181, 267, 369, 487, 621, 771 ] 720 H611 hmzi r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 x rztz,mxr 1 z t 1 z [ 7, 26, 61, 112, 181, 268, 372, 492, 628, 780 ] 721 H452 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y rztz,mzt 1 z [ 7, 26, 61, 113, 183, 270, 373, 492, 627, 778 ] 722 H305 1 i it 1 x , ty, t 1 y ,mxr 1 z tz, r 2 y rzt 1 z ,mxr 1 z t 1 z [ 7, 26, 62, 109, 171, 247, 337, 439, 558, 688 ] 723 H486 1 r 2 z r 2 x tx, r 2 y t 1 x , r 2 x t 1 x ,myt 1 y ,mztz, it 1 z 724 H479 1 r 2 z r 2 x tx, r 2 y t 1 x , r 2 x t 1 x ,myt 1 y , r 2 x tz, r 2 y t 1 z [ 7, 26, 62, 110, 174, 250, 342, 446, 566, 698 ] 725 H616 hmzi r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz,mxt 1 z [ 7, 26, 62, 116, 187, 275, 379, 499, 635, 787 ] 726 H504 1 r 2 z r 2 z t 1 x , ryty, r 2 x ryt 1 y , r 1 y t 1 y , tz, t 1 z [ 7, 26, 63, 115, 188, 274, 380, 498, 636, 786 ] 727 H308 1 i r 2 z t 1 x ,mzty,mzt 1 y ,mxr 1 z tz, r 2 y rzt 1 z ,mxr 1 z t 1 z 728 H323 1 i r 2 z tx, r 2 z t 1 x ,mxryty, r 2 x ryt 1 y ,mxryt 1 y , r 2 x t 1 z [ 7, 26, 63, 116, 187, 275, 379, 499, 635, 787 ] 729 H540 1 r 2 z it 1 x , ryty, r 2 x ryt 1 y , r 1 y t 1 y ,mytz,myt 1 z 730 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z 731 H529 1 r 2 z mytx,mxt 1 x ,myt 1 x ,myty, r 2 y rztz, r 2 y t 1 z [ 7, 26, 64, 113, 181, 267, 370, 490, 626, 778 ] 732 H452 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 x rztz, it 1 z [ 7, 26, 64, 114, 177, 253, 345, 449, 569, 701 ] 733 H486 1 r 2 z r 2 y t 1 x ,mxty,mxt 1 y ,myt 1 y ,mztz, it 1 z 734 H479 1 r 2 z r 2 y t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz, r 2 y t 1 z [ 7, 26, 64, 117, 188, 275, 378, 498, 634, 786 ] 735 H529 1 r 2 z mxt 1 x ,mxty,myty,mxt 1 y , r 2 y rztz, r 2 y t 1 z [ 7, 26, 64, 120, 191, 279, 383, 503, 639, 791 ] 736 H453 1 r 2 z mztx, it 1 x ,mzt 1 x , it 1 y , r 2 x rztz, r 2 y rzt 1 z [ 7, 26, 65, 114, 177, 251, 343, 447, 566, 697 ] 737 H511 1 r 2 z r 2 z rxtx, r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z [ 7, 26, 65, 121, 192, 280, 384, 504, 640, 792 ] 738 H489 1 r 2 z mztx, it 1 x ,mzt 1 x , it 1 y , rztz, r 1 z t 1 z [ 7, 26, 66, 127, 196, 284, 389, 509, 643, 795 ] 739 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 x ty, r 2 x tz, r 2 y t 1 z 62 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X [ 7, 27, 61, 108, 167, 241, 328, 427, 541, 668 ] 740 H461 1 r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz, it 1 z [ 7, 27, 62, 112, 175, 251, 343, 447, 567, 699 ] 741 H477 1 r 2 z r 2 y t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y ,mztz, it 1 z 742 H429 1 r 2 z r 2 y t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z [ 7, 27, 63, 110, 172, 248, 338, 440, 559, 689 ] 743 H466 1 r 2 z r 2 z t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y ,mztz, it 1 z 744 H427 1 r 2 z r 2 z t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z [ 7, 27, 63, 111, 175, 251, 343, 447, 567, 699 ] 745 H617 hmzi r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , itz, r 2 z t 1 z [ 7, 27, 63, 115, 185, 272, 376, 496, 632, 784 ] 746 H503 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y rztz, r 2 y t 1 z [ 7, 27, 64, 112, 176, 252, 344, 448, 568, 700 ] 747 H461 1 r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, it 1 z [ 7, 27, 64, 117, 187, 274, 378, 498, 634, 786 ] 748 H527 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz,mzt 1 z [ 7, 27, 65, 113, 174, 249, 341, 445, 561, 690 ] 749 H541 1 r 2 z r 2 z rxtx,mxty,myty,mxt 1 y ,mztz, it 1 z [ 7, 27, 65, 115, 178, 254, 346, 450, 570, 702 ] 750 H475 1 r 2 z it 1 x ,mxty,mxt 1 y ,myt 1 y ,mztz, it 1 z 751 H485 1 r 2 z it 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz, r 2 y t 1 z [ 7, 27, 65, 119, 191, 279, 383, 503, 639, 791 ] 752 H622 hmzi r 2 z mxt 1 x ,mxty,myty,mxt 1 y , r 2 y rztz,mxrzt 1 z [ 7, 27, 65, 121, 192, 280, 383, 505, 640, 792 ] 753 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,myt 1 y ,mztz, it 1 z [ 7, 27, 65, 122, 193, 281, 385, 505, 641, 793 ] 754 H505 1 r 2 z r 2 y t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x rztz, r 2 y rzt 1 z 755 H506 1 r 2 z r 2 y t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , rztz, r 1 z t 1 z [ 7, 27, 66, 117, 186, 273, 376, 496, 632, 784 ] 756 H503 1 r 2 z r 2 z t 1 x , ty, r 2 z t 1 y , t 1 y , r 2 y rztz, r 2 y t 1 z [ 7, 27, 68, 119, 189, 276, 379, 499, 635, 787 ] 757 H527 1 r 2 z mxt 1 x ,mxty,myty,mxt 1 y , r 2 y rztz, it 1 z [ 7, 28, 63, 110, 171, 244, 333, 434, 549, 676 ] 758 H305 1 i tx, it 1 x , t 1 x , ity, it 1 y , r 2 y rzt 1 z [ 7, 28, 63, 110, 173, 248, 339, 442, 559, 688 ] 759 H504 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 x ryt 1 y , r 2 z tz, r 2 z t 1 z [ 7, 28, 63, 113, 176, 252, 344, 448, 568, 700 ] 760 H473 1 r 2 z it 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y ,mztz, it 1 z 761 H474 1 r 2 z it 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z [ 7, 28, 64, 112, 176, 252, 344, 448, 568, 700 ] 762 H615 hmzi r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y , itz, r 2 z t 1 z 763 H326 1 i r 2 z rxt 1 x , r 2 y ty, r 2 y t 1 y ,myt 1 y , r 2 z tz, r 2 z t 1 z [ 7, 28, 64, 113, 177, 254, 346, 450, 570, 702 ] 764 H541 1 r 2 z r 2 z rxtx,mxty,mxt 1 y ,myt 1 y , itz, it 1 z [ 7, 28, 65, 113, 176, 251, 343, 447, 566, 697 ] 765 H511 1 r 2 z r 2 z rxtx, r 2 y ty, r 2 x ty, r 2 y t 1 y , r 2 x tz, r 2 y t 1 z [ 7, 28, 65, 115, 183, 269, 372, 492, 628, 780 ] 766 H452 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y rztz, it 1 z [ 7, 28, 65, 118, 188, 275, 379, 499, 635, 787 ] 767 H529 1 r 2 z mytx,mxt 1 x ,myt 1 x ,myt 1 y , r 2 y rztz, r 2 y t 1 z [ 7, 28, 66, 119, 188, 275, 378, 498, 634, 786 ] 768 H529 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, r 2 y t 1 z [ 7, 28, 66, 120, 191, 279, 383, 503, 639, 791 ] 769 H438 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 x rztz, r 2 y rzt 1 z [ 7, 28, 66, 121, 192, 280, 384, 504, 640, 792 ] 770 H323 1 i r 2 z t 1 x ,mxryty, r 2 x ryt 1 y ,mxryt 1 y ,mxtz,mxt 1 z [ 7, 28, 67, 121, 192, 280, 384, 504, 640, 792 ] 771 H446 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , rztz, r 1 z t 1 z [ 7, 28, 67, 123, 194, 282, 386, 506, 642, 794 ] 772 H540 1 r 2 z it 1 x , ryty, r 2 x ryt 1 y , r 1 y t 1 y ,mxtz,mxt 1 z [ 7, 28, 68, 122, 195, 281, 387, 505, 643, 793 ] 773 H305 1 i it 1 x , ity, it 1 y ,mxr 1 z tz, r 2 y rzt 1 z ,mxr 1 z t 1 z 774 H326 1 i mzrxtx, r 2 z rxt 1 x ,mzrxt 1 x ,myt 1 y , r 2 z tz, r 2 z t 1 z [ 7, 28, 68, 123, 194, 282, 386, 506, 642, 794 ] 775 H504 1 r 2 z r 2 z t 1 x , ryty, r 2 x ryt 1 y , r 1 y t 1 y , r 2 z tz, r 2 z t 1 z 776 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,myt 1 y , itz, it 1 z [ 7, 28, 68, 127, 196, 284, 387, 509, 644, 796 ] 777 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z [ 7, 29, 68, 120, 189, 276, 379, 499, 635, 787 ] 778 H527 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, it 1 z [ 7, 29, 69, 124, 195, 283, 387, 507, 643, 795 ] 779 H496 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x rztz, r 2 y rzt 1 z 780 H514 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , rztz, r 1 z t 1 z [ 7, 30, 67, 121, 192, 280, 384, 504, 640, 792 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 63 Nbr. gr Hi L m X 781 H611 hmzi r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y rztz,mxrzt 1 z [ 7, 31, 69, 124, 195, 283, 387, 507, 643, 795 ] 782 H622 hmzi r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz,mxrzt 1 z 12 [ 7, 23, 51, 91, 143, 207, 283, 371, 471, 583 ] 783*, [ 7, 23, 52, 95, 151, 219, 300, 395, 503, 623 ] 784*, [ 7, 24, 56, 102, 160, 232, 318, 416, 528, 654 ] 786*, [ 7, 24, 56, 105, 173, 260, 364, 484, 620, 772 ] 785*, [ 7, 24, 57, 104, 164, 239, 327, 428, 544, 673 ] 787*, [ 7, 24, 58, 106, 165, 240, 330, 431, 547, 678 ] 788*, [ 7, 25, 58, 107, 169, 243, 332, 435, 552, 683 ] 789*, [ 7, 25, 59, 107, 169, 245, 335, 439, 557, 689 ] 790*, [ 7, 25, 59, 109, 173, 249, 339, 443, 561, 693 ] 791*, [ 7, 25, 60, 109, 169, 242, 331, 435, 551, 679 ] 793*, [ 7, 25, 60, 109, 170, 245, 335, 439, 556, 687 ] 795*, [ 7, 25, 60, 109, 171, 246, 335, 439, 556, 687 ] 794*, [ 7, 25, 60, 109, 172, 249, 339, 443, 561, 693 ] 796*, [ 7, 25, 60, 111, 174, 249, 339, 443, 561, 693 ] 797*, [ 7, 25, 61, 116, 188, 276, 380, 500, 636, 788 ] 792*, [ 7, 26, 62, 115, 185, 272, 376, 496, 632, 784 ] 798*, [ 7, 26, 63, 117, 188, 276, 380, 500, 636, 788 ] 799*, 800*, 801*, 802*, [ 7, 26, 64, 119, 190, 279, 383, 502, 639, 791 ] 803*, [ 7, 27, 63, 111, 173, 249, 339, 443, 561, 693 ] 804*, [ 7, 27, 65, 120, 192, 280, 384, 504, 640, 792 ] 805*, [ 7, 27, 66, 120, 189, 276, 380, 500, 636, 788 ] 806*, 807*, 808*, 809*, [ 7, 27, 66, 124, 197, 284, 388, 508, 644, 796 ] 810*, [ 7, 29, 69, 124, 196, 284, 388, 508, 644, 796 ] 811*, 812*, 813*, 814*, [ 8, 26, 56, 98, 152, 218, 296, 386, 488, 602 ] 783 H749 hmy, r 2 z i i r 2 y tx, r 2 z tx,mzt 1 x , r 2 z t 1 x , r 2 z ty, r 2 z t 1 y , mzt 1 z [ 8, 26, 60, 106, 164, 234, 320, 418, 528, 650 ] 784 H750 hmy, r 2 z i i r 2 y tx, r 2 z tx,mzt 1 x , r 2 z t 1 x , r 2 z ty, r 2 z t 1 y , r 2 x rzt 1 z [ 8, 27, 60, 109, 176, 261, 364, 484, 620, 772 ] 785 H690 hmxi r 2 z mxtx,mxt 1 x ,myt 1 y , r 2 y rztz,mzr 1 z tz, r 2 x t 1 z , r 2 y t 1 z [ 8, 27, 61, 108, 167, 241, 328, 427, 541, 668 ] 786 H680 hmxi r 2 z r 2 z tx,mxtx,myt 1 x ,mxt 1 x ,myt 1 y , r 2 y tz, r 2 x t 1 z [ 8, 27, 62, 109, 171, 246, 336, 438, 555, 685 ] 787 H359 1 mz itx, r 2 z tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , mzt 1 z [ 8, 27, 63, 110, 172, 248, 338, 440, 559, 689 ] 788 H682 hmxi r 2 z itx, r 2 y tx, r 2 x t 1 x , r 2 y t 1 x ,myt 1 y , r 2 y tz, r 2 x t 1 z [ 8, 28, 63, 113, 176, 252, 344, 448, 568, 700 ] 789 H683 hmxi r 2 z itx, r 2 y tx, r 2 x t 1 x , r 2 y t 1 x , r 2 x t 1 y , r 2 y tz, r 2 x t 1 z [ 8, 28, 64, 112, 176, 252, 344, 448, 568, 700 ] 790 H749 hmy, r 2 z i i r 2 y tx, r 2 z tx,mzt 1 x , r 2 z t 1 x , r 2 x ty,mzt 1 y , mzt 1 z [ 8, 28, 64, 114, 178, 254, 346, 450, 570, 702 ] 791 H370 1 mz itx, r 2 z tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , r 2 x t 1 z [ 8, 28, 64, 117, 188, 276, 380, 500, 636, 788 ] 792 H750 hmy,mzi r 2 z mxt 1 x ,myty,myt 1 y , r 2 y rztz,mzrztz,mxrzt 1 z , rzt 1 z [ 8, 28, 65, 113, 174, 249, 341, 445, 561, 690 ] 793 H541 1 r 2 z r 2 z rxtx,mxty,myty,mxt 1 y ,myt 1 y ,mztz, it 1 z [ 8, 28, 65, 113, 176, 251, 343, 447, 566, 697 ] 64 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 794 H511 1 r 2 z r 2 z rxtx, r 2 y ty, r 2 x ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z [ 8, 28, 65, 113, 176, 252, 344, 448, 568, 700 ] 795 H344 1 mz r 2 z tx, it 1 x , ity, r 2 z ty, it 1 y , r 2 z t 1 y , r 2 y rzt 1 z [ 8, 28, 65, 113, 178, 254, 346, 450, 570, 702 ] 796 H378 1 mz r 2 y tx,mxtx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , r 2 y rzt 1 z [ 8, 28, 65, 115, 178, 254, 346, 450, 570, 702 ] 797 H682 hmyi mx r 2 y t 1 x , r 2 z ty,myty,mxt 1 y , t 1 y ,mztz, r 2 y t 1 z [ 8, 29, 66, 119, 188, 275, 378, 498, 634, 786 ] 798 H690 hmxi r 2 z r 2 z tx,myt 1 x ,myt 1 y , r 2 y rztz,mzr 1 z tz, r 2 x t 1 z , r 2 y t 1 z [ 8, 29, 66, 119, 189, 276, 380, 500, 636, 788 ] 799 H452 1 r 2 z r 2 z t 1 x , r 2 z ty, ty, r 2 z t 1 y , t 1 y , r 2 y rztz, mzt 1 z 800 H503 1 r 2 z r 2 z t 1 x , r 2 z ty, ty, r 2 z t 1 y , t 1 y , r 2 y rztz, r 2 x t 1 z 801 H527 1 r 2 z mxt 1 x ,mxty,myty,mxt 1 y ,myt 1 y , r 2 y rztz, mzt 1 z 802 H529 1 r 2 z mxtx,mytx,mxt 1 x ,myt 1 x ,myt 1 y , r 2 y rztz, r 2 y t 1 z [ 8, 29, 67, 121, 192, 280, 384, 504, 640, 792 ] 803 H541 1 r 2 z r 2 z rxtx, rxtx, r 2 y rxt 1 x , r 1 x t 1 x ,myt 1 y ,mztz, it 1 z [ 8, 30, 66, 114, 178, 254, 346, 450, 570, 702 ] 804 H750 hmy, r 2 z i i r 2 y tx, r 2 z tx,mzt 1 x , r 2 z t 1 x , r 2 x ty,mzt 1 y , r 2 x rzt 1 z [ 8, 30, 68, 123, 194, 282, 386, 506, 642, 794 ] 805 H750 hmy,mzi r 2 z mxt 1 x , r 2 z ty,mxt 1 y , r 2 y rztz,mzrztz,mxrzt 1 z , rzt 1 z [ 8, 30, 69, 121, 190, 277, 380, 500, 636, 788 ] 806 H452 1 r 2 z r 2 z t 1 x , r 2 z ty, ty, r 2 z t 1 y , t 1 y , r 2 y rztz, it 1 z 807 H503 1 r 2 z r 2 z t 1 x , r 2 z ty, ty, r 2 z t 1 y , t 1 y , r 2 y rztz, r 2 y t 1 z 808 H527 1 r 2 z mxt 1 x ,mxty,myty,mxt 1 y ,myt 1 y , r 2 y rztz, it 1 z 809 H529 1 r 2 z mxt 1 x ,mxty,myty,mxt 1 y ,myt 1 y , r 2 y rztz, r 2 y t 1 z [ 8, 30, 69, 126, 196, 284, 388, 508, 644, 796 ] 810 H511 1 r 2 z r 2 z rxtx, rxtx, r 2 y rxt 1 x , r 1 x t 1 x , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z [ 8, 32, 70, 125, 196, 284, 388, 508, 644, 796 ] 811 H604 hr 2 y i r2 z itx,mxtx,myt 1 x ,mxt 1 x ,myt 1 y , r 2 x rztz, rzt 1 z 812 H598 hr 2 y i r2 z r 2 z tx, r 2 y tx, r 2 x t 1 x , r 2 y t 1 x , r 2 x t 1 y , r 2 x rztz, rzt 1 z 813 H611 hmzi r 2 z r 2 z t 1 x , r 2 z ty, ty, r 2 z t 1 y , t 1 y , r 2 y rztz, mxrzt 1 z 814 H622 hmzi r 2 z mxt 1 x ,mxty,myty,mxt 1 y ,myt 1 y , r 2 y rztz, mxrzt 1 z 13 [ 6, 19, 44, 83, 134, 196, 272, 361, 460, 573 ] 816*, [ 6, 19, 44, 84, 138, 201, 272, 357, 460, 577 ] 815*, [ 6, 20, 45, 78, 121, 176, 238, 310, 395, 486 ] 817*, 818*, [ 6, 20, 45, 80, 125, 180, 245, 320, 405, 500 ] 819*, 820*, [ 6, 20, 47, 85, 132, 191, 261, 340, 431, 533 ] 821*, [ 6, 20, 47, 87, 137, 195, 263, 343, 434, 535 ] 825*, [ 6, 20, 47, 87, 139, 203, 280, 370, 471, 585 ] 823*, [ 6, 20, 47, 87, 141, 211, 292, 382, 486, 602 ] 822*, [ 6, 20, 47, 87, 141, 211, 297, 397, 509, 635 ] 826*, [ 6, 20, 47, 89, 147, 216, 292, 384, 496, 615 ] 824*, [ 6, 20, 49, 95, 154, 221, 296, 386, 492, 607 ] 827*, [ 6, 20, 52, 105, 170, 243, 329, 429, 544, 675 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 65 Nbr. gr Hi L m X 828*, [ 6, 21, 50, 90, 143, 212, 290, 381, 490, 606 ] 829*, [ 6, 21, 51, 93, 146, 214, 293, 383, 489, 605 ] 830*, [ 6, 21, 52, 97, 152, 218, 295, 386, 492, 607 ] 831*, [ 6, 21, 52, 99, 159, 229, 307, 401, 513, 631 ] 832*, [ 6, 21, 53, 103, 168, 245, 331, 431, 546, 675 ] 835*, [ 6, 21, 53, 104, 169, 243, 329, 429, 544, 675 ] 833*, [ 6, 21, 53, 106, 169, 236, 320, 421, 534, 661 ] 834*, [ 6, 21, 53, 107, 173, 243, 327, 433, 555, 682 ] 836*, [ 6, 21, 54, 107, 172, 243, 327, 433, 555, 682 ] 837*, [ 6, 21, 58, 112, 175, 249, 336, 440, 559, 690 ] 838*, [ 6, 22, 53, 95, 149, 217, 295, 386, 492, 607 ] 839*, [ 6, 22, 54, 97, 153, 225, 305, 399, 511, 629 ] 840*, [ 6, 22, 55, 100, 156, 228, 315, 414, 527, 655 ] 842*, [ 6, 22, 55, 106, 166, 237, 325, 427, 543, 673 ] 847*, [ 6, 22, 55, 107, 176, 255, 345, 447, 563, 694 ] 844*, [ 6, 22, 56, 102, 157, 227, 307, 401, 513, 631 ] 841*, [ 6, 22, 56, 105, 166, 240, 327, 428, 544, 673 ] 845*, [ 6, 22, 56, 106, 165, 237, 325, 427, 543, 673 ] 849*, [ 6, 22, 56, 107, 166, 234, 320, 421, 534, 661 ] 843*, [ 6, 22, 57, 108, 170, 244, 333, 436, 552, 683 ] 846*, [ 6, 22, 57, 109, 174, 250, 337, 439, 556, 687 ] 850*, [ 6, 22, 58, 109, 171, 246, 334, 436, 552, 682 ] 848*, [ 6, 22, 59, 111, 170, 243, 332, 434, 550, 680 ] 851*, [ 6, 23, 56, 101, 159, 232, 319, 420, 535, 664 ] 852*, [ 6, 23, 56, 107, 173, 249, 339, 442, 559, 691 ] 853*, [ 6, 23, 58, 108, 171, 246, 335, 439, 557, 689 ] 854*, [ 6, 23, 58, 109, 173, 251, 343, 445, 561, 693 ] 855*, [ 6, 23, 58, 110, 177, 254, 343, 445, 561, 693 ] 856*, [ 6, 23, 59, 114, 180, 252, 339, 443, 561, 693 ] 857*, [ 6, 23, 60, 112, 175, 250, 339, 443, 561, 693 ] 858*, [ 6, 23, 60, 114, 177, 250, 339, 443, 561, 693 ] 859*, [ 6, 24, 56, 100, 159, 232, 319, 420, 535, 664 ] 860*, [ 6, 24, 59, 110, 171, 243, 332, 435, 552, 683 ] 861*, [ 6, 24, 60, 108, 169, 245, 335, 439, 557, 689 ] 862*, [ 6, 24, 60, 109, 171, 247, 337, 441, 559, 691 ] 863*, [ 6, 24, 60, 111, 173, 247, 337, 441, 559, 691 ] 864*, 865*, [ 6, 24, 60, 113, 177, 250, 339, 443, 561, 693 ] 866*, [ 6, 24, 61, 112, 172, 245, 335, 439, 556, 687 ] 867*, [ 6, 24, 62, 112, 173, 249, 339, 443, 561, 693 ] 868*, [ 6, 24, 62, 114, 175, 249, 339, 443, 561, 693 ] 869*, [ 6, 24, 62, 115, 177, 250, 339, 443, 561, 693 ] 870*, [ 7, 23, 52, 95, 150, 216, 293, 381, 485, 603 ] 815 H412 1 r 2 z rx r 2 z rxt 1 x ,myty, r 2 z t 1 y , r 1 x t 1 y , rxtz,mzt 1 z [ 7, 23, 52, 95, 150, 218, 301, 393, 497, 619 ] 816 H413 1 r 2 z rx mxt 1 x ,myty, r 2 z t 1 y , r 1 x t 1 y , rxtz,mzt 1 z [ 7, 24, 52, 90, 141, 202, 273, 358, 452, 556 ] 817 H388 1 r 2 z rx r 2 z rxt 1 x ,mzrxty, it 1 y ,mzrxt 1 y ,mzrxtz,mzrxt 1 z [ 7, 24, 52, 91, 143, 204, 276, 362, 457, 563 ] 818 H400 1 r 2 z rx mxt 1 x ,mzrxty, it 1 y ,mzrxt 1 y ,mzrxtz,mzrxt 1 z 66 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X [ 7, 24, 53, 93, 145, 208, 283, 369, 467, 576 ] 819 H389 1 r 2 z rx it 1 x ,mzrxty, it 1 y ,mzrxt 1 y ,mzrxtz,mzrxt 1 z [ 7, 24, 53, 94, 147, 210, 287, 375, 474, 584 ] 820 H398 1 r 2 z rx r 2 y rxt 1 x ,mzrxty, it 1 y ,mzrxt 1 y ,mzrxtz,mzrxt 1 z [ 7, 24, 54, 95, 148, 213, 288, 376, 477, 588 ] 821 H418 1 r 2 z rx mxt 1 x , r 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, rxt 1 z [ 7, 24, 54, 96, 152, 221, 301, 395, 501, 618 ] 822 H407 1 r 2 z rx r 2 z rxt 1 x , r 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, rxt 1 z [ 7, 24, 54, 98, 154, 222, 307, 400, 504, 630 ] 823 H395 1 r 2 z rx mxt 1 x , r 2 x ty, it 1 y ,mzrxt 1 y ,mzrxtz, r 2 x t 1 z [ 7, 24, 54, 98, 156, 225, 306, 402, 510, 629 ] 824 H394 1 r 2 z rx r 2 z rxt 1 x , r 2 x ty, it 1 y ,mzrxt 1 y ,mzrxtz, r 2 x t 1 z [ 7, 24, 55, 99, 154, 219, 298, 389, 493, 610 ] 825 H410 1 r 2 z rx r 2 y rxt 1 x , r 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, rxt 1 z [ 7, 24, 55, 99, 157, 231, 320, 423, 543, 677 ] 826 H414 1 r 2 z rx it 1 x , r 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, rxt 1 z [ 7, 24, 56, 103, 161, 227, 306, 401, 506, 623 ] 827 H419 1 r 2 z rx mxt 1 x ,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mzt 1 z [ 7, 24, 60, 112, 175, 249, 337, 438, 557, 689 ] 828 H422 1 r 2 z rx r 2 z tx,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mzt 1 z [ 7, 25, 56, 99, 157, 226, 307, 405, 512, 631 ] 829 H395 1 r 2 z rx mxt 1 x , r 2 y rxty,mzrxt 1 y ,mzrxtz, r 2 y rxt 1 z , r 2 x t 1 z [ 7, 25, 57, 100, 156, 225, 305, 399, 505, 622 ] 830 H413 1 r 2 z rx mxt 1 x ,mxr 1 x ty,myty, r 1 x t 1 y , rxtz,mxrxt 1 z [ 7, 25, 58, 103, 159, 226, 306, 401, 506, 623 ] 831 H402 1 r 2 z rx r 2 z rxt 1 x ,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mzt 1 z [ 7, 25, 58, 106, 166, 234, 316, 416, 524, 644 ] 832 H401 1 r 2 z rx mxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z [ 7, 25, 60, 111, 175, 249, 337, 438, 557, 689 ] 833 H422 1 r 2 z rx r 2 z tx,myty,mxr 1 x t 1 y ,mxrxtz,mztz,mzt 1 z [ 7, 25, 60, 113, 172, 243, 332, 433, 548, 674 ] 834 H409 1 r 2 z rx r 2 z tx, r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z [ 7, 25, 61, 111, 177, 253, 343, 447, 566, 698 ] 835 H418 1 r 2 z rx mxt 1 x , r 2 z ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z [ 7, 25, 62, 116, 177, 249, 344, 452, 567, 694 ] 836 H415 1 r 2 z rx it 1 x ,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mzt 1 z [ 7, 25, 63, 115, 177, 249, 344, 452, 567, 694 ] 837 H404 1 r 2 z rx r 2 y rxt 1 x ,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mzt 1 z [ 7, 25, 65, 114, 178, 251, 343, 448, 567, 699 ] 838 H407 1 r 2 z rx r 2 z rxt 1 x , r 2 z ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z [ 7, 26, 58, 101, 158, 226, 306, 401, 506, 623 ] 839 H412 1 r 2 z rx r 2 z rxt 1 x ,mxr 1 x ty,myty, r 1 x t 1 y , rxtz,mxrxt 1 z [ 7, 26, 59, 103, 163, 233, 315, 415, 523, 643 ] 840 H394 1 r 2 z rx r 2 z rxt 1 x , r 2 y rxty,mzrxt 1 y ,mzrxtz, r 2 y rxt 1 z , r 2 x t 1 z [ 7, 26, 61, 106, 164, 234, 316, 416, 524, 644 ] 841 H391 1 r 2 z rx r 2 z rxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z [ 7, 26, 62, 107, 168, 243, 332, 433, 551, 680 ] 842 H395 1 r 2 z rx mxt 1 x , r 2 y rxty, it 1 y ,mzrxt 1 y ,mzrxtz, r 2 y rxt 1 z [ 7, 26, 62, 112, 170, 243, 332, 433, 548, 674 ] 843 H409 1 r 2 z rx r 2 z tx, r 2 x ty, r 2 y rxt 1 y , r 2 y rxtz, r 2 x tz, r 2 x t 1 z [ 7, 26, 62, 114, 181, 257, 349, 451, 571, 702 ] 844 H413 1 r 2 z rx mxt 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , r 2 y tz,mxrxt 1 z [ 7, 26, 63, 111, 174, 249, 338, 441, 559, 688 ] 845 H413 1 r 2 z rx mxt 1 x ,mxr 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz,mxrxt 1 z [ 7, 26, 63, 112, 175, 250, 341, 444, 563, 694 ] 846 H394 1 r 2 z rx r 2 z rxt 1 x , r 2 y rxty, it 1 y ,mzrxt 1 y ,mzrxtz, r 2 y rxt 1 z [ 7, 26, 63, 114, 175, 251, 343, 447, 567, 699 ] 847 H396 1 r 2 z rx it 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z [ 7, 26, 64, 112, 175, 250, 341, 444, 562, 692 ] 848 H412 1 r 2 z rx r 2 z rxt 1 x ,mxr 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz,mxrxt 1 z [ 7, 26, 64, 113, 175, 251, 343, 447, 567, 699 ] 849 H392 1 r 2 z rx r 2 y rxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z [ 7, 26, 64, 114, 179, 254, 344, 448, 568, 700 ] 850 H400 1 r 2 z rx mxt 1 x ,mzrxty, it 1 y , itz, it 1 z ,mzrxt 1 z [ 7, 26, 65, 113, 174, 249, 339, 442, 560, 690 ] 851 H388 1 r 2 z rx r 2 z rxt 1 x ,mzrxty, it 1 y , itz, it 1 z ,mzrxt 1 z [ 7, 27, 62, 108, 170, 245, 335, 438, 556, 687 ] 852 H401 1 r 2 z rx mxt 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 y rxt 1 z [ 7, 27, 62, 114, 178, 254, 346, 449, 569, 701 ] 853 H409 1 r 2 z rx r 2 z tx, r 2 y rxty, r 2 y rxt 1 y , r 2 y rxtz, r 2 y rxt 1 z , r 2 x t 1 z [ 7, 27, 64, 113, 177, 252, 344, 448, 568, 700 ] 854 H419 1 r 2 z rx mxt 1 x ,mxr 1 x ty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mxrxt 1 z [ 7, 27, 64, 114, 178, 256, 348, 450, 570, 702 ] 855 H422 1 r 2 z rx r 2 z tx,mxr 1 x ty,mxr 1 x t 1 y ,mxrxtz,mxrxt 1 z ,mzt 1 z [ 7, 27, 64, 115, 181, 256, 348, 450, 570, 702 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 67 Nbr. gr Hi L m X 856 H395 1 r 2 z rx mxt 1 x , r 2 y rxty, it 1 y , itz, r 2 y rxt 1 z , r 2 x t 1 z [ 7, 27, 65, 118, 181, 254, 346, 450, 570, 702 ] 857 H412 1 r 2 z rx r 2 z rxt 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , r 2 y tz,mxrxt 1 z [ 7, 27, 66, 115, 179, 254, 346, 450, 570, 702 ] 858 H410 1 r 2 z rx r 2 y rxt 1 x , r 2 z ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z [ 7, 27, 66, 117, 179, 254, 346, 450, 570, 702 ] 859 H414 1 r 2 z rx it 1 x , r 2 z ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z [ 7, 28, 61, 108, 170, 245, 335, 438, 556, 687 ] 860 H392 1 r 2 z rx r 2 y rxt 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 y rxt 1 z [ 7, 28, 64, 115, 176, 252, 344, 448, 568, 700 ] 861 H389 1 r 2 z rx it 1 x ,mzrxty, it 1 y , itz, it 1 z ,mzrxt 1 z [ 7, 28, 65, 112, 176, 252, 344, 448, 568, 700 ] 862 H404 1 r 2 z rx r 2 y rxt 1 x ,mxr 1 x ty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mxrxt 1 z [ 7, 28, 65, 113, 177, 253, 345, 449, 569, 701 ] 863 H409 1 r 2 z rx r 2 z tx, r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 y rxt 1 z [ 7, 28, 65, 115, 177, 253, 345, 449, 569, 701 ] 864 H396 1 r 2 z rx it 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 y rxt 1 z 865 H391 1 r 2 z rx r 2 z rxt 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 y rxt 1 z [ 7, 28, 65, 117, 179, 254, 346, 450, 570, 702 ] 866 H402 1 r 2 z rx r 2 z rxt 1 x ,mxr 1 x ty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mxrxt 1 z [ 7, 28, 66, 115, 176, 252, 344, 448, 568, 700 ] 867 H398 1 r 2 z rx r 2 y rxt 1 x ,mzrxty, it 1 y , itz, it 1 z ,mzrxt 1 z [ 7, 28, 67, 114, 178, 254, 346, 450, 570, 702 ] 868 H422 1 r 2 z rx r 2 z tx,mxr 1 x ty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mxrxt 1 z [ 7, 28, 67, 116, 178, 254, 346, 450, 570, 702 ] 869 H415 1 r 2 z rx it 1 x ,mxr 1 x ty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mxrxt 1 z [ 7, 28, 67, 117, 179, 254, 346, 450, 570, 702 ] 870 H394 1 r 2 z rx r 2 z rxt 1 x , r 2 y rxty, it 1 y , itz, r 2 y rxt 1 z , r 2 x t 1 z 14 [ 7, 24, 54, 95, 149, 217, 295, 386, 492, 607 ] 871*, [ 7, 24, 55, 99, 156, 226, 308, 404, 514, 636 ] 872*, [ 7, 24, 55, 99, 156, 227, 311, 408, 519, 643 ] 873*, [ 7, 24, 56, 103, 163, 235, 320, 419, 531, 656 ] 874*, [ 7, 24, 56, 103, 165, 241, 329, 429, 544, 675 ] 875*, [ 7, 24, 56, 105, 169, 242, 327, 433, 555, 682 ] 876*, [ 7, 25, 56, 98, 155, 224, 304, 400, 507, 626 ] 877*, [ 7, 25, 57, 99, 155, 227, 307, 401, 513, 631 ] 878*, [ 7, 25, 57, 100, 157, 229, 312, 408, 519, 642 ] 879*, [ 7, 25, 57, 102, 161, 233, 318, 417, 529, 654 ] 880*, [ 7, 25, 57, 103, 163, 237, 325, 427, 543, 673 ] 887*, [ 7, 25, 58, 103, 162, 239, 330, 429, 541, 672 ] 881*, [ 7, 25, 58, 104, 163, 234, 320, 421, 534, 661 ] 883*, [ 7, 25, 58, 104, 163, 235, 320, 419, 531, 656 ] 882*, [ 7, 25, 58, 104, 163, 237, 325, 426, 541, 670 ] 884*, [ 7, 25, 58, 105, 166, 241, 330, 433, 550, 681 ] 890*, [ 7, 25, 59, 106, 164, 235, 320, 419, 531, 656 ] 885*, 886*, [ 7, 25, 60, 108, 168, 242, 329, 431, 547, 675 ] 888*, [ 7, 25, 60, 109, 170, 245, 334, 436, 552, 682 ] 889*, [ 7, 25, 62, 111, 172, 249, 339, 443, 561, 693 ] 891*, [ 7, 25, 63, 113, 173, 249, 339, 443, 561, 693 ] 892*, [ 7, 26, 60, 105, 163, 235, 320, 419, 531, 656 ] 893*, 894*, 895*, 896*, [ 7, 26, 61, 107, 167, 242, 329, 431, 547, 675 ] 897*, 900*, [ 7, 26, 61, 108, 168, 243, 332, 435, 552, 683 ] 902*, 903*, [ 7, 26, 62, 109, 168, 243, 332, 434, 550, 680 ] 898*, 899*, [ 7, 26, 62, 110, 170, 245, 335, 438, 554, 685 ] 901*, [ 7, 26, 62, 110, 172, 249, 339, 443, 561, 693 ] 904*, 905*, 906*, [ 7, 26, 63, 112, 173, 249, 339, 443, 561, 693 ] 907*, 908*, [ 7, 26, 63, 113, 175, 250, 339, 443, 561, 693 ] 68 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 909*, 910*, [ 7, 27, 60, 104, 163, 235, 320, 419, 531, 656 ] 911*, 912*, [ 7, 27, 61, 107, 168, 243, 332, 435, 552, 683 ] 915*, 916*, [ 7, 27, 62, 108, 168, 243, 332, 434, 550, 680 ] 913*, 914*, 917*, [ 7, 27, 62, 109, 170, 245, 335, 439, 556, 687 ] 918*, 919*, [ 7, 27, 63, 111, 173, 249, 339, 443, 561, 693 ] 920*, [ 7, 27, 65, 113, 173, 249, 339, 443, 561, 693 ] 921*, 922*, 923*, 924*, 925*, 926*, [ 7, 28, 65, 112, 173, 249, 339, 443, 561, 693 ] 927*, 928*, 929*, 930*, 931*, 932*, [ 7, 29, 65, 111, 173, 249, 339, 443, 561, 693 ] 933*, 934*, [ 8, 27, 58, 101, 158, 226, 306, 401, 506, 623 ] 871 H650 hmzrxi mx mxt 1 x ,myty, r 2 z t 1 y ,myt 1 y , r 1 x tz,mxr 1 x t 1 z , r 1 x t 1 z [ 8, 27, 59, 104, 163, 234, 318, 416, 526, 650 ] 872 H359 1 mz itx, r 2 z tx,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , mzt 1 z [ 8, 27, 60, 106, 166, 240, 326, 426, 540, 666 ] 873 H370 1 mz itx, r 2 z tx,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z [ 8, 27, 61, 109, 169, 242, 329, 429, 542, 669 ] 874 H368 1 mz mxtx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 y t 1 z [ 8, 27, 61, 109, 173, 249, 337, 438, 557, 689 ] 875 H422 1 r 2 z rx r 2 z tx,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mztz, mzt 1 z [ 8, 27, 63, 113, 176, 249, 344, 452, 567, 694 ] 876 H652 hmzrxi mx r 2 z rxt 1 x ,myty, r 2 z t 1 y ,myt 1 y , r 1 x tz,mxr 1 x t 1 z , r 1 x t 1 z [ 8, 28, 59, 104, 163, 231, 316, 412, 519, 643 ] 877 H359 1 mz r 2 z tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , mzt 1 z [ 8, 28, 60, 104, 164, 234, 316, 416, 524, 644 ] 878 H648 hmzrxi mx mxt 1 x , r 2 x ty, it 1 y , r 2 x t 1 y ,mzr 1 x tz, r 2 z rxt 1 z , mzr 1 x t 1 z [ 8, 28, 60, 105, 165, 236, 321, 420, 531, 655 ] 879 H350 1 mz r 2 z tx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,myt 1 y , mzt 1 z [ 8, 28, 61, 108, 169, 242, 329, 429, 542, 669 ] 880 H365 1 mz r 2 z tx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z [ 8, 28, 62, 108, 172, 249, 336, 436, 555, 687 ] 881 H372 1 mz r 2 z tx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 y t 1 z [ 8, 28, 62, 109, 169, 242, 329, 429, 542, 669 ] 882 H370 1 mz r 2 z tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z [ 8, 28, 62, 109, 170, 243, 332, 433, 548, 674 ] 883 H409 1 r 2 z rx r 2 z tx, r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 x t 1 z [ 8, 28, 62, 109, 172, 247, 337, 440, 558, 688 ] 884 H361 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 z ty, it 1 y , r 2 z t 1 y , r 2 y t 1 z [ 8, 28, 63, 110, 169, 242, 329, 429, 542, 669 ] 885 H359 1 mz itx, r 2 z tx,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , mzt 1 z 886 H353 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , mzt 1 z [ 8, 28, 63, 111, 175, 251, 343, 447, 567, 699 ] 887 H649 hmzrxi mx r 2 z rxt 1 x , r 2 x ty, it 1 y , r 2 x t 1 y ,mzr 1 x tz, r 2 z rxt 1 z , mzr 1 x t 1 z [ 8, 28, 64, 111, 173, 247, 337, 440, 556, 686 ] 888 H359 1 mz itx, r 2 z tx, r 2 y t 1 x ,myty, r 2 x t 1 y ,myt 1 y , mzt 1 z [ 8, 28, 64, 112, 174, 250, 341, 444, 562, 692 ] 889 H377 1 mz r 2 y tx,mzr 1 z t 1 x , r 1 z t 1 x ,mzrzty, rzty,myt 1 y , mzt 1 z [ 8, 28, 64, 112, 176, 252, 344, 448, 568, 700 ] 890 H364 1 mz r 2 z tx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,myt 1 y , it 1 z [ 8, 28, 67, 113, 178, 254, 346, 450, 570, 702 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 69 Nbr. gr Hi L m X 891 H370 1 mz itx, r 2 z tx, r 2 y t 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z [ 8, 28, 68, 114, 178, 254, 346, 450, 570, 702 ] 892 H368 1 mz mxtx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 y t 1 z [ 8, 29, 63, 109, 169, 242, 329, 429, 542, 669 ] 893 H350 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , mzt 1 z 894 H370 1 mz itx, r 2 z tx,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z 895 H359 1 mz r 2 z tx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , mzt 1 z 896 H368 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 y t 1 z [ 8, 29, 64, 110, 173, 247, 337, 440, 556, 686 ] 897 H359 1 mz itx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , mzt 1 z [ 8, 29, 65, 111, 174, 249, 339, 442, 560, 690 ] 898 H330 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , ity, it 1 y , r 2 z t 1 y , mzt 1 z 899 H350 1 mz itx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,myt 1 y , mzt 1 z [ 8, 29, 65, 111, 175, 250, 339, 443, 560, 689 ] 900 H382 1 mz r 2 y tx,mzr 1 z t 1 x , r 1 z t 1 x ,mzrzty, rzty,myt 1 y , r 2 y rzt 1 z [ 8, 29, 65, 112, 175, 251, 342, 445, 564, 695 ] 901 H343 1 mz r 2 z tx, r 2 y rzt 1 x ,mxrzt 1 x , ity, r 2 y rzt 1 y ,mxrzt 1 y , mzt 1 z [ 8, 29, 65, 112, 176, 252, 344, 448, 568, 700 ] 902 H361 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , ity, it 1 y , r 2 z t 1 y , r 2 y t 1 z 903 H365 1 mz itx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z [ 8, 29, 66, 113, 178, 254, 346, 450, 570, 702 ] 904 H370 1 mz itx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z 905 H378 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 y rzt 1 z 906 H378 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x , r 2 x ty,myty,myt 1 y , r 2 y rzt 1 z [ 8, 29, 67, 114, 178, 254, 346, 450, 570, 702 ] 907 H372 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 y t 1 z 908 H373 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , it 1 z [ 8, 29, 67, 115, 179, 254, 346, 450, 570, 702 ] 909 H359 1 mz itx, r 2 z tx, r 2 y t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , mzt 1 z 910 H650 hmzrxi mx mxt 1 x , r 2 z ty, r 2 z t 1 y ,myt 1 y ,mxr 1 x tz,mxr 1 x t 1 z , r 1 x t 1 z [ 8, 30, 62, 109, 169, 242, 329, 429, 542, 669 ] 911 H365 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z 912 H370 1 mz r 2 z tx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z [ 8, 30, 64, 111, 174, 249, 339, 442, 560, 690 ] 913 H361 1 mz itx, it 1 x , r 2 z t 1 x , r 2 z ty, it 1 y , r 2 z t 1 y , r 2 y t 1 z 914 H372 1 mz itx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 y t 1 z [ 8, 30, 64, 112, 176, 252, 344, 448, 568, 700 ] 915 H332 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , ity, it 1 y , r 2 z t 1 y , it 1 z 916 H364 1 mz itx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,myt 1 y , it 1 z [ 8, 30, 65, 112, 175, 251, 341, 444, 563, 693 ] 917 H347 1 mz r 2 z tx, r 2 y rzt 1 x ,mxrzt 1 x , ity, r 2 y rzt 1 y ,mxrzt 1 y , r 2 x rzt 1 z [ 8, 30, 65, 113, 176, 252, 344, 448, 568, 700 ] 918 H344 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , ity, it 1 y , r 2 z t 1 y , r 2 y rzt 1 z 70 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 919 H344 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , ity, r 2 z ty, it 1 y , r 2 y rzt 1 z [ 8, 30, 66, 114, 178, 254, 346, 450, 570, 702 ] 920 H364 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , it 1 z [ 8, 30, 68, 114, 178, 254, 346, 450, 570, 702 ] 921 H648 hmzrxi mx mxt 1 x , ity, it 1 y , r 2 x t 1 y , r 2 z rxtz, r 2 z rxt 1 z , mzr 1 x t 1 z 922 H350 1 mz itx, it 1 x , r 2 z t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , mzt 1 z 923 H370 1 mz itx, r 2 z tx, r 2 y t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z 924 H359 1 mz itx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , mzt 1 z 925 H368 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 y t 1 z 926 H422 1 r 2 z rx r 2 z tx,mxr 1 x ty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mztz, mxrxt 1 z [ 8, 31, 67, 114, 178, 254, 346, 450, 570, 702 ] 927 H361 1 mz itx, it 1 x , r 2 z t 1 x , ity, it 1 y , r 2 z t 1 y , r 2 y t 1 z 928 H372 1 mz itx, it 1 x , r 2 z t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 y t 1 z 929 H365 1 mz itx, it 1 x , r 2 z t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z 930 H370 1 mz itx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 x t 1 z 931 H652 hmzrxi mx r 2 z rxt 1 x , r 2 z ty, r 2 z t 1 y ,myt 1 y ,mxr 1 x tz,mxr 1 x t 1 z , r 1 x t 1 z 932 H409 1 r 2 z rx r 2 z tx, r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 y rxt 1 z [ 8, 32, 66, 114, 178, 254, 346, 450, 570, 702 ] 933 H649 hmzrxi mx r 2 z rxt 1 x , ity, it 1 y , r 2 x t 1 y , r 2 z rxtz, r 2 z rxt 1 z , mzr 1 x t 1 z 934 H364 1 mz itx, it 1 x , r 2 z t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , it 1 z 15 [ 7, 23, 50, 87, 135, 194, 263, 343, 434, 535 ] 936*, [ 7, 23, 51, 91, 143, 207, 283, 371, 471, 583 ] 935*, [ 7, 23, 51, 92, 147, 214, 292, 383, 487, 604 ] 937*, [ 7, 23, 52, 94, 149, 217, 298, 392, 499, 619 ] 939*, [ 7, 23, 52, 95, 152, 221, 303, 399, 507, 627 ] 940*, [ 7, 23, 52, 96, 155, 227, 312, 411, 523, 648 ] 938*, [ 7, 24, 53, 93, 146, 210, 285, 373, 472, 583 ] 941*, [ 7, 24, 54, 95, 148, 214, 291, 379, 480, 593 ] 944*, [ 7, 24, 54, 96, 150, 216, 294, 384, 486, 600 ] 950*, [ 7, 24, 54, 96, 151, 219, 300, 394, 500, 619 ] 945*, [ 7, 24, 54, 97, 153, 222, 304, 399, 507, 628 ] 949*, [ 7, 24, 54, 97, 154, 224, 308, 407, 519, 644 ] 942*, [ 7, 24, 54, 97, 154, 225, 308, 404, 515, 639 ] 948*, [ 7, 24, 55, 97, 151, 220, 300, 392, 499, 616 ] 943*, [ 7, 24, 55, 98, 152, 218, 296, 386, 488, 602 ] 953*, 954*, [ 7, 24, 55, 99, 158, 233, 321, 421, 535, 663 ] 946*, [ 7, 24, 55, 100, 158, 229, 314, 413, 525, 650 ] 956*, [ 7, 24, 55, 100, 159, 231, 316, 415, 527, 652 ] 947*, 951*, 952*, [ 7, 24, 55, 100, 161, 239, 329, 428, 543, 679 ] 955*, [ 7, 24, 55, 101, 162, 237, 326, 429, 546, 677 ] 957*, [ 7, 24, 56, 101, 159, 231, 316, 414, 525, 649 ] 960*, [ 7, 24, 57, 109, 180, 268, 372, 492, 628, 780 ] 958*, 959*, K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 71 Nbr. gr Hi L m X [ 7, 24, 58, 105, 165, 241, 331, 435, 553, 685 ] 961*, [ 7, 24, 59, 108, 169, 245, 335, 439, 557, 689 ] 962*, [ 7, 24, 59, 109, 171, 247, 337, 441, 559, 691 ] 963*, [ 7, 25, 56, 98, 152, 218, 296, 386, 488, 602 ] 964*, [ 7, 25, 57, 99, 152, 218, 296, 386, 488, 602 ] 968*, [ 7, 25, 57, 102, 161, 233, 318, 417, 529, 654 ] 965*, 966*, [ 7, 25, 58, 102, 158, 228, 311, 407, 516, 638 ] 967*, 969*, [ 7, 25, 58, 102, 159, 231, 316, 414, 525, 649 ] 972*, [ 7, 25, 58, 104, 163, 235, 320, 419, 531, 656 ] 970*, 971*, [ 7, 25, 58, 105, 166, 241, 330, 433, 550, 681 ] 977*, 978*, [ 7, 25, 59, 105, 163, 235, 321, 421, 534, 660 ] 982*, 983*, [ 7, 25, 59, 105, 165, 241, 331, 435, 553, 685 ] 973*, [ 7, 25, 59, 106, 166, 241, 330, 432, 548, 678 ] 974*, 975*, [ 7, 25, 59, 107, 169, 245, 335, 439, 557, 689 ] 976*, 985*, 986*, [ 7, 25, 59, 109, 177, 264, 368, 488, 624, 776 ] 979*, [ 7, 25, 60, 108, 169, 245, 335, 439, 557, 689 ] 984*, [ 7, 25, 60, 109, 169, 242, 331, 435, 551, 679 ] 990*, 991*, [ 7, 25, 60, 109, 171, 247, 337, 441, 559, 691 ] 992*, [ 7, 25, 60, 112, 182, 269, 373, 493, 629, 781 ] 987*, [ 7, 25, 60, 113, 184, 272, 376, 496, 632, 784 ] 980*, 981*, 988*, 989*, [ 7, 25, 61, 111, 173, 249, 339, 443, 561, 693 ] 993*, [ 7, 25, 62, 116, 185, 272, 376, 496, 632, 784 ] 994*, 995*, [ 7, 26, 57, 98, 152, 218, 296, 386, 488, 602 ] 998*, [ 7, 26, 58, 101, 158, 228, 311, 407, 516, 638 ] 996*, 997*, [ 7, 26, 59, 104, 163, 235, 320, 419, 531, 656 ] 999*, 1000*, [ 7, 26, 59, 104, 163, 235, 321, 421, 534, 660 ] 1001*, 1002*, [ 7, 26, 60, 107, 168, 243, 332, 435, 552, 683 ] 1006*, 1007*, [ 7, 26, 60, 107, 169, 245, 335, 439, 557, 689 ] 1003*, [ 7, 26, 61, 108, 168, 243, 332, 434, 550, 680 ] 1004*, 1005*, [ 7, 26, 61, 109, 171, 246, 335, 439, 556, 687 ] 1008*, 1009*, [ 7, 26, 61, 109, 171, 247, 337, 441, 559, 691 ] 1010*, [ 7, 26, 61, 115, 186, 274, 378, 498, 634, 786 ] 1015*, [ 7, 26, 62, 110, 171, 247, 337, 441, 559, 691 ] 1017*, 1018*, 1019*, 1020*, [ 7, 26, 62, 111, 173, 249, 339, 443, 561, 693 ] 1021*, [ 7, 26, 62, 112, 175, 250, 339, 443, 561, 693 ] 1022*, 1023*, [ 7, 26, 62, 113, 181, 268, 372, 492, 628, 780 ] 1011*, 1012*, [ 7, 26, 62, 115, 185, 272, 376, 496, 632, 784 ] 1024*, 1025*, [ 7, 26, 62, 115, 186, 274, 378, 498, 634, 786 ] 1013*, 1014*, [ 7, 26, 62, 118, 190, 278, 382, 502, 638, 790 ] 1016*, [ 7, 26, 63, 112, 173, 249, 339, 443, 561, 693 ] 1028*, 1029*, [ 7, 26, 63, 116, 186, 274, 378, 498, 634, 786 ] 1030*, 1031*, [ 7, 26, 63, 118, 190, 278, 382, 502, 638, 790 ] 1026*, 1027*, [ 7, 26, 65, 123, 195, 282, 385, 505, 642, 793 ] 1032*, 1033*, [ 7, 26, 67, 126, 197, 284, 388, 508, 644, 796 ] 1034*, [ 7, 27, 62, 109, 171, 247, 337, 441, 559, 691 ] 1036*, 1037*, 1038*, [ 7, 27, 62, 112, 181, 268, 372, 492, 628, 780 ] 1035*, [ 7, 27, 63, 111, 173, 249, 339, 443, 561, 693 ] 72 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 1040*, [ 7, 27, 63, 115, 185, 272, 376, 496, 632, 784 ] 1039*, [ 7, 27, 64, 112, 173, 249, 339, 443, 561, 693 ] 1041*, 1042*, 1043*, 1044*, [ 7, 27, 65, 118, 187, 274, 378, 498, 634, 786 ] 1045*, 1046*, [ 7, 27, 65, 122, 194, 282, 386, 506, 642, 794 ] 1047*, [ 7, 27, 66, 122, 194, 282, 386, 506, 642, 794 ] 1048*, 1049*, [ 7, 27, 70, 127, 196, 284, 388, 508, 644, 796 ] 1050*, [ 7, 28, 64, 111, 173, 249, 339, 443, 561, 693 ] 1051*, 1052*, 1053*, 1054*, 1055*, [ 7, 28, 65, 117, 187, 274, 378, 498, 634, 786 ] 1056*, 1057*, [ 7, 28, 65, 117, 188, 276, 380, 500, 636, 788 ] 1058*, [ 7, 28, 66, 120, 192, 280, 384, 504, 640, 792 ] 1059*, 1060*, 1061*, [ 7, 28, 67, 124, 196, 284, 388, 508, 644, 796 ] 1062*, [ 7, 28, 68, 124, 196, 284, 388, 508, 644, 796 ] 1063*, [ 7, 28, 68, 126, 198, 284, 388, 508, 644, 796 ] 1064*, [ 7, 28, 69, 126, 197, 284, 388, 508, 644, 796 ] 1065*, [ 7, 29, 68, 122, 194, 282, 386, 506, 642, 794 ] 1066*, 1067*, 1068*, [ 7, 29, 70, 125, 196, 284, 388, 508, 644, 796 ] 1069*, [ 7, 30, 70, 124, 196, 284, 388, 508, 644, 796 ] 1070*, [ 8, 26, 56, 98, 152, 218, 296, 386, 488, 602 ] 935 H431 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y ,mztz, it 1 z , mzt 1 z 936 H461 1 r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y , r 2 y tz, r 2 x tz, mzt 1 z [ 8, 26, 57, 102, 159, 227, 308, 402, 509, 629 ] 937 H468 1 r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y ,mztz, it 1 z , mzt 1 z [ 8, 26, 57, 102, 161, 234, 321, 421, 534, 661 ] 938 H460 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y tz, r 2 x tz, mzt 1 z [ 8, 26, 59, 104, 163, 234, 319, 416, 527, 650 ] 939 H433 1 r 2 z it 1 x , ty, r 2 z t 1 y , t 1 y ,mztz, it 1 z , mzt 1 z [ 8, 26, 59, 105, 165, 235, 319, 417, 529, 651 ] 940 H466 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 x t 1 y ,mztz, it 1 z , mzt 1 z [ 8, 27, 58, 102, 159, 227, 309, 403, 509, 629 ] 941 H469 1 r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 27, 58, 103, 161, 232, 318, 417, 530, 657 ] 942 H460 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y tz, it 1 z , mzt 1 z [ 8, 27, 59, 102, 160, 229, 310, 406, 513, 631 ] 943 H433 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 z t 1 y ,mztz, it 1 z , mzt 1 z [ 8, 27, 59, 103, 162, 232, 315, 411, 520, 641 ] 944 H485 1 r 2 z mztx, it 1 x ,mzt 1 x ,myt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 27, 59, 104, 161, 231, 315, 410, 518, 640 ] 945 H461 1 r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y , r 2 y tz, it 1 z , mzt 1 z [ 8, 27, 59, 104, 165, 239, 328, 430, 546, 674 ] 946 H462 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 z t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 27, 59, 105, 165, 238, 325, 425, 538, 665 ] 947 H425 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 27, 59, 105, 166, 238, 325, 425, 538, 663 ] 948 H475 1 r 2 z mztx, it 1 x ,mzt 1 x ,myt 1 y ,mztz, it 1 z , mzt 1 z [ 8, 27, 60, 105, 164, 235, 320, 417, 528, 651 ] 949 H434 1 r 2 z mztx, it 1 x ,mzt 1 x , it 1 y ,mztz, it 1 z , mzt 1 z [ 8, 27, 60, 105, 165, 236, 321, 418, 529, 652 ] 950 H474 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 73 Nbr. gr Hi L m X [ 8, 27, 60, 107, 167, 240, 327, 427, 540, 667 ] 951 H481 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y ,mztz, it 1 z , mzt 1 z 952 H483 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 x tz, mzt 1 z [ 8, 27, 61, 106, 165, 236, 321, 418, 529, 652 ] 953 H427 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 954 H469 1 r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 27, 61, 107, 171, 249, 340, 441, 563, 699 ] 955 H462 1 r 2 z it 1 x , ty, r 2 z t 1 y , t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 27, 61, 108, 170, 243, 332, 433, 548, 675 ] 956 H473 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 x t 1 y ,mztz, it 1 z , mzt 1 z [ 8, 27, 61, 108, 171, 247, 340, 444, 564, 696 ] 957 H471 1 r 2 z mztx, it 1 x ,mzt 1 x , it 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 27, 61, 112, 181, 268, 372, 492, 628, 780 ] 958 H452 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y rztz, r 2 x rztz, mzt 1 z 959 H487 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y ,mzr 1 z tz,mzrztz, mzt 1 z [ 8, 27, 62, 108, 169, 242, 329, 428, 542, 668 ] 960 H461 1 r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y , r 2 x tz, it 1 z , mzt 1 z [ 8, 27, 63, 108, 171, 246, 338, 442, 562, 694 ] 961 H460 1 r 2 z r 2 z t 1 x , ty, r 2 z t 1 y , t 1 y , r 2 y tz, it 1 z , mzt 1 z [ 8, 27, 64, 110, 174, 250, 342, 446, 566, 698 ] 962 H425 1 r 2 z r 2 z t 1 x , ty, r 2 z t 1 y , t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 27, 65, 113, 177, 253, 345, 449, 569, 701 ] 963 H486 1 r 2 z r 2 y t 1 x ,mxty,mxt 1 y ,myt 1 y ,mztz, it 1 z , mzt 1 z [ 8, 28, 61, 106, 165, 236, 321, 418, 529, 652 ] 964 H427 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 28, 61, 108, 168, 241, 328, 428, 541, 668 ] 965 H483 1 r 2 z mytx,mxt 1 x ,myt 1 x ,myt 1 y , r 2 y tz, it 1 z , mzt 1 z 966 H482 1 r 2 z mytx,mxt 1 x ,myt 1 x ,myt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 28, 62, 106, 165, 236, 321, 418, 529, 652 ] 967 H433 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 z t 1 y , itz, it 1 z , mzt 1 z 968 H485 1 r 2 z mztx, it 1 x ,mzt 1 x ,myt 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 969 H431 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , itz, it 1 z , mzt 1 z [ 8, 28, 62, 109, 169, 242, 329, 429, 542, 669 ] 970 H468 1 r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y ,mztz, it 1 z , mzt 1 z 971 H461 1 r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 x tz, mzt 1 z [ 8, 28, 63, 108, 169, 242, 329, 428, 542, 668 ] 972 H461 1 r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y , r 2 y tz, r 2 x tz, it 1 z [ 8, 28, 63, 108, 171, 246, 338, 442, 562, 694 ] 973 H460 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y tz, r 2 x tz, it 1 z [ 8, 28, 63, 110, 173, 248, 338, 441, 559, 689 ] 974 H486 1 r 2 z r 2 x tx, r 2 y t 1 x , r 2 x t 1 x ,myt 1 y ,mztz, it 1 z , mzt 1 z 975 H479 1 r 2 z r 2 x tx, r 2 y t 1 x , r 2 x t 1 x ,myt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 28, 63, 110, 174, 250, 342, 446, 566, 698 ] 976 H462 1 r 2 z it 1 x , ty, r 2 z t 1 y , t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 28, 63, 111, 175, 251, 343, 447, 567, 699 ] 977 H477 1 r 2 z r 2 y t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y ,mztz, it 1 z , mzt 1 z 74 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 978 H429 1 r 2 z r 2 y t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 28, 63, 113, 181, 267, 370, 490, 626, 778 ] 979 H452 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 x rztz, it 1 z , mzt 1 z [ 8, 28, 63, 115, 185, 272, 376, 496, 632, 784 ] 980 H503 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y rztz, r 2 x rztz, r 2 y t 1 z 981 H519 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y ,mzr 1 z tz,mzrztz, r 2 y t 1 z [ 8, 28, 64, 110, 171, 244, 333, 434, 549, 676 ] 982 H475 1 r 2 z mztx, it 1 x ,mzt 1 x ,myt 1 y , itz, it 1 z , mzt 1 z 983 H468 1 r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y , itz, it 1 z , mzt 1 z [ 8, 28, 64, 110, 174, 250, 342, 446, 566, 698 ] 984 H462 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 z t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 28, 64, 112, 176, 252, 344, 448, 568, 700 ] 985 H483 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, it 1 z , mzt 1 z 986 H482 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 28, 64, 115, 185, 271, 375, 494, 631, 782 ] 987 H503 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 x rztz, r 2 y t 1 z , r 2 x t 1 z [ 8, 28, 64, 117, 187, 274, 378, 498, 634, 786 ] 988 H527 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, r 2 x rztz, mzt 1 z 989 H528 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y ,mzr 1 z tz,mzrztz, mzt 1 z [ 8, 28, 65, 113, 174, 249, 341, 445, 561, 690 ] 990 H541 1 r 2 z r 2 z rxtx,mxty,mxt 1 y ,myt 1 y ,mztz, it 1 z , mzt 1 z 991 H541 1 r 2 z r 2 z rxtx,mxty,myty,mxt 1 y ,mztz, it 1 z , mzt 1 z [ 8, 28, 65, 113, 177, 253, 345, 449, 569, 701 ] 992 H479 1 r 2 z r 2 y t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 28, 66, 114, 178, 254, 346, 450, 570, 702 ] 993 H475 1 r 2 z it 1 x ,mxty,mxt 1 y ,myt 1 y ,mztz, it 1 z , mzt 1 z [ 8, 28, 66, 117, 186, 273, 376, 496, 632, 784 ] 994 H503 1 r 2 z r 2 z t 1 x , ty, r 2 z t 1 y , t 1 y , r 2 y rztz, r 2 x rztz, r 2 y t 1 z 995 H519 1 r 2 z r 2 z t 1 x , ty, r 2 z t 1 y , t 1 y ,mzr 1 z tz,mzrztz, r 2 y t 1 z [ 8, 29, 61, 106, 165, 236, 321, 418, 529, 652 ] 996 H434 1 r 2 z mztx, it 1 x ,mzt 1 x , it 1 y , itz, it 1 z , mzt 1 z 997 H433 1 r 2 z it 1 x , ty, r 2 z t 1 y , t 1 y , itz, it 1 z , mzt 1 z 998 H474 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 29, 62, 109, 169, 242, 329, 429, 542, 669 ] 999 H461 1 r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz, it 1 z , mzt 1 z 1000 H469 1 r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 29, 63, 110, 171, 244, 333, 434, 549, 676 ] 1001 H473 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 x t 1 y , itz, it 1 z , mzt 1 z 1002 H466 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 x t 1 y , itz, it 1 z , mzt 1 z [ 8, 29, 63, 110, 174, 250, 342, 446, 566, 698 ] 1003 H471 1 r 2 z it 1 x ,mzty, it 1 y ,mzt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 29, 64, 111, 174, 249, 339, 442, 560, 690 ] 1004 H466 1 r 2 z r 2 z t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y ,mztz, it 1 z , mzt 1 z 1005 H427 1 r 2 z r 2 z t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 29, 64, 112, 176, 252, 344, 448, 568, 700 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 75 Nbr. gr Hi L m X 1006 H473 1 r 2 z it 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y ,mztz, it 1 z , mzt 1 z 1007 H474 1 r 2 z it 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 29, 65, 113, 176, 251, 343, 447, 566, 697 ] 1008 H511 1 r 2 z r 2 z rxtx, r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 1009 H511 1 r 2 z r 2 z rxtx, r 2 y ty, r 2 x ty, r 2 y t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 29, 65, 113, 177, 253, 345, 449, 569, 701 ] 1010 H479 1 r 2 z r 2 y t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 29, 65, 115, 183, 269, 372, 492, 628, 780 ] 1011 H452 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y rztz, r 2 x rztz, it 1 z 1012 H487 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y ,mzr 1 z tz,mzrztz, it 1 z [ 8, 29, 65, 118, 188, 275, 379, 499, 635, 787 ] 1013 H529 1 r 2 z mytx,mxt 1 x ,myt 1 x ,myt 1 y , r 2 y rztz, r 2 x rztz, r 2 y t 1 z 1014 H530 1 r 2 z mytx,mxt 1 x ,myt 1 x ,myt 1 y ,mzr 1 z tz,mzrztz, r 2 y t 1 z [ 8, 29, 65, 121, 189, 278, 379, 500, 634, 788 ] 1015 H539 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 x ty,mzr 1 z tz,mzrztz, r 2 y rzt 1 z [ 8, 29, 65, 121, 190, 280, 382, 504, 638, 792 ] 1016 H453 1 r 2 z mztx, it 1 x ,mzt 1 x , it 1 y , r 2 x rztz, r 2 y rzt 1 z , r 2 x rzt 1 z [ 8, 29, 66, 113, 177, 253, 345, 449, 569, 701 ] 1017 H486 1 r 2 z r 2 x tx, r 2 y t 1 x , r 2 x t 1 x ,myt 1 y , itz, it 1 z , mzt 1 z 1018 H479 1 r 2 z r 2 x tx, r 2 y t 1 x , r 2 x t 1 x ,myt 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 1019 H481 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , itz, it 1 z , mzt 1 z 1020 H483 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 x tz, it 1 z [ 8, 29, 66, 114, 178, 254, 346, 450, 570, 702 ] 1021 H485 1 r 2 z it 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 29, 66, 115, 179, 254, 346, 450, 570, 702 ] 1022 H511 1 r 2 z r 2 z rxtx, r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y tz, r 2 x tz, r 2 y t 1 z 1023 H541 1 r 2 z r 2 z rxtx,mxty,mxt 1 y ,myt 1 y , itz,mztz, it 1 z [ 8, 29, 66, 119, 188, 275, 378, 498, 634, 786 ] 1024 H529 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, r 2 x rztz, r 2 y t 1 z 1025 H530 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y ,mzr 1 z tz,mzrztz, r 2 y t 1 z [ 8, 29, 66, 120, 191, 279, 383, 503, 639, 791 ] 1026 H438 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 x rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 1027 H495 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y ,mzr 1 z tz,mzrztz, r 2 x rzt 1 z [ 8, 29, 67, 114, 178, 254, 346, 450, 570, 702 ] 1028 H461 1 r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, it 1 z , mzt 1 z 1029 H469 1 r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 29, 67, 119, 189, 276, 379, 499, 635, 787 ] 1030 H527 1 r 2 z mxt 1 x ,mxty,myty,mxt 1 y , r 2 y rztz, it 1 z , mzt 1 z 1031 H529 1 r 2 z mxt 1 x ,mxty,myty,mxt 1 y , r 2 y rztz, r 2 y t 1 z , r 2 x t 1 z [ 8, 29, 68, 124, 194, 281, 385, 506, 641, 793 ] 1032 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,myt 1 y ,mztz, it 1 z , mzt 1 z 1033 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,myty,mztz, it 1 z , mzt 1 z [ 8, 29, 71, 126, 196, 284, 388, 508, 644, 796 ] 76 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 1034 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 30, 64, 115, 183, 269, 372, 492, 628, 780 ] 1035 H452 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y rztz, it 1 z , mzt 1 z [ 8, 30, 65, 113, 177, 253, 345, 449, 569, 701 ] 1036 H477 1 r 2 z r 2 y t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , itz, it 1 z , mzt 1 z 1037 H429 1 r 2 z r 2 y t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 1038 H486 1 r 2 z r 2 y t 1 x ,mxty,mxt 1 y ,myt 1 y , itz, it 1 z , mzt 1 z [ 8, 30, 65, 117, 186, 273, 376, 496, 632, 784 ] 1039 H503 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y rztz, r 2 y t 1 z , r 2 x t 1 z [ 8, 30, 66, 114, 178, 254, 346, 450, 570, 702 ] 1040 H485 1 r 2 z it 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 30, 67, 114, 178, 254, 346, 450, 570, 702 ] 1041 H466 1 r 2 z r 2 z t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , itz, it 1 z , mzt 1 z 1042 H427 1 r 2 z r 2 z t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 1043 H468 1 r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y , itz, it 1 z , mzt 1 z 1044 H461 1 r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 x tz, it 1 z [ 8, 30, 68, 120, 189, 276, 379, 499, 635, 787 ] 1045 H527 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, r 2 x rztz, it 1 z 1046 H528 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y ,mzr 1 z tz,mzrztz, it 1 z [ 8, 30, 68, 125, 194, 284, 386, 508, 642, 796 ] 1047 H505 1 r 2 z r 2 y t 1 x , r 2 y ty, r 2 x ty, r 2 y t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z [ 8, 30, 69, 124, 195, 283, 387, 507, 643, 795 ] 1048 H496 1 r 2 z mxt 1 x ,mxty,myty,mxt 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 1049 H502 1 r 2 z mxt 1 x ,mxty,myty,mxt 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z [ 8, 30, 73, 125, 196, 284, 388, 508, 644, 796 ] 1050 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,myt 1 y , itz, it 1 z , mzt 1 z [ 8, 31, 66, 114, 178, 254, 346, 450, 570, 702 ] 1051 H473 1 r 2 z it 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , itz, it 1 z , mzt 1 z 1052 H474 1 r 2 z it 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 1053 H475 1 r 2 z it 1 x ,mxty,mxt 1 y ,myt 1 y , itz, it 1 z , mzt 1 z 1054 H511 1 r 2 z r 2 z rxtx, r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 1055 H541 1 r 2 z r 2 z rxtx,mxty,mxt 1 y ,myt 1 y , itz, it 1 z , mzt 1 z [ 8, 31, 67, 120, 189, 276, 379, 499, 635, 787 ] 1056 H527 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, it 1 z , mzt 1 z 1057 H529 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, r 2 y t 1 z , r 2 x t 1 z [ 8, 31, 67, 121, 191, 278, 381, 500, 636, 788 ] 1058 H539 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 x t 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z [ 8, 31, 67, 121, 192, 280, 384, 504, 640, 792 ] 1059 H453 1 r 2 z mztx, it 1 x ,mzt 1 x , it 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 1060 H438 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 1061 H495 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z [ 8, 31, 69, 126, 195, 285, 387, 509, 643, 797 ] 1062 H539 1 r 2 z it 1 x , r 2 y ty, r 2 x ty, r 2 y t 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z [ 8, 31, 70, 125, 196, 284, 388, 508, 644, 796 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 77 Nbr. gr Hi L m X 1063 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 x t 1 y , r 2 y tz, r 2 x tz, r 2 y t 1 z [ 8, 31, 70, 127, 196, 284, 388, 508, 644, 796 ] 1064 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 x ty, r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 31, 71, 126, 196, 284, 388, 508, 644, 796 ] 1065 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 8, 32, 69, 124, 195, 283, 387, 507, 643, 795 ] 1066 H505 1 r 2 z r 2 y t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 1067 H496 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 1068 H502 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z [ 8, 32, 71, 125, 196, 284, 388, 508, 644, 796 ] 1069 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,myt 1 y , itz,mztz, it 1 z [ 8, 33, 70, 125, 196, 284, 388, 508, 644, 796 ] 1070 H539 1 r 2 z it 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z 16 [ 8, 27, 59, 104, 163, 235, 320, 419, 531, 656 ] 1071*, [ 8, 27, 60, 106, 167, 242, 329, 431, 547, 675 ] 1072*, 1075*, [ 8, 27, 60, 107, 168, 243, 332, 435, 552, 683 ] 1077*, [ 8, 27, 60, 107, 169, 245, 334, 436, 552, 682 ] 1073*, [ 8, 27, 61, 108, 168, 243, 332, 434, 550, 680 ] 1074*, 1078*, [ 8, 27, 61, 109, 169, 242, 331, 435, 551, 679 ] 1079*, [ 8, 27, 61, 109, 170, 245, 335, 438, 554, 685 ] 1076*, [ 8, 27, 61, 109, 170, 245, 335, 439, 556, 687 ] 1081*, [ 8, 27, 61, 109, 171, 246, 335, 439, 556, 687 ] 1080*, [ 8, 27, 61, 109, 172, 249, 339, 443, 561, 693 ] 1082*, 1083*, [ 8, 27, 62, 111, 173, 249, 339, 443, 561, 693 ] 1084*, [ 8, 28, 64, 112, 173, 249, 339, 443, 561, 693 ] 1086*, 1087*, 1088*, 1089*, 1090*, [ 8, 28, 64, 117, 188, 276, 380, 500, 636, 788 ] 1085*, [ 8, 29, 64, 111, 173, 249, 339, 443, 561, 693 ] 1091*, 1092*, 1093*, 1094*, 1095*, 1096*, 1097*, 1098*, 1099*, 1100*, 1101*, [ 8, 29, 67, 120, 189, 276, 380, 500, 636, 788 ] 1103*, [ 8, 29, 67, 121, 193, 282, 385, 505, 642, 793 ] 1102*, [ 8, 29, 68, 124, 196, 284, 388, 508, 644, 796 ] 1104*, [ 8, 30, 67, 119, 189, 276, 380, 500, 636, 788 ] 1105*, 1106*, 1107*, 1108*, [ 8, 30, 70, 125, 196, 284, 388, 508, 644, 796 ] 1109*, [ 8, 31, 70, 124, 196, 284, 388, 508, 644, 796 ] 1110*, 1111*, 1112*, 1113*, 1114*, 1115*, 1116*, 1117*, 1118*, [ 9, 29, 62, 109, 169, 242, 329, 429, 542, 669 ] 1071 H680 hmxi r 2 z r 2 z tx,mxtx,myt 1 x ,mxt 1 x ,myt 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z [ 9, 29, 63, 110, 173, 247, 337, 440, 556, 686 ] 1072 H359 1 mz itx, r 2 z tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , myt 1 y ,mzt 1 z [ 9, 29, 63, 111, 174, 250, 341, 444, 562, 692 ] 1073 H377 1 mz r 2 y tx,mxtx,mzr 1 z t 1 x , r 1 z t 1 x ,mzrzty, rzty, myt 1 y ,mzt 1 z [ 9, 29, 64, 111, 174, 249, 339, 442, 560, 690 ] 1074 H682 hmxi r 2 z itx, r 2 y tx, r 2 x t 1 x , r 2 y t 1 x ,myt 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z [ 9, 29, 64, 111, 175, 250, 339, 443, 560, 689 ] 1075 H382 1 mz r 2 y tx,mxtx,mzr 1 z t 1 x , r 1 z t 1 x ,mzrzty, rzty, myt 1 y , r 2 y rzt 1 z [ 9, 29, 64, 112, 175, 251, 342, 445, 564, 695 ] 1076 H343 1 mz r 2 z tx, r 2 y rzt 1 x ,mxrzt 1 x , ity, r 2 z ty, r 2 y rzt 1 y , mxrzt 1 y ,mzt 1 z [ 9, 29, 64, 112, 176, 252, 344, 448, 568, 700 ] 78 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 1077 H683 hmxi r 2 z itx, r 2 y tx, r 2 x t 1 x , r 2 y t 1 x , r 2 x t 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z [ 9, 29, 65, 112, 175, 251, 341, 444, 563, 693 ] 1078 H347 1 mz r 2 z tx, r 2 y rzt 1 x ,mxrzt 1 x , ity, r 2 z ty, r 2 y rzt 1 y , mxrzt 1 y , r 2 x rzt 1 z [ 9, 29, 65, 113, 174, 249, 341, 445, 561, 690 ] 1079 H541 1 r 2 z r 2 z rxtx,mxty,myty,mxt 1 y ,myt 1 y ,mztz, it 1 z ,mzt 1 z [ 9, 29, 65, 113, 176, 251, 343, 447, 566, 697 ] 1080 H511 1 r 2 z r 2 z rxtx, r 2 y ty, r 2 x ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 9, 29, 65, 113, 176, 252, 344, 448, 568, 700 ] 1081 H344 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , ity, r 2 z ty, it 1 y , r 2 z t 1 y , r 2 y rzt 1 z [ 9, 29, 65, 113, 178, 254, 346, 450, 570, 702 ] 1082 H370 1 mz itx, r 2 z tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , myt 1 y , r 2 x t 1 z 1083 H378 1 mz r 2 y tx,mxtx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , myt 1 y , r 2 y rzt 1 z [ 9, 29, 66, 114, 178, 254, 346, 450, 570, 702 ] 1084 H682 hmyi mx r 2 y t 1 x , r 2 z ty,myty,mxt 1 y , t 1 y ,mztz, r 2 y t 1 z ,mzt 1 z [ 9, 30, 66, 119, 189, 276, 380, 500, 636, 788 ] 1085 H690 hmxi r 2 z r 2 z tx,mxtx,myt 1 x ,mxt 1 x ,myt 1 y , r 2 y rztz, mzr 1 z tz, r 2 y t 1 z [ 9, 30, 67, 114, 178, 254, 346, 450, 570, 702 ] 1086 H682 hmxi r 2 z itx, r 2 y tx, r 2 x t 1 x , r 2 y t 1 x ,myt 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z 1087 H680 hmxi r 2 z r 2 z tx,mxtx,myt 1 x ,mxt 1 x ,myt 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z 1088 H359 1 mz itx, r 2 z tx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y , myt 1 y ,mzt 1 z 1089 H343 1 mz itx, r 2 y rzt 1 x ,mxrzt 1 x , ity, r 2 z ty, r 2 y rzt 1 y , mxrzt 1 y ,mzt 1 z 1090 H377 1 mz r 2 y tx,mxtx,mzr 1 z t 1 x , r 1 z t 1 x ,mzrzty, rzty, r 2 x t 1 y ,mzt 1 z [ 9, 31, 66, 114, 178, 254, 346, 450, 570, 702 ] 1091 H683 hmxi r 2 z itx, r 2 y tx, r 2 x t 1 x , r 2 y t 1 x , r 2 x t 1 y , r 2 x tz, r 2 y tz, r 2 x t 1 z 1092 H344 1 mz itx, it 1 x , r 2 z t 1 x , ity, r 2 z ty, it 1 y , r 2 z t 1 y , r 2 y rzt 1 z 1093 H344 1 mz itx, r 2 z tx, it 1 x , r 2 z t 1 x , ity, r 2 z ty, it 1 y , r 2 y rzt 1 z 1094 H683 hmxi r 2 z itx, r 2 y tx, r 2 x t 1 x , r 2 y t 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z 1095 H682 hmyi mx r 2 y t 1 x , r 2 z ty,myty,mxt 1 y , t 1 y , r 2 y tz, r 2 y t 1 z ,mzt 1 z 1096 H370 1 mz itx, r 2 z tx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y , myt 1 y , r 2 x t 1 z 1097 H347 1 mz itx, r 2 y rzt 1 x ,mxrzt 1 x , ity, r 2 z ty, r 2 y rzt 1 y , mxrzt 1 y , r 2 x rzt 1 z 1098 H378 1 mz r 2 y tx,mxtx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y , myt 1 y , r 2 y rzt 1 z 1099 H382 1 mz r 2 y tx,mxtx,mzr 1 z t 1 x , r 1 z t 1 x ,mzrzty, rzty, r 2 x t 1 y , r 2 y rzt 1 z 1100 H511 1 r 2 z r 2 z rxtx, r 2 y ty, r 2 x ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 1101 H541 1 r 2 z r 2 z rxtx,mxty,myty,mxt 1 y ,myt 1 y , itz, it 1 z ,mzt 1 z [ 9, 31, 68, 122, 194, 281, 385, 506, 641, 793 ] 1102 H541 1 r 2 z r 2 z rxtx, rxtx, r 2 y rxt 1 x , r 1 x t 1 x ,myt 1 y ,mztz, it 1 z ,mzt 1 z [ 9, 31, 69, 121, 190, 277, 380, 500, 636, 788 ] 1103 H690 hmxi r 2 z r 2 z tx,mxtx,myt 1 x ,mxt 1 x ,myt 1 y , r 2 y rztz, mzr 1 z tz, r 2 x t 1 z [ 9, 31, 70, 125, 196, 284, 388, 508, 644, 796 ] 1104 H511 1 r 2 z r 2 z rxtx, rxtx, r 2 y rxt 1 x , r 1 x t 1 x , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z [ 9, 32, 68, 121, 190, 277, 380, 500, 636, 788 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 79 Nbr. gr Hi L m X 1105 H452 1 r 2 z r 2 z t 1 x , r 2 z ty, ty, r 2 z t 1 y , t 1 y , r 2 y rztz, it 1 z ,mzt 1 z 1106 H503 1 r 2 z r 2 z t 1 x , r 2 z ty, ty, r 2 z t 1 y , t 1 y , r 2 y rztz, r 2 y t 1 z , r 2 x t 1 z 1107 H527 1 r 2 z mxt 1 x ,mxty,myty,mxt 1 y ,myt 1 y , r 2 y rztz, it 1 z ,mzt 1 z 1108 H529 1 r 2 z mxt 1 x ,mxty,myty,mxt 1 y ,myt 1 y , r 2 y rztz, r 2 y t 1 z , r 2 x t 1 z [ 9, 32, 71, 125, 196, 284, 388, 508, 644, 796 ] 1109 H541 1 r 2 z r 2 z rxtx, rxtx, r 2 y rxt 1 x , r 1 x t 1 x ,myt 1 y , itz, it 1 z ,mzt 1 z [ 9, 33, 70, 125, 196, 284, 388, 508, 644, 796 ] 1110 H453 1 r 2 z it 1 x , ity,mzty, it 1 y ,mzt 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 1111 H539 1 r 2 z itx,mztx, it 1 x ,mzt 1 x , r 2 x t 1 y ,mzr 1 z tz, mzrztz, r 2 y rzt 1 z 1112 H539 1 r 2 z it 1 x , r 2 y ty, r 2 x ty, r 2 y t 1 y , r 2 x t 1 y ,mzr 1 z tz, mzrztz, r 2 y rzt 1 z 1113 H438 1 r 2 z r 2 z t 1 x , r 2 z ty, ty, r 2 z t 1 y , t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 1114 H495 1 r 2 z r 2 z t 1 x , r 2 z ty, ty, r 2 z t 1 y , t 1 y ,mzr 1 z tz, mzrztz, r 2 y rzt 1 z 1115 H511 1 r 2 z r 2 z rxtx, rxtx, r 2 y rxt 1 x , r 1 x t 1 x , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 1116 H505 1 r 2 z r 2 y t 1 x , r 2 y ty, r 2 x ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 1117 H496 1 r 2 z mxt 1 x ,mxty,myty,mxt 1 y ,myt 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 1118 H502 1 r 2 z mxt 1 x ,mxty,myty,mxt 1 y ,myt 1 y ,mzr 1 z tz, mzrztz, r 2 y rzt 1 z 17A [ 5, 14, 29, 50, 77, 110, 149, 194, 245, 302 ] 1119*, 1120*, [ 5, 14, 30, 52, 79, 114, 155, 200, 254, 314 ] 1121*, [ 5, 14, 33, 67, 114, 168, 227, 302, 391, 481 ] 1122*, [ 5, 16, 35, 60, 93, 134, 181, 236, 299, 368 ] 1123*, 1124*, [ 5, 16, 36, 63, 97, 139, 189, 247, 312, 384 ] 1125*, [ 5, 16, 38, 75, 128, 191, 266, 356, 456, 568 ] 1126*, [ 6, 19, 41, 70, 110, 157, 212, 278, 351, 431 ] 1119 H307 1 i r 2 z rxt 1 x , r 2 z rxt 1 y ,mzrxt 1 y ,mzrxtz, r 2 z rxt 1 z [ 6, 19, 42, 72, 112, 160, 218, 284, 359, 442 ] 1120 H310 1 i mxt 1 x , r 2 z rxt 1 y ,mzrxt 1 y ,mzrxtz, r 2 z rxt 1 z [ 6, 19, 43, 73, 114, 165, 222, 290, 368, 451 ] 1121 H322 1 i mxt 1 x ,mxr 1 x t 1 y , rxt 1 y ,mxrxtz, r 1 x t 1 z [ 6, 19, 45, 82, 132, 195, 268, 356, 455, 559 ] 1122 H325 1 i r 2 z rxt 1 x ,mxrxt 1 y , r 1 x t 1 y , rxtz,mxr 1 x t 1 z [ 6, 21, 45, 78, 122, 175, 236, 310, 391, 481 ] 1123 H306 1 i it 1 x , r 2 z rxt 1 y ,mzrxt 1 y ,mzrxtz, r 2 z rxt 1 z [ 6, 21, 46, 80, 125, 178, 243, 316, 401, 493 ] 1124 H309 1 i r 2 z rxt 1 x , r 2 y rxt 1 y ,mzr 1 x t 1 y , r 2 y rxtz,mzr 1 x t 1 z [ 6, 21, 47, 82, 128, 183, 250, 325, 411, 507 ] 1125 H324 1 i r 2 z rxt 1 x ,mxr 1 x t 1 y , rxt 1 y ,mxrxtz, r 1 x t 1 z [ 6, 21, 48, 89, 144, 207, 288, 382, 486, 602 ] 1126 H321 1 i it 1 x ,mxr 1 x t 1 y , rxt 1 y ,mxrxtz, r 1 x t 1 z 17B [ 5, 15, 37, 74, 122, 178, 244, 322, 410, 508 ] 1127*, [ 5, 15, 38, 78, 132, 200, 282, 378, 488, 612 ] 1128*, [ 5, 15, 40, 82, 134, 200, 278, 368, 472, 590 ] 1129*, [ 5, 15, 40, 83, 142, 216, 300, 393, 500, 622 ] 1130*, [ 5, 15, 40, 89, 152, 230, 319, 422, 541, 670 ] 1131*, [ 5, 15, 41, 87, 146, 221, 306, 403, 513, 637 ] 1132*, [ 5, 16, 39, 74, 119, 174, 239, 314, 399, 494 ] 1133*, 1134*, [ 5, 16, 39, 76, 127, 192, 271, 364, 471, 592 ] 1135*, [ 5, 16, 42, 80, 131, 196, 274, 364, 467, 584 ] 80 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 1138*, [ 5, 16, 42, 81, 128, 185, 253, 332, 421, 520 ] 1136*, 1137*, [ 5, 17, 44, 83, 131, 190, 259, 339, 430, 531 ] 1139*, [ 5, 17, 46, 93, 155, 230, 317, 421, 541, 670 ] 1140*, [ 5, 17, 46, 95, 156, 229, 315, 419, 536, 665 ] 1141*, [ 5, 17, 48, 95, 155, 227, 314, 415, 527, 655 ] 1142*, [ 5, 18, 43, 78, 123, 178, 243, 318, 403, 498 ] 1143*, 1144*, [ 5, 18, 45, 89, 148, 223, 311, 413, 528, 659 ] 1145*, [ 5, 18, 47, 93, 151, 223, 312, 411, 523, 653 ] 1146*, [ 6, 20, 50, 93, 148, 213, 291, 382, 484, 598 ] 1127 H420 1 r 2 z rx mxt 1 x ,mxrxt 1 y ,mytz,mxr 1 x t 1 z ,myt 1 z [ 6, 20, 51, 96, 157, 230, 319, 420, 537, 666 ] 1128 H421 1 r 2 z rx mxt 1 x , rxt 1 y , r 2 z tz, r 2 z t 1 z , r 1 x t 1 z [ 6, 20, 52, 96, 151, 223, 305, 401, 507, 631 ] 1129 H423 1 r 2 z rx r 2 z tx,mxrxt 1 y ,mytz,mxr 1 x t 1 z ,myt 1 z [ 6, 20, 52, 97, 158, 232, 318, 413, 523, 647 ] 1130 H423 1 r 2 z rx r 2 z tx,mzty,mxrxt 1 y ,mzt 1 y ,mxr 1 x t 1 z [ 6, 20, 52, 103, 163, 241, 328, 437, 554, 689 ] 1131 H416 1 r 2 z rx it 1 x ,mxrxt 1 y ,mytz,mxr 1 x t 1 z ,myt 1 z [ 6, 20, 53, 101, 161, 237, 320, 422, 533, 661 ] 1132 H411 1 r 2 z rx r 2 y rxt 1 x , rxt 1 y , r 2 z tz, r 2 z t 1 z , r 1 x t 1 z [ 6, 21, 50, 90, 142, 205, 280, 366, 464, 573 ] 1133 H387 1 r 2 z rx it 1 x , ty, r 2 z rxt 1 y , t 1 y , r 2 z rxt 1 z [ 6, 21, 51, 92, 145, 209, 286, 374, 474, 585 ] 1134 H384 1 r 2 z rx mxt 1 x , ty, r 2 z rxt 1 y , t 1 y , r 2 z rxt 1 z [ 6, 21, 51, 94, 154, 225, 313, 412, 528, 655 ] 1135 H386 1 r 2 z rx mxt 1 x ,mxty,mxt 1 y ,mzr 1 x t 1 y ,mzr 1 x t 1 z [ 6, 21, 53, 94, 148, 213, 291, 380, 481, 593 ] 1136 H406 1 r 2 z rx r 2 z tx, ty, r 2 z rxt 1 y , t 1 y , r 2 z rxt 1 z 1137 H406 1 r 2 z rx r 2 z tx, r 2 z rxt 1 y , tz, r 2 z rxt 1 z , t 1 z [ 6, 21, 53, 94, 153, 220, 305, 398, 509, 628 ] 1138 H399 1 r 2 z rx r 2 y rxt 1 x ,mxty,mxt 1 y ,mzr 1 x t 1 y ,mzr 1 x t 1 z [ 6, 22, 55, 97, 153, 220, 299, 391, 495, 610 ] 1139 H405 1 r 2 z rx r 2 y rxt 1 x ,mxrxt 1 y ,mytz,mxr 1 x t 1 z ,myt 1 z [ 6, 22, 56, 102, 164, 239, 330, 435, 556, 687 ] 1140 H403 1 r 2 z rx r 2 z rxt 1 x ,mxrxt 1 y ,mytz,mxr 1 x t 1 z ,myt 1 z [ 6, 22, 57, 106, 171, 245, 337, 442, 563, 693 ] 1141 H417 1 r 2 z rx it 1 x , rxt 1 y , r 2 z tz, r 2 z t 1 z , r 1 x t 1 z [ 6, 22, 58, 104, 168, 238, 330, 430, 546, 674 ] 1142 H408 1 r 2 z rx r 2 z rxt 1 x , rxt 1 y , r 2 z tz, r 2 z t 1 z , r 1 x t 1 z [ 6, 23, 52, 92, 144, 207, 282, 368, 466, 575 ] 1143 H383 1 r 2 z rx r 2 z rxt 1 x , ty, r 2 z rxt 1 y , t 1 y , r 2 z rxt 1 z [ 6, 23, 53, 94, 147, 211, 288, 376, 476, 587 ] 1144 H390 1 r 2 z rx r 2 y rxt 1 x , ty, r 2 z rxt 1 y , t 1 y , r 2 z rxt 1 z [ 6, 23, 55, 102, 167, 243, 335, 438, 559, 691 ] 1145 H397 1 r 2 z rx it 1 x ,mxty,mxt 1 y ,mzr 1 x t 1 y ,mzr 1 x t 1 z [ 6, 23, 56, 104, 164, 238, 328, 428, 544, 674 ] 1146 H385 1 r 2 z rx r 2 z rxt 1 x ,mxty,mxt 1 y ,mzr 1 x t 1 y ,mzr 1 x t 1 z 18A [ 7, 22, 47, 82, 127, 182, 247, 322, 407, 502 ] 1147*, 1148*, [ 7, 22, 48, 85, 132, 189, 257, 336, 425, 524 ] 1149*, 1150*, [ 7, 23, 50, 87, 135, 194, 263, 343, 434, 535 ] 1152*, [ 7, 23, 51, 90, 140, 202, 275, 359, 455, 562 ] 1151*, [ 7, 23, 51, 91, 143, 206, 280, 366, 464, 574 ] 1153*, [ 7, 23, 51, 91, 143, 207, 283, 371, 471, 583 ] 1154*, [ 7, 23, 52, 94, 148, 214, 292, 382, 484, 598 ] 1156*, [ 7, 23, 52, 94, 150, 220, 302, 396, 504, 626 ] 1157*, [ 7, 23, 52, 95, 152, 222, 305, 401, 510, 632 ] 1155*, [ 7, 23, 52, 96, 155, 226, 311, 411, 523, 647 ] 1158*, [ 7, 23, 52, 97, 159, 234, 321, 425, 545, 674 ] 1159*, [ 7, 23, 54, 99, 156, 228, 315, 414, 527, 655 ] 1160*, [ 7, 23, 55, 103, 165, 241, 331, 435, 553, 685 ] 1161*, K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 81 Nbr. gr Hi L m X [ 7, 24, 54, 96, 150, 216, 294, 384, 486, 600 ] 1163*, 1165*, [ 7, 24, 55, 99, 156, 226, 309, 405, 514, 636 ] 1162*, 1164*, [ 7, 24, 55, 100, 159, 232, 319, 420, 535, 664 ] 1166*, [ 7, 24, 56, 102, 161, 233, 319, 419, 532, 658 ] 1167*, 1168*, [ 7, 24, 57, 104, 164, 239, 327, 428, 544, 673 ] 1171*, [ 7, 24, 57, 105, 167, 243, 333, 437, 555, 687 ] 1169*, 1170*, [ 7, 24, 58, 107, 168, 243, 333, 436, 552, 683 ] 1172*, [ 7, 25, 56, 98, 152, 218, 296, 386, 488, 602 ] 1175*, [ 7, 25, 57, 101, 158, 228, 311, 407, 516, 638 ] 1173*, 1174*, [ 7, 25, 58, 104, 163, 235, 321, 421, 534, 660 ] 1176*, 1177*, [ 7, 25, 59, 107, 169, 245, 334, 436, 552, 682 ] 1179*, [ 7, 25, 59, 107, 169, 245, 335, 439, 557, 689 ] 1178*, 1180*, 1181*, 1182*, [ 7, 25, 59, 108, 171, 247, 337, 441, 559, 691 ] 1183*, [ 7, 25, 60, 113, 183, 269, 371, 489, 623, 773 ] 1184*, [ 7, 25, 62, 117, 190, 278, 382, 502, 638, 790 ] 1185*, [ 7, 26, 61, 109, 171, 247, 337, 441, 559, 691 ] 1186*, 1187*, 1188*, 1189*, [ 7, 26, 61, 110, 173, 249, 339, 443, 561, 693 ] 1190*, [ 7, 26, 63, 118, 190, 278, 382, 502, 638, 790 ] 1191*, [ 7, 26, 65, 121, 194, 282, 386, 506, 642, 794 ] 1192*, [ 7, 27, 63, 111, 173, 249, 339, 443, 561, 693 ] 1193*, 1194*, [ 7, 27, 65, 120, 192, 280, 384, 504, 640, 792 ] 1195*, 1196*, [ 7, 28, 67, 122, 194, 282, 386, 506, 642, 794 ] 1197*, 1198*, [ 8, 25, 54, 94, 146, 209, 284, 370, 468, 577 ] 1147 H654 hmzrxi r 2 z rx r 2 z rxt 1 x ,mzrxty, it 1 y , t 1 y , tz, r 2 z rxt 1 z , mzrxt 1 z [ 8, 25, 55, 96, 149, 213, 290, 378, 478, 589 ] 1148 H655 hmzrxi r 2 z rx mxt 1 x ,mzrxty, it 1 y , t 1 y , tz, r 2 z rxt 1 z , mzrxt 1 z 1149 H301 1 i tx, it 1 x , t 1 x , ty, it 1 y , t 1 y , r 2 y t 1 z [ 8, 25, 55, 96, 150, 215, 293, 382, 483, 595 ] 1150 H406 1 r 2 z rx r 2 z tx, ty, r 2 z rxt 1 y , t 1 y , tz, r 2 z rxt 1 z , t 1 z [ 8, 26, 57, 99, 154, 221, 300, 391, 495, 610 ] 1151 H303 1 i tx, it 1 x , t 1 x ,mxty,mxt 1 y , r 2 x t 1 y , r 2 x t 1 z [ 8, 26, 57, 99, 155, 222, 301, 393, 497, 612 ] 1152 H660 hmzrxi r 2 z rx mxt 1 x , r 1 x ty, r 2 z t 1 y ,mzt 1 y ,mytz,mxr 1 x t 1 z , rxt 1 z [ 8, 26, 58, 102, 159, 227, 309, 403, 510, 629 ] 1153 H314 1 i tx, it 1 x , t 1 x ,myt 1 y ,mxtz,mxt 1 z , r 2 x t 1 z [ 8, 26, 58, 102, 160, 230, 314, 410, 520, 642 ] 1154 H302 1 i tx, it 1 x , t 1 x ,mzty, r 2 z t 1 y ,mzt 1 y , mzt 1 z [ 8, 26, 58, 102, 160, 231, 316, 413, 524, 647 ] 1155 H301 1 i it 1 x , ty, it 1 y , t 1 y ,mytz, r 2 y t 1 z , myt 1 z [ 8, 26, 58, 102, 161, 232, 317, 414, 525, 648 ] 1156 H315 1 i r 2 z t 1 x ,mzty, r 2 z t 1 y ,mzt 1 y ,mytz, r 2 y t 1 z , myt 1 z [ 8, 26, 58, 102, 161, 233, 319, 415, 525, 649 ] 1157 H423 1 r 2 z rx r 2 z tx,mzty,mxrxt 1 y ,mzt 1 y ,mytz,mxr 1 x t 1 z , myt 1 z [ 8, 26, 58, 104, 165, 237, 326, 427, 541, 667 ] 1158 H312 1 i tx, it 1 x , t 1 x , r 2 z t 1 y ,mxtz,mxt 1 z , r 2 x t 1 z [ 8, 26, 58, 104, 166, 241, 332, 437, 558, 689 ] 1159 H658 hmzrxi r 2 z rx r 2 z rxt 1 x , r 1 x ty, r 2 z t 1 y ,mzt 1 y ,mytz,mxr 1 x t 1 z , rxt 1 z [ 8, 26, 61, 107, 168, 243, 332, 433, 551, 680 ] 82 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 1160 H395 1 r 2 z rx mxt 1 x , r 2 y rxty, it 1 y ,mzrxt 1 y ,mzrxtz, r 2 y rxt 1 z , r 2 x t 1 z [ 8, 26, 61, 108, 172, 248, 340, 444, 564, 696 ] 1161 H311 1 i tx, it 1 x , t 1 x , r 2 z t 1 y ,mytz, r 2 y t 1 z , myt 1 z [ 8, 27, 60, 105, 164, 235, 320, 417, 528, 651 ] 1162 H302 1 i tx, it 1 x , t 1 x , r 2 z t 1 y , r 2 z tz, r 2 z t 1 z , mzt 1 z 1163 H312 1 i tx, it 1 x , t 1 x ,mzty, r 2 z t 1 y ,mzt 1 y , r 2 x t 1 z 1164 H304 1 i mztx, r 2 z t 1 x ,mzt 1 x , r 2 z t 1 y , r 2 z tz, r 2 z t 1 z , mzt 1 z 1165 H317 1 i mztx, r 2 z t 1 x ,mzt 1 x ,myt 1 y ,mytz, r 2 y t 1 z , myt 1 z [ 8, 27, 61, 108, 170, 245, 335, 438, 556, 687 ] 1166 H657 hmzrxi r 2 z rx mxt 1 x , r 2 y rxty,mxt 1 y , r 2 x t 1 y ,mxtz, r 2 y rxt 1 z , mzr 1 x t 1 z [ 8, 27, 62, 109, 170, 243, 332, 433, 548, 675 ] 1167 H313 1 i tx, it 1 x , t 1 x ,myt 1 y , r 2 z tz, r 2 z t 1 z , mzt 1 z 1168 H316 1 i mztx, r 2 z t 1 x ,mzt 1 x ,myt 1 y , r 2 z tz, r 2 z t 1 z , mzt 1 z [ 8, 27, 62, 109, 173, 249, 341, 445, 565, 697 ] 1169 H301 1 i tx, it 1 x , t 1 x , it 1 y ,mytz, r 2 y t 1 z , myt 1 z 1170 H315 1 i mztx, r 2 z t 1 x ,mzt 1 x , r 2 z t 1 y ,mytz, r 2 y t 1 z , myt 1 z [ 8, 27, 63, 110, 173, 249, 338, 441, 559, 688 ] 1171 H413 1 r 2 z rx mxt 1 x ,mxr 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz,mxrxt 1 z , mzt 1 z [ 8, 27, 63, 111, 174, 250, 341, 444, 563, 694 ] 1172 H394 1 r 2 z rx r 2 z rxt 1 x , r 2 y rxty, it 1 y ,mzrxt 1 y ,mzrxtz, r 2 y rxt 1 z , r 2 x t 1 z [ 8, 28, 61, 106, 165, 236, 321, 418, 529, 652 ] 1173 H300 1 i tx, it 1 x , t 1 x , it 1 y , r 2 z tz, r 2 z t 1 z , mzt 1 z 1174 H302 1 i it 1 x ,mzty, r 2 z t 1 y ,mzt 1 y , r 2 z tz, r 2 z t 1 z , mzt 1 z 1175 H319 1 i r 2 z t 1 x ,mxty,mxt 1 y , r 2 x t 1 y ,mytz, r 2 y t 1 z , myt 1 z [ 8, 28, 63, 110, 171, 244, 333, 434, 549, 676 ] 1176 H314 1 i tx, it 1 x , t 1 x , r 2 y ty, r 2 y t 1 y ,myt 1 y , r 2 x t 1 z 1177 H317 1 i r 2 z t 1 x , r 2 y ty, r 2 y t 1 y ,myt 1 y ,mytz, r 2 y t 1 z , myt 1 z [ 8, 28, 63, 110, 174, 250, 342, 446, 566, 698 ] 1178 H312 1 i it 1 x ,mzty, r 2 z t 1 y ,mzt 1 y ,mxtz,mxt 1 z , r 2 x t 1 z [ 8, 28, 63, 111, 174, 250, 341, 444, 562, 692 ] 1179 H412 1 r 2 z rx r 2 z rxt 1 x ,mxr 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz,mxrxt 1 z , mzt 1 z [ 8, 28, 64, 112, 176, 252, 344, 448, 568, 700 ] 1180 H315 1 i mztx, r 2 z t 1 x ,mzt 1 x ,mzty, r 2 z t 1 y ,mzt 1 y , r 2 y t 1 z 1181 H316 1 i mztx, r 2 z t 1 x ,mzt 1 x , r 2 y ty, r 2 y t 1 y ,myt 1 y , mzt 1 z 1182 H661 hmzrxi r 2 z rx mxt 1 x ,mxr 1 x ty, r 2 y t 1 y ,myt 1 y , r 2 z tz,mxrxt 1 z , r 1 x t 1 z [ 8, 28, 64, 113, 177, 253, 345, 449, 569, 701 ] 1183 H409 1 r 2 z rx r 2 z tx, r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 y rxt 1 z , r 2 x t 1 z [ 8, 28, 64, 117, 187, 274, 377, 496, 631, 782 ] 1184 H305 1 i tx, it 1 x , t 1 x , it 1 y ,mxr 1 z tz, r 2 y rzt 1 z , mxr 1 z t 1 z [ 8, 28, 66, 119, 192, 278, 384, 502, 640, 790 ] 1185 H308 1 i mztx, r 2 z t 1 x ,mzt 1 x , r 2 z t 1 y ,mxr 1 z tz, r 2 y rzt 1 z , mxr 1 z t 1 z [ 8, 29, 65, 113, 177, 253, 345, 449, 569, 701 ] 1186 H656 hmzrxi r 2 z rx r 2 z rxt 1 x , r 2 y rxty,mxt 1 y , r 2 x t 1 y ,mxtz, r 2 y rxt 1 z , mzr 1 x t 1 z 1187 H316 1 i r 2 z t 1 x , r 2 y ty, r 2 y t 1 y ,myt 1 y , r 2 z tz, r 2 z t 1 z , mzt 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 83 Nbr. gr Hi L m X 1188 H317 1 i mztx, r 2 z t 1 x ,mzt 1 x , r 2 y ty, r 2 y t 1 y ,myt 1 y , r 2 y t 1 z 1189 H318 1 i r 2 z t 1 x , r 2 y ty, r 2 y t 1 y ,myt 1 y ,mxtz,mxt 1 z , r 2 x t 1 z [ 8, 29, 65, 114, 178, 254, 346, 450, 570, 702 ] 1190 H422 1 r 2 z rx r 2 z tx,mxr 1 x ty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mxrxt 1 z , mzt 1 z [ 8, 29, 67, 122, 193, 281, 385, 505, 641, 793 ] 1191 H323 1 i r 2 z t 1 x ,mxryty, r 2 x ryt 1 y ,mxryt 1 y ,mxtz,mxt 1 z , r 2 x t 1 z [ 8, 29, 69, 123, 196, 282, 388, 506, 644, 794 ] 1192 H326 1 i mzrxtx, r 2 z rxt 1 x ,mzrxt 1 x ,myt 1 y , r 2 z tz, r 2 z t 1 z , mzt 1 z [ 8, 30, 66, 114, 178, 254, 346, 450, 570, 702 ] 1193 H659 hmzrxi r 2 z rx r 2 z rxt 1 x ,mxr 1 x ty, r 2 y t 1 y ,myt 1 y , r 2 z tz,mxrxt 1 z , r 1 x t 1 z 1194 H314 1 i it 1 x , r 2 y ty, r 2 y t 1 y ,myt 1 y ,mxtz,mxt 1 z , r 2 x t 1 z [ 8, 30, 67, 121, 192, 280, 384, 504, 640, 792 ] 1195 H305 1 i tx, it 1 x , t 1 x , ity,mxr 1 z tz, r 2 y rzt 1 z , mxr 1 z t 1 z 1196 H308 1 i mztx, r 2 z t 1 x ,mzt 1 x , r 2 z ty,mxr 1 z tz, r 2 y rzt 1 z , mxr 1 z t 1 z [ 8, 31, 69, 124, 195, 283, 387, 507, 643, 795 ] 1197 H323 1 i r 2 z t 1 x ,mxryty, r 2 x ryt 1 y ,mxryt 1 y ,mxtz, r 2 x tz, mxt 1 z 1198 H326 1 i mzrxtx, r 2 z rxt 1 x ,mzrxt 1 x ,myt 1 y , r 2 z tz,mztz, r 2 z t 1 z 18B [ 7, 25, 57, 103, 163, 237, 325, 427, 543, 673 ] 1200*, [ 7, 25, 58, 103, 161, 235, 322, 421, 535, 663 ] 1199*, [ 7, 26, 61, 107, 167, 242, 329, 431, 547, 675 ] 1201*, [ 7, 26, 62, 109, 168, 243, 332, 434, 550, 680 ] 1202*, [ 7, 27, 61, 107, 169, 245, 334, 436, 552, 682 ] 1203*, [ 7, 27, 62, 109, 170, 245, 335, 438, 554, 685 ] 1204*, [ 7, 27, 65, 113, 173, 249, 339, 443, 561, 693 ] 1205*, [ 7, 29, 65, 111, 173, 249, 339, 443, 561, 693 ] 1206*, [ 8, 28, 62, 108, 170, 244, 332, 434, 550, 678 ] 1199 H759 hmz, r 2 x i r2 z rx r 2 y rxt 1 x , r 2 z ty, r 1 x ty,mxt 1 y ,mzr 1 x t 1 y ,mzr 1 x tz, rxt 1 z [ 8, 28, 62, 110, 174, 250, 342, 446, 566, 698 ] 1200 H760 hmz, r 2 x i r2 z rx mxt 1 x , r 2 z ty, r 1 x ty,mxt 1 y ,mzr 1 x t 1 y ,mzr 1 x tz, rxt 1 z [ 8, 29, 65, 111, 175, 250, 339, 443, 560, 689 ] 1201 H413 1 r 2 z rx mxt 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , r 1 x t 1 y , rxtz, mxrxt 1 z [ 8, 29, 66, 112, 175, 251, 341, 444, 563, 693 ] 1202 H395 1 r 2 z rx mxt 1 x , r 2 y rxty, r 2 x ty, it 1 y ,mzrxt 1 y ,mzrxtz, r 2 y rxt 1 z [ 8, 30, 63, 111, 174, 250, 341, 444, 562, 692 ] 1203 H412 1 r 2 z rx r 2 z rxt 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , r 1 x t 1 y , rxtz, mxrxt 1 z [ 8, 30, 64, 112, 175, 251, 342, 445, 564, 695 ] 1204 H394 1 r 2 z rx r 2 z rxt 1 x , r 2 y rxty, r 2 x ty, it 1 y ,mzrxt 1 y ,mzrxtz, r 2 y rxt 1 z [ 8, 30, 68, 114, 178, 254, 346, 450, 570, 702 ] 1205 H758 hmz, r 2 x i r2 z rx mxt 1 x ,mxrxty, r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y ,mxr 1 x tz, r 2 y rxt 1 z [ 8, 32, 66, 114, 178, 254, 346, 450, 570, 702 ] 1206 H757 hmz, r 2 x i r2 z rx r 2 y rxt 1 x ,mxrxty, r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y ,mxr 1 x tz, r 2 y rxt 1 z 19 [ 8, 26, 56, 98, 152, 218, 296, 386, 488, 602 ] 1208*, [ 8, 26, 57, 101, 158, 228, 311, 407, 516, 638 ] 1207*, [ 8, 26, 57, 101, 159, 231, 316, 414, 525, 649 ] 1209*, [ 8, 26, 58, 104, 162, 232, 316, 414, 524, 646 ] 1213*, 84 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X [ 8, 26, 58, 104, 163, 235, 321, 421, 534, 660 ] 1211*, 1212*, [ 8, 26, 58, 104, 164, 238, 326, 428, 543, 671 ] 1214*, [ 8, 26, 58, 104, 165, 241, 331, 435, 553, 685 ] 1215*, 1210*, [ 8, 26, 59, 107, 169, 245, 335, 439, 557, 689 ] 1216*, [ 8, 27, 61, 109, 171, 247, 337, 441, 559, 691 ] 1219*, 1220*, 1221*, [ 8, 27, 61, 112, 181, 268, 372, 492, 628, 780 ] 1217*, 1218*, [ 8, 27, 62, 115, 185, 272, 376, 496, 632, 784 ] 1222*, 1223*, [ 8, 28, 63, 111, 173, 249, 339, 443, 561, 693 ] 1224*, 1225*, 1226*, 1227*, 1228*, 1229*, 1230*, [ 8, 28, 64, 117, 187, 274, 378, 498, 634, 786 ] 1231*, 1232*, 1233*, 1234*, [ 8, 28, 64, 117, 188, 276, 380, 500, 636, 788 ] 1235*, [ 8, 28, 65, 120, 192, 280, 384, 504, 640, 792 ] 1236*, 1237*, 1238*, [ 8, 29, 67, 122, 194, 282, 386, 506, 642, 794 ] 1239*, 1240*, 1241*, [ 8, 30, 69, 124, 196, 284, 388, 508, 644, 796 ] 1242*, 1243*, 1244*, 1245*, 1246*, 1247*, 1248*, 1249*, 1250*, [ 9, 28, 61, 106, 165, 236, 321, 418, 529, 652 ] 1207 H610 hmzi r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , itz,mztz, r 2 z t 1 z , t 1 z 1208 H616 hmzi r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y , r 2 y tz, r 2 x tz, mxt 1 z ,myt 1 z [ 9, 28, 62, 108, 169, 242, 329, 428, 542, 668 ] 1209 H461 1 r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y , r 2 y tz, r 2 x tz, it 1 z ,mzt 1 z [ 9, 28, 62, 108, 171, 246, 338, 442, 562, 694 ] 1210 H460 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y tz, r 2 x tz, it 1 z ,mzt 1 z [ 9, 28, 63, 110, 171, 244, 333, 434, 549, 676 ] 1211 H615 hmzi r 2 z tx, r 2 z t 1 x , t 1 x ,myt 1 y , itz,mztz, r 2 z t 1 z , t 1 z 1212 H305 1 i tx, it 1 x , t 1 x , ity, ty, it 1 y , t 1 y , r 2 y rzt 1 z 1213 H540 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 x ryt 1 y ,mxtz,mytz, mxt 1 z ,myt 1 z [ 9, 28, 63, 110, 173, 248, 339, 442, 559, 688 ] 1214 H504 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 x ryt 1 y , r 2 z tz, tz, r 2 z t 1 z , t 1 z [ 9, 28, 63, 110, 173, 248, 339, 442, 562, 694 ] 1215 H323 1 i r 2 z tx,mztx, r 2 z t 1 x ,mzt 1 x , r 2 x ryt 1 y ,mxtz, mxt 1 z , r 2 x t 1 z [ 9, 28, 63, 110, 174, 250, 342, 446, 566, 698 ] 1216 H614 hmzi r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y tz, r 2 x tz, mxt 1 z ,myt 1 z [ 9, 29, 64, 115, 183, 269, 372, 492, 628, 780 ] 1217 H452 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y rztz, r 2 x rztz, it 1 z ,mzt 1 z 1218 H487 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y ,mzr 1 z tz,mzrztz, it 1 z ,mzt 1 z [ 9, 29, 65, 113, 177, 253, 345, 449, 569, 701 ] 1219 H617 hmzi r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , itz,mztz, r 2 z t 1 z , t 1 z 1220 H618 hmzi r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 x tz, mxt 1 z ,myt 1 z 1221 H483 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 x tz, it 1 z ,mzt 1 z [ 9, 29, 65, 117, 186, 273, 376, 496, 632, 784 ] 1222 H503 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y rztz, r 2 x rztz, r 2 y t 1 z , r 2 x t 1 z 1223 H519 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y ,mzr 1 z tz,mzrztz, r 2 y t 1 z , r 2 x t 1 z [ 9, 30, 66, 114, 178, 254, 346, 450, 570, 702 ] 1224 H615 hmzi r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y , itz,mztz, r 2 z t 1 z , t 1 z 1225 H616 hmzi r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 x tz, mxt 1 z ,myt 1 z 1226 H308 1 i mztx, r 2 z t 1 x ,mzt 1 x , r 2 z ty,mzty, r 2 z t 1 y , mzt 1 y , r 2 y rzt 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 85 Nbr. gr Hi L m X 1227 H326 1 i r 2 z rxt 1 x , r 2 y ty,myty, r 2 y t 1 y ,myt 1 y , r 2 z tz, r 2 z t 1 z ,mzt 1 z 1228 H461 1 r 2 z r 2 z t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 x tz, it 1 z ,mzt 1 z 1229 H511 1 r 2 z r 2 z rxtx, r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y tz, r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 1230 H541 1 r 2 z r 2 z rxtx,mxty,mxt 1 y ,myt 1 y , itz,mztz, it 1 z ,mzt 1 z [ 9, 30, 67, 120, 189, 276, 379, 499, 635, 787 ] 1231 H527 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, r 2 x rztz, it 1 z ,mzt 1 z 1232 H528 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y ,mzr 1 z tz,mzrztz, it 1 z ,mzt 1 z 1233 H529 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, r 2 x rztz, r 2 y t 1 z , r 2 x t 1 z 1234 H530 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y ,mzr 1 z tz,mzrztz, r 2 y t 1 z , r 2 x t 1 z [ 9, 30, 67, 121, 191, 278, 381, 500, 636, 788 ] 1235 H539 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 x t 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z , r 2 x rzt 1 z [ 9, 30, 67, 121, 192, 280, 384, 504, 640, 792 ] 1236 H611 hmzi r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y , r 2 y rztz, r 2 x rztz, mxrzt 1 z ,mxr 1 z t 1 z 1237 H619 hmzi r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y ,mzr 1 z tz,mzrztz, r 1 z t 1 z , rzt 1 z 1238 H495 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z t 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z , r 2 x rzt 1 z [ 9, 31, 69, 124, 195, 283, 387, 507, 643, 795 ] 1239 H622 hmzi r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, r 2 x rztz, mxrzt 1 z ,mxr 1 z t 1 z 1240 H623 hmzi r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y ,mzr 1 z tz,mzrztz, r 1 z t 1 z , rzt 1 z 1241 H502 1 r 2 z mxt 1 x ,mxty,mxt 1 y ,myt 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z , r 2 x rzt 1 z [ 9, 32, 70, 125, 196, 284, 388, 508, 644, 796 ] 1242 H305 1 i it 1 x , ity, ty, it 1 y , t 1 y ,mxr 1 z tz, r 2 y rzt 1 z ,mxr 1 z t 1 z 1243 H308 1 i r 2 z t 1 x , r 2 z ty,mzty, r 2 z t 1 y ,mzt 1 y ,mxr 1 z tz, r 2 y rzt 1 z ,mxr 1 z t 1 z 1244 H323 1 i r 2 z tx,mztx, r 2 z t 1 x ,mzt 1 x ,mxryty, r 2 x ryt 1 y , mxryt 1 y , r 2 x t 1 z 1245 H326 1 i mzrxtx, r 2 z rxt 1 x ,mzrxt 1 x ,myt 1 y , r 2 z tz,mztz, r 2 z t 1 z ,mzt 1 z 1246 H539 1 r 2 z it 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z , r 2 x rzt 1 z 1247 H540 1 r 2 z it 1 x , ryty, r 2 x ryt 1 y , r 1 y t 1 y ,mxtz,mytz, mxt 1 z ,myt 1 z 1248 H504 1 r 2 z r 2 z t 1 x , ryty, r 2 x ryt 1 y , r 1 y t 1 y , r 2 z tz, tz, r 2 z t 1 z , t 1 z 1249 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 x t 1 y , r 2 y tz, r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 1250 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,myt 1 y , itz,mztz, it 1 z ,mzt 1 z 20 [ 8, 27, 59, 105, 165, 239, 327, 429, 545, 675 ] 1252*, [ 8, 27, 60, 106, 167, 242, 329, 431, 547, 675 ] 1254*, [ 8, 27, 60, 107, 168, 243, 332, 435, 552, 683 ] 1257*, [ 8, 27, 60, 107, 169, 245, 334, 436, 552, 682 ] 1251*, [ 8, 27, 61, 108, 168, 243, 332, 434, 550, 680 ] 1253*, 1258*, [ 8, 27, 61, 108, 170, 247, 336, 440, 559, 690 ] 1255*, [ 8, 27, 61, 109, 170, 245, 335, 438, 554, 685 ] 1256*, [ 8, 27, 61, 109, 170, 245, 335, 439, 556, 687 ] 1259*, 1260*, [ 8, 27, 61, 109, 172, 249, 339, 443, 561, 693 ] 1261*, [ 8, 27, 62, 111, 173, 249, 339, 443, 561, 693 ] 1262*, 86 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X [ 8, 28, 64, 112, 173, 249, 339, 443, 561, 693 ] 1263*, 1264*, 1265*, 1266*, 1267*, 1268*, [ 8, 29, 64, 111, 173, 249, 339, 443, 561, 693 ] 1269*, 1270*, 1271*, 1272*, 1273*, 1274*, 1275*, 1276*, 1277*, 1278*, [ 9, 29, 63, 111, 174, 250, 341, 444, 562, 692 ] 1251 H412 1 r 2 z rx r 2 z rxt 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , r 1 x t 1 y , rxtz, mxrxt 1 z ,mzt 1 z [ 9, 29, 63, 111, 174, 250, 342, 446, 566, 698 ] 1252 H418 1 r 2 z rx mxt 1 x , r 2 z ty, r 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z , rxt 1 z [ 9, 29, 64, 111, 174, 249, 339, 442, 560, 690 ] 1253 H388 1 r 2 z rx r 2 z rxt 1 x ,mzrxty, it 1 y ,mzrxt 1 y , itz,mzrxtz, it 1 z ,mzrxt 1 z [ 9, 29, 64, 111, 175, 250, 339, 443, 560, 689 ] 1254 H413 1 r 2 z rx mxt 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , r 1 x t 1 y , rxtz, mxrxt 1 z ,mzt 1 z [ 9, 29, 64, 111, 176, 251, 343, 448, 567, 699 ] 1255 H407 1 r 2 z rx r 2 z rxt 1 x , r 2 z ty, r 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z , rxt 1 z [ 9, 29, 64, 112, 175, 251, 342, 445, 564, 695 ] 1256 H394 1 r 2 z rx r 2 z rxt 1 x , r 2 y rxty, r 2 x ty, it 1 y ,mzrxt 1 y ,mzrxtz, r 2 y rxt 1 z , r 2 x t 1 z [ 9, 29, 64, 112, 176, 252, 344, 448, 568, 700 ] 1257 H389 1 r 2 z rx it 1 x ,mzrxty, it 1 y ,mzrxt 1 y , itz,mzrxtz, it 1 z ,mzrxt 1 z [ 9, 29, 65, 112, 175, 251, 341, 444, 563, 693 ] 1258 H395 1 r 2 z rx mxt 1 x , r 2 y rxty, r 2 x ty, it 1 y ,mzrxt 1 y ,mzrxtz, r 2 y rxt 1 z , r 2 x t 1 z [ 9, 29, 65, 113, 176, 252, 344, 448, 568, 700 ] 1259 H398 1 r 2 z rx r 2 y rxt 1 x ,mzrxty, it 1 y ,mzrxt 1 y , itz,mzrxtz, it 1 z ,mzrxt 1 z 1260 H400 1 r 2 z rx mxt 1 x ,mzrxty, it 1 y ,mzrxt 1 y , itz,mzrxtz, it 1 z ,mzrxt 1 z [ 9, 29, 65, 113, 178, 254, 346, 450, 570, 702 ] 1261 H410 1 r 2 z rx r 2 y rxt 1 x , r 2 z ty, r 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z , rxt 1 z [ 9, 29, 66, 114, 178, 254, 346, 450, 570, 702 ] 1262 H414 1 r 2 z rx it 1 x , r 2 z ty, r 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z , rxt 1 z [ 9, 30, 67, 114, 178, 254, 346, 450, 570, 702 ] 1263 H422 1 r 2 z rx r 2 z tx,mxr 1 x ty,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz, mxrxt 1 z ,mzt 1 z 1264 H409 1 r 2 z rx r 2 z tx, r 2 y rxty, r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 y rxt 1 z , r 2 x t 1 z 1265 H395 1 r 2 z rx mxt 1 x , r 2 y rxty, it 1 y ,mzrxt 1 y , itz,mzrxtz, r 2 y rxt 1 z , r 2 x t 1 z 1266 H413 1 r 2 z rx mxt 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , r 1 x t 1 y , r 2 y tz, rxtz,mxrxt 1 z 1267 H419 1 r 2 z rx mxt 1 x ,mxr 1 x ty,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz, mxrxt 1 z ,mzt 1 z 1268 H401 1 r 2 z rx mxt 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 y rxt 1 z , r 2 x t 1 z [ 9, 31, 66, 114, 178, 254, 346, 450, 570, 702 ] 1269 H415 1 r 2 z rx it 1 x ,mxr 1 x ty,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz, mxrxt 1 z ,mzt 1 z 1270 H396 1 r 2 z rx it 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 y rxt 1 z , r 2 x t 1 z 1271 H422 1 r 2 z rx r 2 z tx,mxr 1 x ty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mztz, mxrxt 1 z ,mzt 1 z 1272 H409 1 r 2 z rx r 2 z tx, r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 y rxt 1 z , r 2 x t 1 z 1273 H394 1 r 2 z rx r 2 z rxt 1 x , r 2 y rxty, it 1 y ,mzrxt 1 y , itz,mzrxtz, r 2 y rxt 1 z , r 2 x t 1 z 1274 H412 1 r 2 z rx r 2 z rxt 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , r 1 x t 1 y , r 2 y tz, rxtz,mxrxt 1 z 1275 H402 1 r 2 z rx r 2 z rxt 1 x ,mxr 1 x ty,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz, mxrxt 1 z ,mzt 1 z 1276 H391 1 r 2 z rx r 2 z rxt 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 y rxt 1 z , r 2 x t 1 z 1277 H404 1 r 2 z rx r 2 y rxt 1 x ,mxr 1 x ty,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz, mxrxt 1 z ,mzt 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 87 Nbr. gr Hi L m X 1278 H392 1 r 2 z rx r 2 y rxt 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 y rxt 1 z , r 2 x t 1 z 21 [ 9, 29, 63, 111, 173, 249, 339, 443, 561, 693 ] 1279*, [ 9, 30, 66, 119, 189, 276, 380, 500, 636, 788 ] 1280*, [ 9, 31, 69, 124, 196, 284, 388, 508, 644, 796 ] 1281*, [ 10, 30, 66, 114, 178, 254, 346, 450, 570, 702 ] 1279 H779 hmz,mzrx, r 2 x i r2 z rx mxt 1 x , r 2 z ty, r 2 x ty,mxt 1 y ,mzr 1 x t 1 y ,mxr 1 x tz, mzr 1 x tz, r 2 y rxt 1 z , r 2 x t 1 z [ 10, 31, 68, 121, 190, 277, 380, 500, 636, 788 ] 1280 H690 hmxi r 2 z r 2 z tx,mxtx,myt 1 x ,mxt 1 x ,myt 1 y , r 2 y rztz, mzr 1 z tz, r 2 x t 1 z , r 2 y t 1 z [ 10, 32, 70, 125, 196, 284, 388, 508, 644, 796 ] 1281 H750 hmy,mzi r 2 z mxt 1 x , r 2 z ty,myty,mxt 1 y ,myt 1 y , r 2 y rztz, mzrztz,mxrzt 1 z , rzt 1 z 22 [ 5, 14, 34, 78, 153, 246, 345, 461, 602, 756 ] 1298*, [ 5, 15, 42, 101, 174, 248, 340, 443, 558, 697 ] 1306*, [ 5, 16, 43, 89, 147, 216, 296, 388, 499, 621 ] 1317*, [ 5, 16, 45, 98, 167, 247, 340, 447, 569, 706 ] 1320*, [ 5, 16, 45, 102, 182, 272, 371, 491, 635, 789 ] 1321*, [ 5, 16, 46, 101, 170, 248, 340, 448, 570, 705 ] 1318*, [ 5, 16, 48, 106, 172, 247, 339, 445, 567, 703 ] 1319*, [ 5, 17, 44, 88, 146, 216, 298, 394, 504, 626 ] 1323*, [ 5, 17, 44, 91, 163, 258, 365, 481, 615, 771 ] 1322*, [ 5, 17, 47, 101, 172, 252, 344, 452, 575, 713 ] 1324*, [ 5, 17, 48, 103, 169, 244, 333, 439, 557, 687 ] 1325*, [ 5, 17, 50, 108, 173, 247, 336, 441, 563, 694 ] 1326*, [ 5, 17, 51, 109, 177, 257, 356, 468, 591, 732 ] 1327*, [ 5, 17, 53, 111, 178, 259, 357, 470, 597, 739 ] 1328*, [ 5, 18, 47, 98, 171, 258, 362, 482, 618, 770 ] 1329*, [ 5, 18, 48, 103, 171, 246, 339, 446, 566, 702 ] 1331*, [ 5, 18, 49, 102, 169, 244, 336, 443, 560, 695 ] 1330*, [ 5, 18, 51, 110, 180, 260, 359, 470, 596, 738 ] 1333*, [ 5, 18, 52, 109, 174, 255, 361, 482, 618, 770 ] 1332*, [ 5, 18, 53, 114, 183, 259, 355, 468, 595, 738 ] 1334*, [ 5, 19, 51, 100, 161, 233, 321, 424, 539, 669 ] 1335*, [ 5, 19, 52, 108, 172, 243, 337, 444, 559, 697 ] 1336*, [ 5, 19, 53, 110, 178, 254, 349, 460, 581, 719 ] 1337*, [ 5, 19, 53, 110, 183, 272, 376, 495, 632, 785 ] 1338*, [ 5, 19, 57, 114, 178, 256, 351, 460, 586, 726 ] 1339*, [ 5, 19, 57, 116, 184, 264, 361, 474, 601, 742 ] 1340*, [ 5, 20, 50, 98, 156, 230, 316, 418, 532, 662 ] 1341*, [ 5, 20, 56, 115, 180, 258, 353, 464, 589, 730 ] 1342*, [ 5, 20, 56, 115, 185, 270, 373, 496, 632, 784 ] 1343*, [ 5, 20, 56, 117, 186, 266, 363, 476, 604, 748 ] 1346*, [ 5, 20, 57, 113, 177, 254, 349, 460, 583, 720 ] 1344*, [ 5, 20, 57, 113, 182, 272, 376, 496, 632, 784 ] 1345*, [ 5, 20, 63, 117, 177, 260, 356, 466, 592, 732 ] 1347*, [ 6, 18, 39, 72, 120, 184, 264, 360, 472, 600 ] 1282 H650 hmxi r 2 y rx mxtx,mxt 1 x ,myty,myt 1 y , r 2 y t 1 z [ 6, 19, 44, 82, 130, 188, 258, 338, 428, 530 ] 88 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 1283 H650 hmxi r 2 y rx mxtx,mxt 1 x ,mxrxty,myt 1 y ,mxr 1 x tz [ 6, 19, 44, 83, 136, 205, 291, 392, 509, 643 ] 1284 H422 1 r 2 z rx r 2 z tx, r 2 y t 1 x ,myty,myt 1 y ,mzt 1 z 1285 H580 1 mzrx itx,mxt 1 x ,myty,myt 1 y ,mzt 1 z [ 6, 19, 45, 89, 152, 232, 328, 440, 568, 712 ] 1286 H580 1 mzrx r 2 z rxtx,mxt 1 x ,myty,myt 1 y ,mzt 1 z 1287 H578 1 mzrx mxtx, r 2 y rxt 1 x ,myty,myt 1 y ,mzt 1 z [ 6, 20, 47, 87, 139, 203, 279, 367, 467, 579 ] 1288 H648 hmxi r 2 y rx mxtx,mxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 z rxtz [ 6, 20, 47, 88, 144, 216, 304, 408, 528, 664 ] 1289 H652 hmxi r 2 y rx itx, r 2 x t 1 x ,myty,myt 1 y , r 2 y t 1 z 1290 H648 hmxi r 2 y rx mxtx,mxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 6, 20, 48, 92, 153, 232, 328, 440, 568, 712 ] 1291 H574 1 mzrx r 2 z rxtx, it 1 x ,myty,myt 1 y ,mzt 1 z 1292 H580 1 mzrx itx, r 2 y rxt 1 x ,myty,myt 1 y ,mzt 1 z [ 6, 20, 49, 93, 147, 215, 301, 397, 503, 627 ] 1293 H578 1 mzrx mxtx, r 2 y rxt 1 x , rxty,myt 1 y , r 1 x tz [ 6, 20, 49, 96, 160, 240, 336, 448, 576, 720 ] 1294 H652 hmxi r 2 y rx r 2 z rxtx, r 2 z rxt 1 x ,myty,myt 1 y , r 2 y t 1 z 1295 H422 1 r 2 z rx r 1 x tx, rxt 1 x ,myty,myt 1 y ,mzt 1 z 1296 H650 hmxi r 2 y rx r 2 y rxtx,mzrxt 1 x ,myty,myt 1 y , r 2 y t 1 z 1297 H580 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,myty,myt 1 y ,mzt 1 z [ 6, 20, 50, 103, 177, 263, 365, 490, 628, 774 ] 1298 H571 1 mzrx mxtx, r 2 y rxt 1 x ,myty, r 2 z t 1 y ,mztz [ 6, 21, 50, 93, 152, 227, 317, 424, 547, 685 ] 1299 H409 1 r 2 z rx r 2 z tx, r 2 y t 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z 1300 H560 1 mzrx itx,mxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 6, 21, 51, 95, 152, 224, 311, 411, 524, 652 ] 1301 H557 1 mzrx mxtx, r 2 y rxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x tz [ 6, 21, 51, 96, 154, 227, 317, 421, 539, 673 ] 1302 H572 1 mzrx r 2 z rxtx, it 1 x ,mxrxty, r 2 z t 1 y ,mxr 1 x tz [ 6, 21, 51, 97, 160, 240, 336, 448, 576, 720 ] 1303 H560 1 mzrx r 2 z rxtx,mxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z 1304 H557 1 mzrx mxtx, r 2 y rxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 6, 21, 52, 96, 152, 228, 316, 412, 532, 664 ] 1305 H650 hmxi r 2 y rx r 2 y rxtx,mzrxt 1 x ,mxrxty,myt 1 y ,mxr 1 x tz [ 6, 21, 57, 116, 177, 256, 353, 452, 575, 717 ] 1306 H571 1 mzrx mxtx, r 2 y rxt 1 x , rxty, r 2 z t 1 y , r 1 x tz [ 6, 22, 51, 96, 156, 232, 324, 432, 556, 696 ] 1307 H649 hmxi r 2 y rx itx, r 2 x t 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 6, 22, 52, 97, 156, 230, 319, 422, 539, 671 ] 1308 H551 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxty, it 1 y , r 2 z rxtz [ 6, 22, 52, 98, 158, 234, 323, 428, 547, 682 ] 1309 H393 1 r 2 z rx r 2 y rztx, r 2 x ryt 1 x , r 2 y rzty, r 2 y rztz, r 2 x ryt 1 z [ 6, 22, 52, 98, 161, 240, 336, 448, 576, 720 ] 1310 H555 1 mzrx r 2 z rxtx, it 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z 1311 H560 1 mzrx itx, r 2 y rxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 6, 22, 53, 97, 157, 233, 321, 425, 545, 677 ] 1312 H648 hmxi r 2 y rx r 2 y rxtx,mzrxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 z rxtz [ 6, 22, 53, 100, 164, 244, 340, 452, 580, 724 ] 1313 H649 hmxi r 2 y rx r 2 z rxtx, r 2 z rxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z 1314 H409 1 r 2 z rx r 1 x tx, rxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z 1315 H648 hmxi r 2 y rx r 2 y rxtx,mzrxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z 1316 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 6, 22, 55, 101, 159, 233, 316, 412, 528, 648 ] 1317 H580 1 mzrx r 2 z rxtx,mxt 1 x , rxty,myt 1 y , r 1 x tz [ 6, 22, 58, 111, 176, 255, 349, 459, 582, 719 ] 1318 H580 1 mzrx itx,mxt 1 x , rxty,myt 1 y , r 1 x tz [ 6, 22, 59, 115, 177, 256, 349, 457, 581, 717 ] 1319 H422 1 r 2 z rx r 2 z tx, r 2 y t 1 x ,myty,mxr 1 x t 1 y ,mxrxtz [ 6, 22, 60, 111, 179, 261, 355, 465, 591, 729 ] 1320 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mzr 1 x ty, it 1 y ,mzr 1 x tz [ 6, 22, 60, 115, 191, 278, 381, 506, 643, 790 ] 1321 H561 1 mzrx mxtx, r 2 y rxt 1 x , r 2 x ty, it 1 y , r 2 x tz [ 6, 23, 54, 106, 182, 272, 374, 493, 632, 785 ] 1322 H571 1 mzrx mxtx, r 2 y rxt 1 x ,myty, r 2 z t 1 y , r 2 y t 1 z [ 6, 23, 56, 102, 162, 235, 320, 420, 533, 658 ] 1323 H560 1 mzrx r 2 z rxtx,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x tz [ 6, 23, 59, 112, 180, 260, 354, 464, 588, 728 ] 1324 H560 1 mzrx itx,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x tz [ 6, 23, 60, 112, 176, 255, 344, 455, 571, 705 ] 1325 H580 1 mzrx r 2 z rxtx, r 2 y rxt 1 x , rxty,myt 1 y , r 1 x tz [ 6, 23, 62, 115, 176, 257, 345, 455, 578, 708 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 89 Nbr. gr Hi L m X 1326 H422 1 r 2 z rx r 1 x tx, rxt 1 x ,myty,mxr 1 x t 1 y ,mxrxtz [ 6, 23, 63, 116, 182, 264, 364, 474, 598, 744 ] 1327 H580 1 mzrx itx, r 2 y rxt 1 x , rxty,myt 1 y , r 1 x tz [ 6, 23, 64, 116, 184, 266, 364, 477, 605, 749 ] 1328 H409 1 r 2 z rx r 2 z tx, r 2 y t 1 x , r 2 y rxt 1 y , r 2 y rxtz, r 2 x t 1 z [ 6, 24, 56, 111, 184, 270, 376, 494, 632, 782 ] 1329 H576 1 mzrx mxtx, r 2 y rxt 1 x , r 2 z ty, r 2 z t 1 y , r 2 y t 1 z [ 6, 24, 59, 113, 177, 253, 351, 455, 574, 715 ] 1330 H571 1 mzrx mxtx, r 2 y rxt 1 x ,myty,mxrxt 1 y ,mxr 1 x t 1 z [ 6, 24, 60, 114, 179, 257, 352, 460, 581, 720 ] 1331 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x tz [ 6, 24, 61, 115, 183, 269, 376, 494, 632, 782 ] 1332 H572 1 mzrx r 2 z rxtx, it 1 x , r 2 z ty, r 2 z t 1 y , r 2 y t 1 z [ 6, 24, 63, 118, 185, 268, 367, 478, 606, 749 ] 1333 H560 1 mzrx itx, r 2 y rxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x tz [ 6, 24, 65, 119, 185, 265, 363, 476, 603, 748 ] 1334 H409 1 r 2 z rx r 1 x tx, rxt 1 x , r 2 y rxt 1 y , r 2 y rxtz, r 2 x t 1 z [ 6, 25, 60, 108, 172, 246, 338, 442, 558, 692 ] 1335 H652 hmxi r 2 y rx r 2 z rxtx, r 2 z rxt 1 x ,mxrxty,myt 1 y ,mxr 1 x tz [ 6, 25, 60, 118, 176, 253, 353, 454, 574, 718 ] 1336 H576 1 mzrx mxtx, r 2 y rxt 1 x ,mxrxty, r 2 z t 1 y ,mxr 1 x tz [ 6, 25, 63, 118, 183, 262, 361, 470, 592, 735 ] 1337 H561 1 mzrx mxtx, r 2 y rxt 1 x , r 2 z rxt 1 y , r 2 x tz, r 2 z rxt 1 z [ 6, 25, 63, 119, 192, 280, 382, 505, 641, 792 ] 1338 H561 1 mzrx mxtx, r 2 y rxt 1 x , it 1 y , r 2 x tz, it 1 z [ 6, 25, 65, 118, 182, 263, 359, 469, 597, 736 ] 1339 H574 1 mzrx r 2 z rxtx, it 1 x , rxty,myt 1 y , r 1 x tz [ 6, 25, 66, 120, 187, 268, 368, 481, 608, 751 ] 1340 H652 hmxi r 2 y rx itx, r 2 x t 1 x ,mxrxty,myt 1 y ,mxr 1 x tz [ 6, 26, 59, 109, 169, 247, 335, 441, 557, 691 ] 1341 H649 hmxi r 2 y rx r 2 z rxtx, r 2 z rxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 z rxtz [ 6, 26, 64, 121, 183, 266, 361, 474, 599, 742 ] 1342 H555 1 mzrx r 2 z rxtx, it 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x tz [ 6, 26, 64, 122, 192, 278, 383, 505, 639, 793 ] 1343 H551 1 mzrx r 2 z rxtx, it 1 x , it 1 y , itz, it 1 z [ 6, 26, 65, 119, 182, 263, 360, 471, 594, 734 ] 1344 H556 1 mzrx mxtx, r 2 y rxt 1 x , r 2 z rxty, it 1 y , r 2 z rxtz [ 6, 26, 65, 120, 191, 281, 383, 505, 639, 793 ] 1345 H556 1 mzrx mxtx, r 2 y rxt 1 x , it 1 y , itz, it 1 z [ 6, 26, 65, 123, 188, 272, 369, 484, 612, 757 ] 1346 H649 hmxi r 2 y rx itx, r 2 x t 1 x , r 2 z rxty, r 2 x t 1 y , r 2 z rxtz [ 6, 26, 71, 119, 185, 271, 367, 479, 609, 750 ] 1347 H393 1 r 2 z rx ryrxtx, r 1 y r 1 z t 1 x , ryrxty, r 2 y rztz, r 1 y r 1 z t 1 z 23 [ 6, 19, 44, 85, 146, 229, 332, 452, 588, 740 ] 1348*, [ 6, 19, 44, 87, 154, 241, 341, 460, 602, 756 ] 1349*, [ 6, 20, 49, 95, 154, 224, 308, 406, 516, 640 ] 1351*, [ 6, 20, 49, 95, 156, 234, 333, 452, 588, 740 ] 1350*, [ 6, 20, 49, 95, 157, 237, 336, 453, 588, 740 ] 1352*, [ 6, 20, 50, 101, 172, 260, 364, 484, 620, 772 ] 1353*, 1354*, [ 6, 20, 52, 108, 172, 244, 339, 443, 558, 697 ] 1355*, [ 6, 21, 49, 94, 165, 259, 365, 481, 615, 771 ] 1356*, [ 6, 21, 49, 97, 170, 258, 362, 482, 618, 770 ] 1357*, [ 6, 21, 52, 100, 162, 237, 326, 429, 546, 677 ] 1359*, [ 6, 21, 52, 100, 164, 244, 341, 456, 589, 740 ] 1358*, [ 6, 21, 52, 101, 168, 253, 356, 476, 612, 764 ] 1361*, [ 6, 21, 53, 103, 168, 247, 340, 448, 570, 705 ] 1360*, [ 6, 21, 53, 104, 173, 260, 364, 484, 620, 772 ] 1362*, 1363*, 1364*, [ 6, 21, 54, 105, 167, 244, 339, 442, 559, 697 ] 1365*, [ 6, 21, 56, 107, 169, 247, 340, 447, 569, 706 ] 1366*, [ 6, 21, 56, 110, 180, 267, 369, 491, 635, 789 ] 1367*, [ 6, 22, 54, 103, 172, 260, 364, 484, 620, 772 ] 1368*, [ 6, 22, 54, 104, 170, 246, 337, 443, 560, 695 ] 1369*, [ 6, 22, 54, 107, 171, 243, 337, 444, 559, 697 ] 90 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 1370*, [ 6, 22, 55, 105, 170, 249, 343, 452, 575, 713 ] 1371*, [ 6, 22, 55, 105, 172, 256, 357, 476, 612, 764 ] 1372*, [ 6, 22, 55, 105, 173, 257, 358, 477, 612, 764 ] 1373*, [ 6, 22, 56, 107, 172, 253, 348, 456, 580, 718 ] 1375*, [ 6, 22, 56, 108, 175, 256, 354, 467, 591, 732 ] 1374*, [ 6, 22, 56, 109, 180, 268, 372, 492, 628, 780 ] 1376*, 1377*, [ 6, 22, 58, 111, 171, 245, 339, 442, 559, 697 ] 1378*, [ 6, 23, 54, 102, 172, 260, 364, 484, 620, 772 ] 1379*, [ 6, 23, 56, 108, 176, 260, 361, 478, 613, 764 ] 1381*, [ 6, 23, 57, 106, 170, 255, 361, 482, 618, 770 ] 1380*, [ 6, 23, 57, 109, 177, 259, 357, 469, 596, 738 ] 1384*, [ 6, 23, 57, 110, 181, 268, 372, 492, 628, 780 ] 1386*, 1387*, [ 6, 23, 58, 106, 168, 245, 333, 439, 557, 687 ] 1382*, [ 6, 23, 58, 111, 178, 255, 349, 460, 581, 719 ] 1385*, [ 6, 23, 58, 111, 183, 273, 376, 495, 632, 785 ] 1388*, [ 6, 23, 59, 110, 170, 245, 339, 442, 559, 697 ] 1389*, [ 6, 23, 59, 110, 175, 260, 364, 484, 620, 772 ] 1383*, [ 6, 23, 59, 112, 176, 254, 349, 460, 583, 720 ] 1390*, [ 6, 23, 59, 112, 181, 272, 376, 496, 632, 784 ] 1391*, [ 6, 23, 60, 112, 176, 256, 350, 458, 582, 720 ] 1392*, [ 6, 23, 60, 114, 184, 272, 376, 496, 632, 784 ] 1394*, [ 6, 23, 61, 114, 178, 257, 354, 467, 591, 732 ] 1393*, [ 6, 24, 59, 108, 172, 249, 341, 448, 568, 704 ] 1397*, [ 6, 24, 59, 109, 170, 245, 339, 442, 559, 697 ] 1396*, [ 6, 24, 59, 109, 173, 252, 347, 457, 584, 726 ] 1395*, [ 6, 24, 60, 111, 176, 256, 351, 460, 586, 726 ] 1398*, [ 6, 24, 62, 113, 176, 256, 350, 458, 582, 720 ] 1400*, [ 6, 24, 62, 115, 184, 272, 376, 496, 632, 784 ] 1403*, [ 6, 24, 62, 116, 182, 262, 359, 471, 598, 740 ] 1401*, [ 6, 24, 62, 117, 184, 264, 361, 474, 601, 742 ] 1399*, [ 6, 24, 63, 117, 183, 264, 361, 474, 601, 742 ] 1402*, [ 6, 25, 59, 110, 173, 252, 344, 452, 573, 710 ] 1404*, [ 6, 25, 60, 112, 177, 258, 353, 464, 589, 730 ] 1405*, [ 6, 25, 61, 114, 185, 272, 376, 496, 632, 784 ] 1406*, [ 6, 25, 62, 112, 176, 256, 350, 458, 582, 720 ] 1407*, [ 6, 25, 62, 114, 184, 272, 376, 496, 632, 784 ] 1408*, [ 6, 25, 62, 116, 184, 270, 373, 496, 632, 784 ] 1409*, [ 6, 25, 62, 118, 185, 266, 363, 476, 604, 748 ] 1410*, [ 6, 25, 63, 118, 184, 266, 363, 476, 604, 748 ] 1411*, [ 7, 23, 52, 98, 165, 252, 356, 476, 612, 764 ] 1348 H578 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y ,mzt 1 z [ 7, 23, 54, 106, 177, 261, 364, 490, 628, 774 ] 1349 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, r 2 z t 1 y ,mztz [ 7, 24, 56, 104, 169, 253, 356, 476, 612, 764 ] 1350 H580 1 mzrx itx, r 2 z rxtx,mxt 1 x ,myty,myt 1 y ,mzt 1 z [ 7, 24, 57, 106, 166, 238, 326, 426, 538, 666 ] 1351 H578 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , rxty,myt 1 y , r 1 x tz [ 7, 24, 57, 106, 171, 255, 357, 476, 612, 764 ] 1352 H580 1 mzrx itx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y ,mzt 1 z [ 7, 24, 58, 112, 184, 272, 376, 496, 632, 784 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 91 Nbr. gr Hi L m X 1353 H580 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y ,mzt 1 z 1354 H578 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y ,mzt 1 z [ 7, 24, 61, 118, 176, 255, 353, 452, 575, 717 ] 1355 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , rxty, r 2 z t 1 y , r 1 x tz [ 7, 25, 55, 107, 184, 273, 374, 493, 632, 785 ] 1356 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, r 2 z t 1 y , r 2 y t 1 z [ 7, 25, 57, 110, 184, 270, 376, 494, 632, 782 ] 1357 H576 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z ty, r 2 z t 1 y , r 2 y t 1 z [ 7, 25, 59, 109, 175, 258, 359, 477, 612, 764 ] 1358 H574 1 mzrx itx, it 1 x , r 2 z rxt 1 x ,myty,myt 1 y ,mzt 1 z [ 7, 25, 60, 110, 173, 250, 341, 446, 565, 698 ] 1359 H557 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x tz [ 7, 25, 60, 111, 176, 255, 349, 459, 582, 719 ] 1360 H580 1 mzrx itx, r 2 z rxtx,mxt 1 x , rxty,myt 1 y , r 1 x tz [ 7, 25, 60, 112, 181, 268, 372, 492, 628, 780 ] 1361 H557 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 7, 25, 60, 113, 184, 272, 376, 496, 632, 784 ] 1362 H574 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x ,myty,myt 1 y ,mzt 1 z 1363 H580 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,myty,myt 1 y ,mzt 1 z 1364 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, r 2 z t 1 y ,mztz [ 7, 25, 62, 114, 175, 256, 352, 452, 576, 716 ] 1365 H578 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , rxty,myt 1 y , r 1 x tz [ 7, 25, 65, 114, 179, 261, 355, 465, 591, 729 ] 1366 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, it 1 y ,mzr 1 x tz [ 7, 25, 65, 118, 189, 276, 381, 506, 643, 790 ] 1367 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty, it 1 y , r 2 x tz [ 7, 26, 60, 112, 184, 272, 376, 496, 632, 784 ] 1368 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, r 2 z t 1 y , r 2 y t 1 z [ 7, 26, 60, 114, 179, 254, 351, 455, 574, 715 ] 1369 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myty,mxrxt 1 y ,mxr 1 x t 1 z [ 7, 26, 61, 117, 176, 253, 353, 454, 574, 718 ] 1370 H576 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,mxrxty, r 2 z t 1 y ,mxr 1 x tz [ 7, 26, 62, 113, 179, 259, 354, 464, 588, 728 ] 1371 H560 1 mzrx itx, r 2 z rxtx,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x tz [ 7, 26, 62, 114, 183, 269, 372, 492, 628, 780 ] 1372 H560 1 mzrx itx, r 2 z rxtx,mxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 7, 26, 63, 114, 185, 269, 373, 492, 628, 780 ] 1373 H560 1 mzrx itx, r 2 y rxt 1 x ,mxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 7, 26, 63, 116, 182, 263, 363, 474, 598, 744 ] 1374 H580 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x , rxty,myt 1 y , r 1 x tz [ 7, 26, 64, 115, 181, 264, 358, 468, 594, 732 ] 1375 H557 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x tz [ 7, 26, 64, 118, 190, 278, 382, 502, 638, 790 ] 1376 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z 1377 H557 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 7, 26, 65, 117, 176, 256, 352, 452, 576, 716 ] 1378 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , rxty, r 2 z t 1 y , r 1 x tz [ 7, 27, 59, 112, 184, 272, 376, 496, 632, 784 ] 1379 H576 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z ty, r 2 z t 1 y , r 2 y t 1 z [ 7, 27, 62, 112, 183, 269, 376, 494, 632, 782 ] 1380 H572 1 mzrx itx, it 1 x , r 2 z rxt 1 x , r 2 z ty, r 2 z t 1 y , r 2 y t 1 z [ 7, 27, 63, 117, 185, 272, 373, 493, 628, 780 ] 1381 H555 1 mzrx itx, it 1 x , r 2 z rxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 7, 27, 64, 112, 177, 255, 344, 455, 571, 705 ] 1382 H580 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x , rxty,myt 1 y , r 1 x tz [ 7, 27, 64, 115, 184, 272, 376, 496, 632, 784 ] 1383 H572 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , r 2 z ty, r 2 z t 1 y , r 2 y t 1 z [ 7, 27, 64, 117, 185, 267, 366, 478, 606, 749 ] 1384 H560 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x tz [ 7, 27, 64, 119, 184, 262, 361, 470, 592, 735 ] 1385 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z rxt 1 y , r 2 x tz, r 2 z rxt 1 z [ 7, 27, 64, 119, 190, 278, 382, 502, 638, 790 ] 1386 H555 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z 1387 H560 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 7, 27, 64, 120, 193, 280, 382, 505, 641, 792 ] 1388 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , it 1 y , r 2 x tz, it 1 z [ 7, 27, 65, 116, 176, 256, 352, 452, 576, 716 ] 1389 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty,mxrxt 1 y ,mxr 1 x t 1 z [ 7, 27, 66, 118, 182, 263, 360, 471, 594, 734 ] 1390 H556 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z rxty, it 1 y , r 2 z rxtz [ 7, 27, 66, 119, 191, 281, 383, 505, 639, 793 ] 1391 H556 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , it 1 y , itz, it 1 z [ 7, 27, 67, 117, 183, 265, 359, 469, 595, 733 ] 1392 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, it 1 y ,mzr 1 x tz [ 7, 27, 67, 118, 183, 263, 363, 474, 598, 744 ] 92 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 1393 H580 1 mzrx itx, r 2 y rxt 1 x ,mxt 1 x , rxty,myt 1 y , r 1 x tz [ 7, 27, 67, 120, 192, 280, 384, 504, 640, 792 ] 1394 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty, it 1 y , r 2 x tz [ 7, 28, 64, 115, 180, 260, 356, 466, 594, 735 ] 1395 H572 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x ,mxrxty, r 2 z t 1 y ,mxr 1 x tz [ 7, 28, 64, 116, 176, 256, 352, 452, 576, 716 ] 1396 H576 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,mxrxty, r 2 z t 1 y ,mxr 1 x tz [ 7, 28, 65, 114, 181, 258, 353, 461, 582, 721 ] 1397 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x tz [ 7, 28, 65, 116, 182, 263, 359, 469, 597, 736 ] 1398 H574 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , rxty,myt 1 y , r 1 x tz [ 7, 28, 67, 121, 187, 268, 368, 481, 608, 751 ] 1399 H572 1 mzrx itx, it 1 x , r 2 z rxt 1 x ,mxrxty, r 2 z t 1 y ,mxr 1 x tz [ 7, 28, 68, 117, 183, 265, 359, 469, 595, 733 ] 1400 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z rxt 1 y , r 2 x tz, r 2 z rxt 1 z [ 7, 28, 68, 120, 187, 268, 367, 479, 607, 750 ] 1401 H560 1 mzrx itx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x tz [ 7, 28, 68, 120, 187, 268, 368, 481, 608, 751 ] 1402 H574 1 mzrx itx, it 1 x , r 2 z rxt 1 x , rxty,myt 1 y , r 1 x tz [ 7, 28, 68, 120, 192, 280, 384, 504, 640, 792 ] 1403 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , it 1 y , r 2 x tz, it 1 z [ 7, 29, 64, 117, 180, 262, 354, 465, 586, 726 ] 1404 H551 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , r 2 z rxty, it 1 y , r 2 z rxtz [ 7, 29, 65, 118, 183, 266, 361, 474, 599, 742 ] 1405 H555 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x tz [ 7, 29, 66, 121, 192, 280, 384, 504, 640, 792 ] 1406 H551 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , it 1 y , itz, it 1 z [ 7, 29, 67, 117, 183, 265, 359, 469, 595, 733 ] 1407 H556 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z rxty, it 1 y , r 2 z rxtz [ 7, 29, 67, 120, 192, 280, 384, 504, 640, 792 ] 1408 H556 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , it 1 y , itz, it 1 z [ 7, 29, 67, 121, 192, 278, 383, 505, 639, 793 ] 1409 H551 1 mzrx itx, it 1 x , r 2 z rxt 1 x , it 1 y , itz, it 1 z [ 7, 29, 67, 123, 188, 272, 369, 484, 612, 757 ] 1410 H551 1 mzrx itx, it 1 x , r 2 z rxt 1 x , r 2 z rxty, it 1 y , r 2 z rxtz [ 7, 29, 68, 122, 188, 272, 369, 484, 612, 757 ] 1411 H555 1 mzrx itx, it 1 x , r 2 z rxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x tz 24 [ 6, 18, 39, 72, 120, 184, 264, 360, 472, 600 ] 1412*, [ 6, 18, 39, 73, 128, 209, 308, 417, 545, 700 ] 1413*, [ 6, 19, 44, 82, 130, 188, 258, 338, 428, 530 ] 1414*, [ 6, 19, 45, 89, 152, 232, 328, 440, 568, 712 ] 1415*, 1416*, [ 6, 19, 47, 97, 161, 235, 328, 432, 548, 686 ] 1417*, [ 6, 20, 46, 88, 152, 238, 344, 465, 598, 749 ] 1418*, [ 6, 20, 46, 90, 161, 249, 353, 473, 609, 761 ] 1419*, [ 6, 20, 47, 87, 139, 203, 279, 367, 467, 579 ] 1423*, [ 6, 20, 47, 88, 144, 216, 304, 408, 528, 664 ] 1424*, 1420*, [ 6, 20, 48, 92, 153, 232, 328, 440, 568, 712 ] 1421*, 1422*, [ 6, 20, 49, 93, 148, 219, 307, 404, 517, 649 ] 1425*, [ 6, 20, 50, 101, 172, 260, 364, 484, 620, 772 ] 1426*, [ 6, 20, 51, 97, 154, 225, 309, 406, 517, 641 ] 1427*, [ 6, 20, 51, 100, 166, 250, 349, 465, 602, 760 ] 1428*, [ 6, 21, 51, 95, 153, 227, 315, 417, 535, 667 ] 1432*, [ 6, 21, 51, 97, 160, 240, 336, 448, 576, 720 ] 1433*, 1434*, [ 6, 21, 51, 98, 160, 235, 327, 432, 549, 685 ] 1431*, [ 6, 21, 51, 100, 163, 238, 332, 437, 553, 691 ] 1430*, [ 6, 21, 51, 100, 171, 260, 364, 484, 620, 772 ] 1429*, [ 6, 21, 55, 108, 170, 245, 339, 442, 559, 697 ] 1435*, [ 6, 21, 56, 106, 167, 246, 336, 441, 563, 694 ] 1436*, [ 6, 22, 51, 96, 156, 232, 324, 432, 556, 696 ] 1437*, [ 6, 22, 52, 98, 161, 240, 336, 448, 576, 720 ] 1439*, 1440*, K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 93 Nbr. gr Hi L m X [ 6, 22, 53, 102, 172, 260, 364, 484, 620, 772 ] 1438*, [ 6, 22, 55, 106, 172, 250, 344, 454, 576, 714 ] 1444*, [ 6, 22, 55, 106, 175, 263, 368, 488, 623, 775 ] 1446*, [ 6, 22, 56, 106, 170, 255, 362, 482, 618, 770 ] 1441*, [ 6, 22, 56, 107, 169, 245, 339, 442, 559, 697 ] 1442*, [ 6, 22, 56, 107, 171, 249, 343, 453, 577, 715 ] 1443*, [ 6, 22, 56, 107, 174, 262, 367, 487, 623, 775 ] 1445*, [ 6, 22, 57, 109, 175, 256, 350, 458, 582, 720 ] 1448*, [ 6, 22, 57, 111, 184, 273, 376, 496, 632, 784 ] 1449*, [ 6, 22, 58, 110, 175, 260, 364, 484, 620, 772 ] 1447*, [ 6, 22, 59, 112, 177, 258, 355, 468, 595, 738 ] 1450*, [ 6, 23, 57, 107, 171, 250, 345, 455, 582, 724 ] 1451*, [ 6, 23, 58, 109, 170, 245, 339, 442, 559, 697 ] 1452*, [ 6, 23, 59, 111, 176, 256, 350, 458, 582, 720 ] 1454*, [ 6, 23, 59, 111, 176, 256, 351, 460, 586, 726 ] 1453*, [ 6, 23, 59, 113, 184, 272, 376, 496, 632, 784 ] 1456*, [ 6, 23, 60, 115, 183, 264, 361, 474, 601, 742 ] 1455*, [ 6, 23, 62, 117, 183, 264, 361, 474, 601, 742 ] 1457*, [ 6, 24, 57, 107, 169, 247, 338, 445, 565, 701 ] 1458*, [ 6, 24, 59, 112, 177, 258, 353, 464, 589, 730 ] 1459*, [ 6, 24, 60, 114, 185, 272, 376, 496, 632, 784 ] 1461*, [ 6, 24, 61, 112, 176, 256, 350, 458, 582, 720 ] 1462*, [ 6, 24, 61, 113, 178, 258, 354, 465, 589, 729 ] 1460*, [ 6, 24, 61, 114, 184, 272, 376, 496, 632, 784 ] 1463*, [ 6, 24, 61, 116, 183, 266, 363, 476, 604, 748 ] 1464*, [ 6, 24, 61, 116, 184, 270, 374, 496, 632, 784 ] 1465*, [ 6, 24, 63, 118, 183, 266, 363, 476, 604, 748 ] 1466*, [ 6, 25, 62, 115, 184, 267, 366, 481, 611, 756 ] 1467*, [ 7, 22, 49, 92, 152, 228, 320, 428, 552, 692 ] 1412 H402 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,myty,myt 1 y ,mzt 1 z [ 7, 22, 49, 94, 161, 246, 344, 459, 596, 750 ] 1413 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,myty, r 2 z t 1 y ,mzt 1 z [ 7, 23, 54, 99, 153, 222, 305, 395, 502, 623 ] 1414 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,myty, r 1 x t 1 y , rxtz [ 7, 23, 54, 104, 172, 256, 356, 472, 604, 752 ] 1415 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,myty,myt 1 y ,mzt 1 z 1416 H419 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,myty,myt 1 y ,mzt 1 z [ 7, 23, 56, 109, 169, 245, 342, 442, 564, 706 ] 1417 H407 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 1 x ty, r 2 z t 1 y , rxt 1 z [ 7, 24, 54, 103, 173, 261, 366, 485, 620, 773 ] 1418 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,myty, r 2 z t 1 y , r 2 y tz [ 7, 24, 54, 106, 178, 266, 370, 490, 626, 778 ] 1419 H407 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y t 1 z [ 7, 24, 55, 102, 166, 246, 342, 454, 582, 726 ] 1420 H404 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x ,myty,myt 1 y ,mzt 1 z [ 7, 24, 56, 105, 172, 256, 356, 472, 604, 752 ] 1421 H415 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x ,myty,myt 1 y ,mzt 1 z 1422 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,myt 1 y ,mztz,mzt 1 z [ 7, 24, 57, 102, 161, 234, 319, 418, 531, 656 ] 1423 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mzrxt 1 y ,mzrxtz, r 2 x t 1 z [ 7, 24, 57, 104, 168, 248, 344, 456, 584, 728 ] 1424 H391 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 7, 24, 58, 106, 164, 242, 334, 432, 554, 690 ] 1425 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,myty, r 1 x t 1 y , rxtz [ 7, 24, 58, 112, 184, 272, 376, 496, 632, 784 ] 1426 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,myty, r 2 z t 1 y ,mzt 1 z [ 7, 24, 60, 106, 166, 240, 326, 426, 540, 666 ] 94 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 1427 H388 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mzrxty, it 1 y ,mzrxt 1 z [ 7, 24, 60, 110, 180, 268, 369, 488, 626, 780 ] 1428 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 x ty, it 1 y , r 2 x t 1 z [ 7, 25, 58, 111, 184, 272, 376, 496, 632, 784 ] 1429 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,myty, r 2 z t 1 y , r 2 y tz [ 7, 25, 58, 112, 173, 252, 349, 450, 571, 714 ] 1430 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mxr 1 x ty, r 2 z t 1 y ,mxrxt 1 z [ 7, 25, 59, 110, 172, 250, 346, 448, 570, 710 ] 1431 H402 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,myty,mxr 1 x t 1 y ,mxrxtz [ 7, 25, 60, 107, 170, 249, 340, 447, 570, 705 ] 1432 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mzrxt 1 y ,mzrxtz, r 2 x t 1 z [ 7, 25, 60, 110, 178, 262, 362, 478, 610, 758 ] 1433 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 x ty, r 2 x t 1 y , r 2 x t 1 z 1434 H401 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 7, 25, 63, 116, 176, 256, 352, 452, 576, 716 ] 1435 H418 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 1 x ty, r 2 z t 1 y , rxt 1 z [ 7, 25, 64, 112, 175, 257, 345, 455, 578, 708 ] 1436 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,myty,mxr 1 x t 1 y ,mxrxtz [ 7, 26, 59, 110, 174, 258, 354, 470, 598, 746 ] 1437 H392 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 7, 26, 59, 112, 184, 272, 376, 496, 632, 784 ] 1438 H418 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y t 1 z [ 7, 26, 60, 111, 178, 262, 362, 478, 610, 758 ] 1439 H396 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z 1440 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 7, 26, 62, 113, 182, 270, 376, 494, 632, 782 ] 1441 H410 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y t 1 z [ 7, 26, 63, 115, 176, 256, 352, 452, 576, 716 ] 1442 H419 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,myty,mxr 1 x t 1 y ,mxrxtz [ 7, 26, 63, 115, 180, 261, 356, 467, 591, 731 ] 1443 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 y rxty, it 1 y , r 2 y rxt 1 z [ 7, 26, 63, 115, 181, 261, 357, 467, 591, 731 ] 1444 H391 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 y rxt 1 y , r 2 y rxtz, r 2 x t 1 z [ 7, 26, 63, 116, 186, 276, 378, 500, 634, 788 ] 1445 H388 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , it 1 y , itz, it 1 z [ 7, 26, 63, 116, 187, 276, 379, 500, 635, 788 ] 1446 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , it 1 y , itz, r 2 x t 1 z [ 7, 26, 64, 115, 184, 272, 376, 496, 632, 784 ] 1447 H414 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , r 2 z ty, r 2 z t 1 y , r 2 y t 1 z [ 7, 26, 65, 116, 183, 265, 359, 469, 595, 733 ] 1448 H400 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mzrxty, it 1 y ,mzrxt 1 z [ 7, 26, 65, 119, 193, 280, 384, 504, 640, 792 ] 1449 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 x ty, it 1 y , r 2 x t 1 z [ 7, 26, 67, 116, 184, 265, 363, 476, 603, 748 ] 1450 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 y rxtz [ 7, 27, 63, 114, 179, 259, 355, 465, 593, 734 ] 1451 H414 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , r 1 x ty, r 2 z t 1 y , rxt 1 z [ 7, 27, 64, 116, 176, 256, 352, 452, 576, 716 ] 1452 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mxr 1 x ty, r 2 z t 1 y ,mxrxt 1 z [ 7, 27, 65, 116, 182, 263, 359, 469, 597, 736 ] 1453 H415 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x ,myty,mxr 1 x t 1 y ,mxrxtz [ 7, 27, 66, 117, 183, 265, 359, 469, 595, 733 ] 1454 H401 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 y rxt 1 y , r 2 y rxtz, r 2 x t 1 z [ 7, 27, 66, 120, 187, 268, 368, 481, 608, 751 ] 1455 H410 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , r 1 x ty, r 2 z t 1 y , rxt 1 z [ 7, 27, 66, 120, 192, 280, 384, 504, 640, 792 ] 1456 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , it 1 y , itz, r 2 x t 1 z [ 7, 27, 68, 120, 187, 268, 368, 481, 608, 751 ] 1457 H404 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x ,myty,mxr 1 x t 1 y ,mxrxtz [ 7, 28, 63, 115, 178, 259, 351, 461, 582, 721 ] 1458 H389 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x ,mzrxty, it 1 y ,mzrxt 1 z [ 7, 28, 65, 118, 183, 266, 361, 474, 599, 742 ] 1459 H396 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , r 2 y rxt 1 y , r 2 y rxtz, r 2 x t 1 z [ 7, 28, 66, 118, 183, 265, 362, 473, 598, 741 ] 1460 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,mxr 1 x t 1 y ,mxrxtz,mzt 1 z [ 7, 28, 66, 121, 192, 280, 384, 504, 640, 792 ] 1461 H389 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , it 1 y , itz, it 1 z [ 7, 28, 67, 117, 183, 265, 359, 469, 595, 733 ] 1462 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 y rxty, it 1 y , r 2 y rxt 1 z [ 7, 28, 67, 120, 192, 280, 384, 504, 640, 792 ] 1463 H400 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , it 1 y , itz, it 1 z [ 7, 28, 67, 121, 188, 272, 369, 484, 612, 757 ] 1464 H398 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x ,mzrxty, it 1 y ,mzrxt 1 z [ 7, 28, 67, 122, 191, 279, 383, 505, 639, 793 ] 1465 H398 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , it 1 y , itz, it 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 95 Nbr. gr Hi L m X [ 7, 28, 69, 121, 188, 272, 369, 484, 612, 757 ] 1466 H392 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , r 2 y rxt 1 y , r 2 y rxtz, r 2 x t 1 z [ 7, 29, 67, 120, 189, 271, 372, 487, 617, 763 ] 1467 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 y rxt 1 y , r 2 y rxtz, r 2 x t 1 z 25 [ 7, 23, 53, 102, 172, 260, 364, 484, 620, 772 ] 1468*, [ 7, 24, 56, 105, 173, 260, 364, 484, 620, 772 ] 1469*, 1470*, [ 7, 24, 58, 109, 170, 245, 339, 442, 559, 697 ] 1471*, [ 7, 25, 59, 108, 174, 260, 364, 484, 620, 772 ] 1472*, [ 7, 25, 60, 112, 178, 257, 354, 467, 591, 732 ] 1473*, [ 7, 25, 61, 112, 176, 256, 350, 458, 582, 720 ] 1475*, [ 7, 25, 61, 113, 178, 258, 354, 465, 589, 729 ] 1474*, [ 7, 25, 61, 114, 184, 272, 376, 496, 632, 784 ] 1476*, [ 7, 26, 62, 114, 181, 262, 359, 471, 598, 740 ] 1478*, [ 7, 26, 62, 115, 183, 264, 361, 474, 601, 742 ] 1477*, [ 7, 26, 62, 115, 184, 267, 366, 481, 611, 756 ] 1479*, [ 7, 26, 62, 115, 185, 272, 376, 496, 632, 784 ] 1480*, 1481*, [ 7, 27, 63, 116, 183, 266, 363, 476, 604, 748 ] 1482*, [ 7, 27, 63, 116, 186, 272, 376, 496, 632, 784 ] 1483*, [ 8, 26, 59, 112, 184, 272, 376, 496, 632, 784 ] 1468 H650 hmxi r 2 y rx r 2 y rxtx,mxtx,mzrxt 1 x ,mxt 1 x ,myty,myt 1 y , r 2 y t 1 z [ 8, 27, 61, 113, 184, 272, 376, 496, 632, 784 ] 1469 H422 1 r 2 z rx r 2 z tx, r 1 x tx, r 2 y t 1 x , rxt 1 x ,myty,myt 1 y , mzt 1 z 1470 H580 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y , mzt 1 z [ 8, 27, 64, 116, 176, 256, 352, 452, 576, 716 ] 1471 H650 hmxi r 2 y rx r 2 y rxtx,mxtx,mzrxt 1 x ,mxt 1 x ,mxrxty,myt 1 y , mxr 1 x tz [ 8, 28, 63, 114, 184, 272, 376, 496, 632, 784 ] 1472 H652 hmxi r 2 y rx itx, r 2 z rxtx, r 2 x t 1 x , r 2 z rxt 1 x ,myty,myt 1 y , r 2 y t 1 z [ 8, 28, 65, 118, 183, 263, 363, 474, 598, 744 ] 1473 H580 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x , rxty,myt 1 y , r 1 x tz [ 8, 28, 66, 118, 183, 265, 362, 473, 598, 741 ] 1474 H422 1 r 2 z rx r 2 z tx, r 1 x tx, r 2 y t 1 x , rxt 1 x ,myty,mxr 1 x t 1 y , mxrxtz [ 8, 28, 67, 117, 183, 265, 359, 469, 595, 733 ] 1475 H648 hmxi r 2 y rx r 2 y rxtx,mxtx,mzrxt 1 x ,mxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 z rxtz [ 8, 28, 67, 120, 192, 280, 384, 504, 640, 792 ] 1476 H648 hmxi r 2 y rx r 2 y rxtx,mxtx,mzrxt 1 x ,mxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 8, 29, 66, 120, 187, 268, 368, 481, 608, 751 ] 1477 H652 hmxi r 2 y rx itx, r 2 z rxtx, r 2 x t 1 x , r 2 z rxt 1 x ,mxrxty,myt 1 y , mxr 1 x tz [ 8, 29, 67, 119, 187, 268, 367, 479, 607, 750 ] 1478 H560 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y , mzr 1 x tz [ 8, 29, 67, 120, 189, 271, 372, 487, 617, 763 ] 1479 H409 1 r 2 z rx r 2 z tx, r 1 x tx, r 2 y t 1 x , rxt 1 x , r 2 y rxt 1 y , r 2 y rxtz, r 2 x t 1 z [ 8, 29, 67, 121, 192, 280, 384, 504, 640, 792 ] 1480 H409 1 r 2 z rx r 2 z tx, r 1 x tx, r 2 y t 1 x , rxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z 1481 H560 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z [ 8, 30, 67, 121, 188, 272, 369, 484, 612, 757 ] 1482 H649 hmxi r 2 y rx itx, r 2 z rxtx, r 2 x t 1 x , r 2 z rxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 z rxtz [ 8, 30, 67, 122, 192, 280, 384, 504, 640, 792 ] 1483 H649 hmxi r 2 y rx itx, r 2 z rxtx, r 2 x t 1 x , r 2 z rxt 1 x , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z 26A 96 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X [ 5, 14, 29, 50, 77, 110, 149, 194, 245, 302 ] 1484*, [ 6, 19, 42, 74, 112, 162, 221, 288, 362, 447 ] 1484 H393 1 r 2 z rx r 2 y rztx, r 1 y r 1 z t 1 x , r 2 y rzty, ryrxty, r 2 y rztz 26B [ 5, 14, 32, 66, 112, 171, 242, 327, 423, 533 ] 1485*, [ 5, 18, 41, 74, 121, 176, 241, 316, 399, 500 ] 1486*, [ 6, 19, 48, 90, 149, 216, 306, 397, 520, 632 ] 1485 H393 1 r 2 z rx r 2 x ryt 1 x , r 1 y r 1 z t 1 x , ryrxty, r 2 y rztz, r 2 x ryt 1 z [ 6, 23, 51, 93, 146, 210, 284, 370, 472, 581 ] 1486 H393 1 r 2 z rx r 2 y rztx, ryrxtx, r 2 y rzty, r 2 y rztz, r 1 y r 1 z t 1 z 26C [ 5, 18, 54, 122, 212, 299, 398, 519, 649, 804 ] 1487*, [ 6, 23, 66, 131, 213, 291, 398, 517, 649, 806 ] 1487 H393 1 r 2 z rx ryrxtx, r 2 x ryt 1 x , r 2 y rztz, r 2 x ryt 1 z , r 1 y r 1 z t 1 z 27A [ 6, 18, 39, 71, 114, 167, 231, 306, 391, 487 ] 1488*, [ 6, 20, 46, 84, 134, 196, 270, 356, 454, 564 ] 1489*, [ 6, 20, 48, 90, 145, 215, 296, 388, 499, 621 ] 1490*, [ 6, 20, 49, 94, 151, 221, 307, 406, 516, 640 ] 1491*, [ 6, 21, 52, 102, 167, 243, 336, 443, 560, 695 ] 1492*, [ 6, 21, 52, 107, 172, 244, 339, 443, 558, 697 ] 1493*, [ 6, 22, 52, 94, 150, 220, 302, 398, 508, 630 ] 1494*, [ 6, 22, 53, 99, 160, 233, 321, 424, 539, 669 ] 1495*, [ 6, 22, 53, 107, 172, 243, 337, 444, 559, 697 ] 1496*, [ 6, 22, 55, 103, 164, 240, 330, 433, 550, 681 ] 1498*, [ 6, 22, 55, 104, 168, 247, 340, 448, 570, 705 ] 1497*, [ 6, 22, 55, 105, 168, 245, 339, 442, 559, 697 ] 1500*, [ 6, 22, 56, 106, 169, 247, 339, 445, 567, 703 ] 1499*, [ 6, 22, 56, 109, 174, 252, 347, 457, 584, 726 ] 1501*, [ 6, 22, 57, 106, 167, 246, 336, 441, 563, 694 ] 1502*, [ 6, 22, 58, 107, 168, 245, 333, 439, 557, 687 ] 1503*, [ 6, 23, 57, 108, 172, 249, 342, 449, 571, 708 ] 1504*, [ 6, 23, 58, 111, 178, 256, 351, 462, 583, 721 ] 1505*, [ 6, 24, 54, 100, 158, 232, 318, 420, 534, 664 ] 1506*, [ 6, 24, 59, 114, 177, 254, 346, 454, 575, 712 ] 1509*, [ 6, 24, 60, 110, 174, 253, 347, 456, 579, 717 ] 1507*, [ 6, 24, 60, 111, 176, 256, 351, 460, 586, 726 ] 1508*, [ 6, 24, 60, 112, 178, 258, 352, 460, 584, 722 ] 1512*, [ 6, 24, 60, 114, 179, 256, 351, 462, 585, 722 ] 1513*, [ 6, 24, 61, 111, 174, 251, 343, 450, 570, 706 ] 1514*, [ 6, 24, 61, 113, 178, 257, 354, 467, 591, 732 ] 1511*, [ 6, 24, 61, 113, 179, 261, 359, 472, 599, 741 ] 1510*, [ 6, 24, 62, 114, 179, 260, 357, 470, 597, 740 ] 1515*, [ 6, 26, 62, 114, 179, 260, 355, 466, 591, 732 ] 1516*, [ 6, 26, 63, 115, 183, 264, 361, 474, 601, 742 ] 1517*, [ 6, 26, 65, 117, 183, 264, 361, 473, 600, 742 ] 1518*, [ 6, 28, 66, 118, 185, 268, 365, 478, 606, 750 ] 1519*, [ 7, 22, 48, 86, 134, 192, 262, 342, 432, 534 ] 1488 H650 hmxi r 2 y rx mxtx,mxt 1 x ,myt 1 y ,mxrxt 1 y , r 2 y tz,mxr 1 x t 1 z [ 7, 24, 54, 96, 150, 216, 294, 384, 486, 600 ] 1489 H648 hmxi r 2 y rx mxtx,mxt 1 x , r 2 x t 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 z rxt 1 z [ 7, 24, 56, 101, 159, 233, 316, 412, 528, 648 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 97 Nbr. gr Hi L m X 1490 H580 1 mzrx r 2 z rxtx,mxt 1 x ,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z [ 7, 24, 57, 105, 164, 237, 326, 426, 538, 666 ] 1491 H578 1 mzrx mxtx, r 2 y rxt 1 x ,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z [ 7, 25, 59, 113, 177, 253, 351, 455, 574, 715 ] 1492 H571 1 mzrx mxtx, r 2 y rxt 1 x , r 2 z t 1 y ,mxrxt 1 y ,mztz,mxr 1 x t 1 z [ 7, 25, 60, 118, 176, 255, 353, 452, 575, 717 ] 1493 H571 1 mzrx mxtx, r 2 y rxt 1 x , rxty, r 2 z t 1 y , r 1 x tz,mztz [ 7, 26, 59, 104, 164, 237, 322, 422, 535, 660 ] 1494 H560 1 mzrx r 2 z rxtx,mxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz,mzr 1 x t 1 z [ 7, 26, 60, 108, 172, 246, 338, 442, 558, 692 ] 1495 H652 hmxi r 2 y rx r 2 z rxtx, r 2 z rxt 1 x ,myt 1 y ,mxrxt 1 y , r 2 y tz,mxr 1 x t 1 z [ 7, 26, 60, 118, 176, 253, 353, 454, 574, 718 ] 1496 H576 1 mzrx mxtx, r 2 y rxt 1 x , r 2 z t 1 y ,mxrxt 1 y , r 2 y tz,mxr 1 x t 1 z [ 7, 26, 61, 111, 176, 255, 349, 459, 582, 719 ] 1497 H580 1 mzrx itx,mxt 1 x ,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z [ 7, 26, 62, 111, 174, 252, 343, 448, 567, 700 ] 1498 H557 1 mzrx mxtx, r 2 y rxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz,mzr 1 x t 1 z [ 7, 26, 62, 113, 177, 256, 349, 457, 581, 717 ] 1499 H422 1 r 2 z rx r 2 z tx, r 2 y t 1 x ,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mzt 1 z [ 7, 26, 62, 114, 176, 256, 352, 452, 576, 716 ] 1500 H650 hmxi r 2 y rx r 2 y rxtx,mzrxt 1 x ,myt 1 y ,mxrxt 1 y , r 2 y tz,mxr 1 x t 1 z [ 7, 26, 63, 116, 180, 260, 356, 466, 594, 735 ] 1501 H572 1 mzrx r 2 z rxtx, it 1 x , r 2 z t 1 y ,mxrxt 1 y , r 2 y tz,mxr 1 x t 1 z [ 7, 26, 64, 112, 175, 257, 345, 455, 578, 708 ] 1502 H422 1 r 2 z rx r 1 x tx, rxt 1 x ,mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mzt 1 z [ 7, 26, 65, 112, 177, 255, 344, 455, 571, 705 ] 1503 H580 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z [ 7, 27, 64, 116, 180, 262, 356, 466, 592, 730 ] 1504 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mzr 1 x ty, it 1 y , r 2 x tz,mzr 1 x tz [ 7, 27, 64, 119, 184, 263, 362, 471, 593, 736 ] 1505 H561 1 mzrx mxtx, r 2 y rxt 1 x , it 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 z rxt 1 z [ 7, 28, 60, 110, 170, 248, 336, 442, 558, 692 ] 1506 H649 hmxi r 2 y rx r 2 z rxtx, r 2 z rxt 1 x , r 2 x t 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 z rxt 1 z [ 7, 28, 65, 115, 181, 261, 356, 466, 590, 730 ] 1507 H560 1 mzrx itx,mxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz,mzr 1 x t 1 z [ 7, 28, 65, 116, 182, 263, 359, 469, 597, 736 ] 1508 H574 1 mzrx r 2 z rxtx, it 1 x ,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z [ 7, 28, 65, 120, 181, 263, 355, 466, 587, 727 ] 1509 H551 1 mzrx r 2 z rxtx, it 1 x , it 1 y , r 2 z rxt 1 y , itz, r 2 z rxt 1 z [ 7, 28, 66, 117, 185, 267, 365, 478, 606, 750 ] 1510 H409 1 r 2 z rx r 2 z tx, r 2 y t 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z [ 7, 28, 66, 118, 183, 263, 363, 474, 598, 744 ] 1511 H580 1 mzrx itx, r 2 y rxt 1 x ,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z [ 7, 28, 66, 118, 184, 266, 360, 470, 596, 734 ] 1512 H648 hmxi r 2 y rx r 2 y rxtx,mzrxt 1 x , r 2 x t 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 z rxt 1 z [ 7, 28, 66, 120, 183, 264, 361, 472, 595, 735 ] 1513 H556 1 mzrx mxtx, r 2 y rxt 1 x , it 1 y , r 2 z rxt 1 y , itz, r 2 z rxt 1 z [ 7, 28, 67, 115, 182, 259, 354, 462, 583, 722 ] 1514 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz,mzr 1 x t 1 z [ 7, 28, 68, 117, 185, 266, 364, 477, 604, 749 ] 1515 H409 1 r 2 z rx r 1 x tx, rxt 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z [ 7, 30, 66, 119, 184, 267, 362, 475, 600, 743 ] 1516 H555 1 mzrx r 2 z rxtx, it 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz,mzr 1 x t 1 z [ 7, 30, 66, 120, 187, 268, 368, 481, 608, 751 ] 1517 H652 hmxi r 2 y rx itx, r 2 x t 1 x ,myt 1 y ,mxrxt 1 y , r 2 y tz,mxr 1 x t 1 z [ 7, 30, 69, 120, 188, 269, 368, 480, 608, 751 ] 1518 H560 1 mzrx itx, r 2 y rxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz,mzr 1 x t 1 z [ 7, 32, 68, 122, 189, 273, 370, 485, 613, 758 ] 1519 H649 hmxi r 2 y rx itx, r 2 x t 1 x , r 2 x t 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 z rxt 1 z 27B [ 6, 19, 47, 95, 160, 240, 336, 448, 576, 720 ] 1520*, [ 6, 20, 51, 107, 184, 274, 378, 498, 634, 786 ] 1521*, [ 6, 21, 58, 119, 193, 281, 387, 508, 644, 796 ] 1522*, [ 6, 21, 58, 120, 196, 284, 388, 508, 644, 796 ] 1523*, [ 6, 21, 58, 124, 201, 288, 392, 512, 648, 800 ] 1524*, [ 6, 22, 60, 120, 195, 285, 391, 512, 648, 800 ] 1525*, [ 6, 22, 60, 126, 208, 296, 397, 516, 652, 804 ] 1526*, [ 6, 22, 62, 128, 198, 283, 393, 517, 647, 795 ] 1527*, [ 6, 22, 62, 129, 207, 293, 396, 516, 652, 804 ] 1528*, [ 6, 22, 64, 130, 205, 292, 396, 516, 652, 804 ] 98 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 1529*, [ 6, 23, 61, 125, 197, 284, 388, 508, 644, 796 ] 1531*, [ 6, 23, 62, 122, 196, 284, 388, 508, 644, 796 ] 1530*, [ 6, 23, 65, 125, 198, 288, 392, 512, 648, 800 ] 1532*, [ 6, 23, 65, 129, 200, 288, 392, 512, 648, 800 ] 1533*, [ 6, 23, 65, 131, 205, 292, 395, 515, 653, 805 ] 1534*, [ 6, 23, 65, 133, 204, 288, 392, 512, 648, 800 ] 1535*, [ 6, 23, 67, 135, 206, 291, 397, 517, 651, 803 ] 1536*, [ 6, 23, 69, 132, 200, 288, 392, 512, 648, 800 ] 1537*, [ 6, 24, 65, 130, 208, 294, 396, 516, 652, 804 ] 1538*, [ 6, 24, 65, 135, 208, 292, 396, 516, 652, 804 ] 1539*, [ 6, 24, 65, 139, 212, 292, 396, 516, 652, 804 ] 1540*, [ 6, 24, 67, 135, 207, 292, 396, 516, 652, 804 ] 1541*, [ 6, 24, 69, 133, 206, 292, 396, 516, 652, 804 ] 1542*, [ 6, 24, 71, 138, 207, 292, 396, 516, 652, 804 ] 1543*, [ 6, 25, 69, 130, 200, 288, 392, 512, 648, 800 ] 1544*, [ 6, 25, 69, 131, 200, 288, 392, 512, 648, 800 ] 1545*, [ 6, 25, 70, 129, 200, 288, 392, 512, 648, 800 ] 1546*, [ 6, 26, 70, 137, 207, 292, 396, 516, 652, 804 ] 1547*, [ 6, 26, 72, 136, 206, 292, 396, 516, 652, 804 ] 1548*, 1549*, [ 6, 27, 69, 129, 200, 288, 392, 512, 648, 800 ] 1550*, [ 6, 28, 71, 135, 206, 292, 396, 516, 652, 804 ] 1551*, [ 7, 23, 57, 110, 178, 262, 362, 478, 610, 758 ] 1520 H650 hmxi r 2 y rx mxtx,mxt 1 x ,myty,myt 1 y ,mxrxt 1 y ,mxr 1 x t 1 z [ 7, 24, 61, 120, 194, 282, 386, 506, 642, 794 ] 1521 H648 hmxi r 2 y rx mxtx,mxt 1 x , r 2 x ty, r 2 x t 1 y , r 2 z rxt 1 y , r 2 z rxt 1 z [ 7, 25, 67, 127, 198, 287, 392, 512, 648, 800 ] 1522 H578 1 mzrx mxtx, r 2 y rxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z [ 7, 25, 67, 128, 200, 288, 392, 512, 648, 800 ] 1523 H580 1 mzrx itx,mxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z [ 7, 25, 67, 133, 200, 292, 392, 516, 648, 804 ] 1524 H422 1 r 2 z rx r 2 z tx, r 2 y t 1 x ,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz [ 7, 26, 68, 127, 201, 289, 393, 514, 651, 802 ] 1525 H571 1 mzrx mxtx, r 2 y rxt 1 x ,myty, r 2 z t 1 y ,mxrxt 1 y ,mxr 1 x t 1 z [ 7, 26, 69, 133, 207, 293, 396, 516, 652, 804 ] 1526 H560 1 mzrx itx,mxt 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 x t 1 y ,mzr 1 x t 1 z [ 7, 26, 71, 133, 196, 291, 395, 518, 644, 803 ] 1527 H571 1 mzrx mxtx, r 2 y rxt 1 x ,myty, rxty, r 2 z t 1 y , r 1 x tz [ 7, 26, 71, 134, 205, 292, 396, 516, 652, 804 ] 1528 H557 1 mzrx mxtx, r 2 y rxt 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 x t 1 y ,mzr 1 x t 1 z [ 7, 26, 73, 133, 204, 292, 396, 516, 652, 804 ] 1529 H409 1 r 2 z rx r 2 z tx, r 2 y t 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz [ 7, 27, 69, 128, 200, 288, 392, 512, 648, 800 ] 1530 H580 1 mzrx r 2 z rxtx,mxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z [ 7, 27, 69, 132, 197, 291, 389, 515, 645, 803 ] 1531 H576 1 mzrx mxtx, r 2 y rxt 1 x , r 2 z ty, r 2 z t 1 y ,mxrxt 1 y ,mxr 1 x t 1 z [ 7, 27, 73, 128, 202, 290, 394, 514, 650, 802 ] 1532 H650 hmxi r 2 y rx r 2 y rxtx,mzrxt 1 x ,myty,myt 1 y ,mxrxt 1 y ,mxr 1 x t 1 z [ 7, 27, 73, 132, 200, 292, 392, 516, 648, 804 ] 1533 H652 hmxi r 2 y rx itx, r 2 x t 1 x ,myty,myt 1 y ,mxrxt 1 y ,mxr 1 x t 1 z [ 7, 27, 73, 134, 204, 292, 394, 517, 653, 804 ] 1534 H561 1 mzrx mxtx, r 2 y rxt 1 x , r 2 x ty, it 1 y , r 2 z rxt 1 y , r 2 z rxt 1 z [ 7, 27, 73, 136, 200, 292, 392, 516, 648, 804 ] 1535 H572 1 mzrx r 2 z rxtx, it 1 x , r 2 z ty, r 2 z t 1 y ,mxrxt 1 y ,mxr 1 x t 1 z [ 7, 27, 76, 135, 202, 293, 397, 516, 650, 805 ] 1536 H561 1 mzrx mxtx, r 2 y rxt 1 x , r 2 x ty,mzr 1 x ty, it 1 y ,mzr 1 x tz [ 7, 27, 77, 131, 201, 291, 393, 515, 649, 803 ] 1537 H580 1 mzrx itx, r 2 y rxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z [ 7, 28, 72, 134, 206, 292, 396, 516, 652, 804 ] 1538 H560 1 mzrx r 2 z rxtx,mxt 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 x t 1 y ,mzr 1 x t 1 z [ 7, 28, 73, 138, 202, 294, 394, 518, 650, 806 ] 1539 H649 hmxi r 2 y rx itx, r 2 x t 1 x , r 2 x ty, r 2 x t 1 y , r 2 z rxt 1 y , r 2 z rxt 1 z [ 7, 28, 73, 142, 202, 294, 394, 518, 650, 806 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 99 Nbr. gr Hi L m X 1540 H551 1 mzrx r 2 z rxtx, it 1 x , ity, it 1 y , r 2 z rxt 1 y , r 2 z rxt 1 z [ 7, 28, 75, 136, 203, 293, 395, 517, 651, 805 ] 1541 H556 1 mzrx mxtx, r 2 y rxt 1 x , ity, it 1 y , r 2 z rxt 1 y , r 2 z rxt 1 z [ 7, 28, 77, 132, 206, 290, 398, 514, 654, 802 ] 1542 H648 hmxi r 2 y rx r 2 y rxtx,mzrxt 1 x , r 2 x ty, r 2 x t 1 y , r 2 z rxt 1 y , r 2 z rxt 1 z [ 7, 28, 79, 135, 204, 292, 396, 516, 652, 804 ] 1543 H560 1 mzrx itx, r 2 y rxt 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 x t 1 y ,mzr 1 x t 1 z [ 7, 29, 75, 131, 201, 291, 393, 515, 649, 803 ] 1544 H580 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z [ 7, 29, 75, 132, 200, 292, 392, 516, 648, 804 ] 1545 H574 1 mzrx r 2 z rxtx, it 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z [ 7, 29, 76, 129, 203, 289, 395, 513, 651, 801 ] 1546 H422 1 r 2 z rx r 1 x tx, rxt 1 x ,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz [ 7, 30, 76, 137, 202, 294, 394, 518, 650, 806 ] 1547 H555 1 mzrx r 2 z rxtx, it 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 x t 1 y ,mzr 1 x t 1 z [ 7, 30, 78, 134, 204, 292, 396, 516, 652, 804 ] 1548 H409 1 r 2 z rx r 1 x tx, rxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz 1549 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 x t 1 y ,mzr 1 x t 1 z [ 7, 31, 73, 132, 200, 292, 392, 516, 648, 804 ] 1550 H652 hmxi r 2 y rx r 2 z rxtx, r 2 z rxt 1 x ,myty,myt 1 y ,mxrxt 1 y ,mxr 1 x t 1 z [ 7, 32, 75, 136, 202, 294, 394, 518, 650, 806 ] 1551 H649 hmxi r 2 y rx r 2 z rxtx, r 2 z rxt 1 x , r 2 x ty, r 2 x t 1 y , r 2 z rxt 1 y , r 2 z rxt 1 z 28A [ 7, 23, 52, 96, 154, 224, 308, 406, 516, 640 ] 1552*, [ 7, 23, 54, 107, 171, 244, 339, 443, 558, 697 ] 1553*, [ 7, 24, 55, 104, 170, 246, 337, 443, 560, 695 ] 1554*, [ 7, 24, 55, 107, 171, 243, 337, 444, 559, 697 ] 1556*, [ 7, 24, 56, 104, 168, 247, 340, 448, 570, 705 ] 1555*, [ 7, 25, 58, 105, 166, 241, 330, 433, 550, 681 ] 1557*, [ 7, 25, 59, 106, 168, 245, 333, 439, 557, 687 ] 1558*, [ 7, 25, 59, 108, 171, 249, 342, 449, 571, 708 ] 1559*, [ 7, 25, 60, 110, 170, 245, 339, 442, 559, 697 ] 1560*, 1561*, [ 7, 26, 60, 109, 170, 245, 339, 442, 559, 697 ] 1563*, 1564*, [ 7, 26, 60, 109, 173, 252, 347, 457, 584, 726 ] 1562*, [ 7, 26, 61, 110, 174, 253, 347, 456, 579, 717 ] 1565*, [ 7, 26, 61, 111, 176, 256, 351, 460, 586, 726 ] 1566*, [ 7, 26, 61, 113, 180, 257, 351, 462, 583, 721 ] 1567*, [ 7, 26, 62, 113, 178, 257, 354, 467, 591, 732 ] 1568*, [ 7, 26, 62, 114, 178, 256, 351, 462, 585, 722 ] 1569*, [ 7, 27, 62, 110, 174, 251, 343, 450, 570, 706 ] 1571*, [ 7, 27, 62, 112, 178, 257, 354, 467, 591, 732 ] 1570*, [ 7, 27, 65, 115, 178, 258, 352, 460, 584, 722 ] 1572*, 1573*, [ 7, 28, 62, 112, 175, 254, 346, 454, 575, 712 ] 1574*, [ 7, 28, 63, 114, 179, 260, 355, 466, 591, 732 ] 1575*, [ 7, 28, 64, 115, 183, 264, 361, 474, 601, 742 ] 1576*, 1577*, [ 7, 28, 65, 114, 178, 258, 352, 460, 584, 722 ] 1578*, 1579*, [ 7, 28, 66, 117, 183, 264, 361, 473, 600, 742 ] 1580*, [ 7, 29, 66, 116, 183, 264, 361, 473, 600, 742 ] 1581*, [ 7, 30, 67, 118, 185, 268, 365, 478, 606, 750 ] 1582*, 1583*, [ 8, 26, 58, 106, 166, 238, 326, 426, 538, 666 ] 1552 H578 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z [ 8, 26, 61, 117, 176, 255, 353, 452, 575, 717 ] 1553 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , rxty, r 2 z t 1 y , r 1 x tz, mztz [ 8, 27, 60, 114, 179, 254, 351, 455, 574, 715 ] 1554 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z t 1 y ,mxrxt 1 y ,mztz, mxr 1 x t 1 z [ 8, 27, 61, 111, 176, 255, 349, 459, 582, 719 ] 100 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 1555 H580 1 mzrx itx, r 2 z rxtx,mxt 1 x ,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z [ 8, 27, 61, 117, 176, 253, 353, 454, 574, 718 ] 1556 H576 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z t 1 y ,mxrxt 1 y , r 2 y tz, mxr 1 x t 1 z [ 8, 28, 63, 112, 175, 252, 343, 448, 567, 700 ] 1557 H557 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, mzr 1 x t 1 z [ 8, 28, 64, 112, 177, 255, 344, 455, 571, 705 ] 1558 H580 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z [ 8, 28, 65, 115, 180, 262, 356, 466, 592, 730 ] 1559 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, it 1 y , r 2 x tz, mzr 1 x tz [ 8, 28, 65, 116, 176, 256, 352, 452, 576, 716 ] 1560 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z t 1 y ,mxrxt 1 y ,mztz, mxr 1 x t 1 z 1561 H578 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z [ 8, 29, 64, 115, 180, 260, 356, 466, 594, 735 ] 1562 H572 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , r 2 z t 1 y ,mxrxt 1 y , r 2 y tz, mxr 1 x t 1 z [ 8, 29, 64, 116, 176, 256, 352, 452, 576, 716 ] 1563 H576 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z t 1 y ,mxrxt 1 y , r 2 y tz, mxr 1 x t 1 z 1564 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , rxty, r 2 z t 1 y , r 1 x tz, mztz [ 8, 29, 65, 115, 181, 261, 356, 466, 590, 730 ] 1565 H560 1 mzrx itx, r 2 z rxtx,mxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, mzr 1 x t 1 z [ 8, 29, 65, 116, 182, 263, 359, 469, 597, 736 ] 1566 H574 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x ,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z [ 8, 29, 65, 120, 185, 263, 362, 471, 593, 736 ] 1567 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , it 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 z rxt 1 z [ 8, 29, 66, 118, 183, 263, 363, 474, 598, 744 ] 1568 H580 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z [ 8, 29, 67, 119, 183, 264, 361, 472, 595, 735 ] 1569 H556 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , it 1 y , r 2 z rxt 1 y , itz, r 2 z rxt 1 z [ 8, 30, 65, 118, 183, 263, 363, 474, 598, 744 ] 1570 H580 1 mzrx itx, r 2 y rxt 1 x ,mxt 1 x ,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z [ 8, 30, 66, 115, 182, 259, 354, 462, 583, 722 ] 1571 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, mzr 1 x t 1 z [ 8, 30, 69, 118, 184, 266, 360, 470, 596, 734 ] 1572 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , it 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 z rxt 1 z 1573 H557 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, mzr 1 x t 1 z [ 8, 31, 65, 118, 181, 263, 355, 466, 587, 727 ] 1574 H551 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , it 1 y , r 2 z rxt 1 y , itz, r 2 z rxt 1 z [ 8, 31, 66, 119, 184, 267, 362, 475, 600, 743 ] 1575 H555 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, mzr 1 x t 1 z [ 8, 31, 66, 120, 187, 268, 368, 481, 608, 751 ] 1576 H572 1 mzrx itx, it 1 x , r 2 z rxt 1 x , r 2 z t 1 y ,mxrxt 1 y , r 2 y tz, mxr 1 x t 1 z 1577 H574 1 mzrx itx, it 1 x , r 2 z rxt 1 x ,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z [ 8, 31, 68, 118, 184, 266, 360, 470, 596, 734 ] 1578 H556 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , it 1 y , r 2 z rxt 1 y , itz, r 2 z rxt 1 z 1579 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, it 1 y , r 2 x tz, mzr 1 x tz [ 8, 31, 69, 120, 188, 269, 368, 480, 608, 751 ] 1580 H560 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, mzr 1 x t 1 z [ 8, 32, 68, 120, 188, 269, 368, 480, 608, 751 ] 1581 H560 1 mzrx itx, r 2 y rxt 1 x ,mxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, mzr 1 x t 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 101 Nbr. gr Hi L m X [ 8, 33, 68, 122, 189, 273, 370, 485, 613, 758 ] 1582 H551 1 mzrx itx, it 1 x , r 2 z rxt 1 x , it 1 y , r 2 z rxt 1 y , itz, r 2 z rxt 1 z 1583 H555 1 mzrx itx, it 1 x , r 2 z rxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, mzr 1 x t 1 z 28B [ 7, 24, 61, 121, 196, 284, 388, 508, 644, 796 ] 1584*, [ 7, 24, 64, 128, 197, 283, 393, 517, 647, 795 ] 1585*, [ 7, 25, 63, 122, 196, 284, 388, 508, 644, 796 ] 1586*, [ 7, 25, 63, 122, 198, 288, 392, 512, 648, 800 ] 1587*, [ 7, 25, 63, 125, 196, 284, 388, 508, 644, 796 ] 1588*, [ 7, 25, 65, 131, 209, 294, 396, 516, 652, 804 ] 1589*, [ 7, 25, 69, 135, 205, 291, 397, 517, 651, 803 ] 1590*, [ 7, 26, 66, 130, 208, 294, 396, 516, 652, 804 ] 1591*, [ 7, 26, 68, 133, 207, 293, 395, 515, 653, 805 ] 1592*, [ 7, 26, 69, 135, 206, 292, 396, 516, 652, 804 ] 1596*, [ 7, 26, 70, 130, 200, 288, 392, 512, 648, 800 ] 1593*, 1594*, [ 7, 26, 70, 131, 200, 288, 392, 512, 648, 800 ] 1595*, [ 7, 27, 69, 131, 201, 288, 392, 512, 648, 800 ] 1597*, [ 7, 27, 70, 129, 200, 288, 392, 512, 648, 800 ] 1598*, 1599*, [ 7, 27, 70, 130, 200, 288, 392, 512, 648, 800 ] 1600*, [ 7, 27, 70, 131, 200, 288, 392, 512, 648, 800 ] 1601*, [ 7, 27, 72, 137, 207, 292, 396, 516, 652, 804 ] 1602*, [ 7, 27, 74, 136, 206, 292, 396, 516, 652, 804 ] 1603*, 1604*, [ 7, 28, 70, 129, 200, 288, 392, 512, 648, 800 ] 1605*, [ 7, 28, 70, 137, 208, 292, 396, 516, 652, 804 ] 1606*, [ 7, 28, 71, 137, 207, 292, 396, 516, 652, 804 ] 1607*, [ 7, 28, 73, 136, 206, 292, 396, 516, 652, 804 ] 1608*, [ 7, 28, 74, 135, 206, 292, 396, 516, 652, 804 ] 1609*, 1610*, [ 7, 29, 70, 129, 200, 288, 392, 512, 648, 800 ] 1611*, 1612*, [ 7, 29, 73, 135, 206, 292, 396, 516, 652, 804 ] 1613*, [ 7, 30, 72, 135, 206, 292, 396, 516, 652, 804 ] 1614*, 1615*, [ 8, 27, 68, 128, 200, 288, 392, 512, 648, 800 ] 1584 H578 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z [ 8, 27, 72, 132, 196, 291, 395, 518, 644, 803 ] 1585 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, rxty, r 2 z t 1 y , r 1 x tz [ 8, 28, 69, 128, 200, 288, 392, 512, 648, 800 ] 1586 H580 1 mzrx itx, r 2 z rxtx,mxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z [ 8, 28, 69, 128, 203, 290, 393, 514, 651, 802 ] 1587 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, r 2 z t 1 y ,mxrxt 1 y , mxr 1 x t 1 z [ 8, 28, 70, 131, 197, 291, 389, 515, 645, 803 ] 1588 H576 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z ty, r 2 z t 1 y ,mxrxt 1 y , mxr 1 x t 1 z [ 8, 28, 72, 135, 206, 292, 396, 516, 652, 804 ] 1589 H557 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , mzr 1 x t 1 z [ 8, 28, 77, 134, 202, 293, 397, 516, 650, 805 ] 1590 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty,mzr 1 x ty, it 1 y , mzr 1 x tz [ 8, 29, 72, 134, 206, 292, 396, 516, 652, 804 ] 1591 H560 1 mzrx itx, r 2 z rxtx,mxt 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , mzr 1 x t 1 z [ 8, 29, 74, 135, 205, 292, 394, 517, 653, 804 ] 1592 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty, it 1 y , r 2 z rxt 1 y , r 2 z rxt 1 z 102 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X [ 8, 29, 76, 130, 202, 290, 394, 514, 650, 802 ] 1593 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, r 2 z t 1 y ,mxrxt 1 y , mxr 1 x t 1 z 1594 H578 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z [ 8, 29, 76, 131, 201, 291, 393, 515, 649, 803 ] 1595 H580 1 mzrx itx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z [ 8, 29, 76, 135, 203, 293, 395, 517, 651, 805 ] 1596 H556 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , ity, it 1 y , r 2 z rxt 1 y , r 2 z rxt 1 z [ 8, 30, 74, 133, 200, 292, 392, 516, 648, 804 ] 1597 H572 1 mzrx itx, it 1 x , r 2 z rxt 1 x , r 2 z ty, r 2 z t 1 y ,mxrxt 1 y , mxr 1 x t 1 z [ 8, 30, 75, 130, 202, 290, 394, 514, 650, 802 ] 1598 H576 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z ty, r 2 z t 1 y ,mxrxt 1 y , mxr 1 x t 1 z 1599 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, rxty, r 2 z t 1 y , r 1 x tz [ 8, 30, 75, 131, 201, 291, 393, 515, 649, 803 ] 1600 H580 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z [ 8, 30, 75, 132, 200, 292, 392, 516, 648, 804 ] 1601 H574 1 mzrx itx, it 1 x , r 2 z rxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z [ 8, 30, 78, 135, 204, 292, 396, 516, 652, 804 ] 1602 H560 1 mzrx itx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , mzr 1 x t 1 z [ 8, 30, 80, 132, 206, 290, 398, 514, 654, 802 ] 1603 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty, it 1 y , r 2 z rxt 1 y , r 2 z rxt 1 z 1604 H557 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , mzr 1 x t 1 z [ 8, 31, 74, 131, 201, 291, 393, 515, 649, 803 ] 1605 H580 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z [ 8, 31, 75, 138, 202, 294, 394, 518, 650, 806 ] 1606 H551 1 mzrx itx, it 1 x , r 2 z rxt 1 x , ity, it 1 y , r 2 z rxt 1 y , r 2 z rxt 1 z [ 8, 31, 76, 137, 202, 294, 394, 518, 650, 806 ] 1607 H555 1 mzrx itx, it 1 x , r 2 z rxt 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , mzr 1 x t 1 z [ 8, 31, 78, 134, 204, 292, 396, 516, 652, 804 ] 1608 H560 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , mzr 1 x t 1 z [ 8, 31, 79, 132, 206, 290, 398, 514, 654, 802 ] 1609 H556 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , ity, it 1 y , r 2 z rxt 1 y , r 2 z rxt 1 z 1610 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty,mzr 1 x ty, it 1 y , mzr 1 x tz [ 8, 32, 73, 132, 200, 292, 392, 516, 648, 804 ] 1611 H572 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , r 2 z ty, r 2 z t 1 y ,mxrxt 1 y , mxr 1 x t 1 z 1612 H574 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z [ 8, 32, 77, 134, 204, 292, 396, 516, 652, 804 ] 1613 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , mzr 1 x t 1 z [ 8, 33, 75, 136, 202, 294, 394, 518, 650, 806 ] 1614 H551 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , ity, it 1 y , r 2 z rxt 1 y , r 2 z rxt 1 z 1615 H555 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , mzr 1 x t 1 z 29A [ 7, 22, 48, 86, 134, 192, 262, 342, 432, 534 ] 1616*, [ 7, 22, 50, 98, 161, 235, 328, 432, 548, 686 ] 1617*, [ 7, 23, 52, 98, 160, 235, 327, 432, 549, 685 ] 1619*, [ 7, 23, 52, 100, 163, 238, 332, 437, 553, 691 ] 1618*, [ 7, 23, 57, 106, 167, 246, 336, 441, 563, 694 ] 1620*, [ 7, 24, 54, 96, 150, 216, 294, 384, 486, 600 ] 1622*, [ 7, 24, 55, 99, 156, 227, 311, 408, 519, 643 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 103 Nbr. gr Hi L m X 1621*, [ 7, 24, 57, 107, 169, 245, 339, 442, 559, 697 ] 1623*, 1624*, [ 7, 25, 58, 107, 171, 250, 345, 455, 582, 724 ] 1625*, [ 7, 25, 58, 108, 174, 252, 346, 456, 578, 716 ] 1629*, [ 7, 25, 59, 109, 170, 245, 339, 442, 559, 697 ] 1626*, 1627*, [ 7, 25, 59, 109, 173, 251, 345, 455, 579, 717 ] 1628*, [ 7, 25, 60, 109, 173, 252, 345, 453, 576, 713 ] 1631*, [ 7, 25, 60, 111, 176, 256, 351, 460, 586, 726 ] 1630*, [ 7, 25, 62, 114, 179, 260, 357, 470, 597, 740 ] 1632*, [ 7, 26, 62, 113, 178, 258, 352, 460, 584, 722 ] 1634*, 1635*, [ 7, 26, 62, 113, 178, 258, 354, 465, 589, 729 ] 1633*, [ 7, 27, 60, 109, 171, 249, 340, 447, 567, 703 ] 1636*, [ 7, 27, 62, 114, 179, 260, 355, 466, 591, 732 ] 1637*, [ 7, 27, 63, 112, 179, 260, 354, 466, 592, 734 ] 1640*, [ 7, 27, 63, 115, 183, 264, 361, 474, 601, 742 ] 1638*, 1639*, [ 7, 27, 64, 114, 178, 258, 352, 460, 584, 722 ] 1641*, 1642*, [ 7, 28, 65, 117, 186, 269, 368, 483, 613, 758 ] 1643*, [ 7, 29, 66, 118, 185, 268, 365, 478, 606, 750 ] 1644*, 1645*, [ 8, 25, 56, 101, 155, 224, 307, 397, 504, 625 ] 1616 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 z t 1 y , r 1 x t 1 y , rxtz, mzt 1 z [ 8, 25, 57, 109, 169, 245, 342, 442, 564, 706 ] 1617 H407 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z [ 8, 26, 58, 112, 173, 252, 349, 450, 571, 714 ] 1618 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mxr 1 x ty, r 2 z t 1 y ,mxrxt 1 z , mzt 1 z [ 8, 26, 59, 110, 172, 250, 346, 448, 570, 710 ] 1619 H402 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mxr 1 x t 1 y ,myt 1 y ,mxrxtz, mzt 1 z [ 8, 26, 64, 112, 175, 257, 345, 455, 578, 708 ] 1620 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,myty,mxr 1 x t 1 y ,mxrxtz, mztz [ 8, 27, 61, 107, 167, 241, 327, 427, 541, 667 ] 1621 H388 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , it 1 y ,mzrxt 1 y ,mzrxtz, it 1 z 1622 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , it 1 y ,mzrxt 1 y ,mzrxtz, r 2 x t 1 z [ 8, 27, 63, 115, 176, 256, 352, 452, 576, 716 ] 1623 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 z t 1 y , r 1 x t 1 y , rxtz, mzt 1 z 1624 H419 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mxr 1 x t 1 y ,myt 1 y ,mxrxtz, mzt 1 z [ 8, 28, 63, 114, 179, 259, 355, 465, 593, 734 ] 1625 H414 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z [ 8, 28, 64, 116, 176, 256, 352, 452, 576, 716 ] 1626 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mxr 1 x ty, r 2 z t 1 y ,mxrxt 1 z , mzt 1 z 1627 H418 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z [ 8, 28, 64, 116, 181, 262, 357, 468, 592, 732 ] 1628 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 y rxty, it 1 y , r 2 y rxt 1 z , r 2 x t 1 z [ 8, 28, 64, 116, 182, 262, 358, 468, 592, 732 ] 1629 H391 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z [ 8, 28, 65, 116, 182, 263, 359, 469, 597, 736 ] 1630 H415 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x ,mxr 1 x t 1 y ,myt 1 y ,mxrxtz, mzt 1 z [ 8, 28, 66, 115, 183, 262, 359, 467, 594, 731 ] 1631 H393 1 r 2 z rx r 2 y rztx, r 2 x ryt 1 x , r 1 y r 1 z t 1 x , r 2 y rzty, ryrxty, r 2 y rztz, r 2 x ryt 1 z [ 8, 28, 68, 117, 185, 266, 364, 477, 604, 749 ] 104 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 1632 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 y rxtz, r 2 x tz [ 8, 29, 66, 118, 183, 265, 362, 473, 598, 741 ] 1633 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,mxr 1 x t 1 y ,myt 1 y ,mxrxtz, mzt 1 z [ 8, 29, 67, 118, 184, 266, 360, 470, 596, 734 ] 1634 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , it 1 y ,mzrxt 1 y ,mzrxtz, r 2 x t 1 z 1635 H401 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z [ 8, 30, 64, 116, 179, 260, 352, 462, 583, 722 ] 1636 H389 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , it 1 y ,mzrxt 1 y ,mzrxtz, it 1 z [ 8, 30, 66, 119, 184, 267, 362, 475, 600, 743 ] 1637 H396 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z [ 8, 30, 66, 120, 187, 268, 368, 481, 608, 751 ] 1638 H410 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z 1639 H404 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x ,mxr 1 x t 1 y ,myt 1 y ,mxrxtz, mzt 1 z [ 8, 30, 67, 119, 187, 269, 367, 479, 609, 752 ] 1640 H393 1 r 2 z rx r 2 y rztx, ryrxtx, r 1 y r 1 z t 1 x , r 2 y rzty, ryrxty, r 2 y rztz, r 1 y r 1 z t 1 z [ 8, 30, 68, 118, 184, 266, 360, 470, 596, 734 ] 1641 H400 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , it 1 y ,mzrxt 1 y ,mzrxtz, it 1 z 1642 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 y rxty, it 1 y , r 2 y rxt 1 z , r 2 x t 1 z [ 8, 31, 68, 121, 190, 272, 373, 488, 618, 764 ] 1643 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z [ 8, 32, 68, 122, 189, 273, 370, 485, 613, 758 ] 1644 H398 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , it 1 y ,mzrxt 1 y ,mzrxtz, it 1 z 1645 H392 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z 29B [ 7, 23, 57, 110, 182, 273, 376, 493, 630, 785 ] 1646*, [ 7, 23, 60, 119, 192, 282, 385, 505, 641, 793 ] 1647*, [ 7, 24, 60, 114, 184, 272, 376, 496, 632, 784 ] 1649*, [ 7, 24, 60, 116, 190, 281, 384, 503, 639, 792 ] 1648*, [ 7, 24, 61, 121, 196, 284, 387, 507, 644, 796 ] 1650*, [ 7, 24, 65, 126, 197, 283, 387, 507, 643, 795 ] 1651*, [ 7, 25, 65, 127, 198, 284, 388, 508, 644, 796 ] 1653*, [ 7, 25, 66, 127, 197, 283, 388, 508, 643, 795 ] 1652*, [ 7, 25, 67, 127, 199, 288, 392, 512, 648, 800 ] 1654*, 1655*, [ 7, 26, 67, 129, 200, 288, 392, 512, 648, 800 ] 1656*, [ 7, 26, 69, 129, 200, 288, 392, 512, 648, 800 ] 1657*, 1658*, [ 7, 26, 69, 131, 200, 288, 392, 512, 648, 800 ] 1659*, [ 7, 26, 70, 129, 200, 288, 392, 512, 648, 800 ] 1660*, [ 7, 26, 71, 134, 206, 292, 396, 516, 652, 804 ] 1661*, 1662*, [ 7, 27, 69, 135, 206, 292, 396, 516, 652, 804 ] 1664*, [ 7, 27, 70, 128, 200, 288, 392, 512, 648, 800 ] 1663*, [ 7, 27, 71, 137, 206, 292, 396, 516, 652, 804 ] 1665*, [ 7, 27, 72, 136, 206, 292, 396, 516, 652, 804 ] 1666*, [ 7, 27, 73, 135, 206, 292, 396, 516, 652, 804 ] 1667*, 1668*, [ 7, 28, 69, 129, 200, 288, 392, 512, 648, 800 ] 1669*, 1670*, [ 7, 28, 72, 135, 206, 292, 396, 516, 652, 804 ] 1671*, [ 7, 29, 71, 135, 206, 292, 396, 516, 652, 804 ] 1672*, 1673*, [ 7, 29, 71, 135, 210, 291, 395, 518, 651, 803 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 105 Nbr. gr Hi L m X 1674*, [ 7, 29, 75, 135, 206, 291, 395, 518, 651, 803 ] 1675*, [ 8, 26, 66, 120, 193, 282, 385, 504, 641, 794 ] 1646 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,myty, r 2 z t 1 y , r 1 x t 1 y , rxtz [ 8, 26, 68, 126, 197, 288, 388, 512, 644, 800 ] 1647 H407 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 z ty, r 2 z t 1 y , r 1 x t 1 y , rxtz [ 8, 27, 67, 124, 198, 287, 388, 510, 645, 799 ] 1648 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , mxrxt 1 z [ 8, 27, 68, 122, 194, 282, 386, 506, 642, 794 ] 1649 H402 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,myty,mxr 1 x t 1 y ,myt 1 y , mxrxtz [ 8, 27, 70, 127, 200, 287, 392, 511, 648, 799 ] 1650 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 x ty, it 1 y ,mzrxt 1 y , mzrxtz [ 8, 27, 73, 128, 200, 286, 392, 510, 648, 798 ] 1651 H388 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , ity, it 1 y ,mzrxt 1 y , mzrxtz [ 8, 28, 73, 129, 200, 287, 392, 511, 648, 799 ] 1652 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 y rxty, r 2 x ty, it 1 y , r 2 y rxt 1 z [ 8, 28, 73, 130, 200, 288, 392, 512, 648, 800 ] 1653 H391 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz [ 8, 28, 74, 129, 202, 290, 394, 514, 650, 802 ] 1654 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,myty, r 2 z t 1 y , r 1 x t 1 y , rxtz 1655 H419 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,myty,mxr 1 x t 1 y ,myt 1 y , mxrxtz [ 8, 29, 73, 132, 200, 292, 392, 516, 648, 804 ] 1656 H410 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , r 2 z ty, r 2 z t 1 y , r 1 x t 1 y , rxtz [ 8, 29, 75, 130, 202, 290, 394, 514, 650, 802 ] 1657 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , mxrxt 1 z 1658 H418 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 z ty, r 2 z t 1 y , r 1 x t 1 y , rxtz [ 8, 29, 75, 132, 200, 292, 392, 516, 648, 804 ] 1659 H404 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x ,myty,mxr 1 x t 1 y ,myt 1 y , mxrxtz [ 8, 29, 76, 129, 203, 289, 395, 513, 651, 801 ] 1660 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,myty,mxr 1 x t 1 y ,myt 1 y , mxrxtz [ 8, 29, 78, 132, 206, 290, 398, 514, 654, 802 ] 1661 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 x ty, it 1 y ,mzrxt 1 y , mzrxtz 1662 H401 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz [ 8, 30, 75, 129, 203, 289, 395, 513, 651, 801 ] 1663 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,mxr 1 x t 1 y ,mxrxtz,mztz, mzt 1 z [ 8, 30, 75, 136, 202, 294, 394, 518, 650, 806 ] 1664 H398 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , ity, it 1 y ,mzrxt 1 y , mzrxtz [ 8, 30, 77, 136, 202, 294, 394, 518, 650, 806 ] 1665 H392 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz [ 8, 30, 78, 134, 204, 292, 396, 516, 652, 804 ] 1666 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz [ 8, 30, 79, 132, 206, 290, 398, 514, 654, 802 ] 1667 H400 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , ity, it 1 y ,mzrxt 1 y , mzrxtz 1668 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 y rxty, r 2 x ty, it 1 y , r 2 y rxt 1 z [ 8, 31, 73, 132, 200, 292, 392, 516, 648, 804 ] 1669 H414 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , r 2 z ty, r 2 z t 1 y , r 1 x t 1 y , rxtz 1670 H415 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x ,myty,mxr 1 x t 1 y ,myt 1 y , mxrxtz [ 8, 31, 77, 134, 204, 292, 396, 516, 652, 804 ] 1671 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 y rxt 1 y , r 2 y rxtz, r 2 x tz, r 2 x t 1 z [ 8, 32, 75, 136, 202, 294, 394, 518, 650, 806 ] 106 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 1672 H389 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , ity, it 1 y ,mzrxt 1 y , mzrxtz 1673 H396 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz [ 8, 32, 75, 137, 204, 291, 397, 516, 651, 805 ] 1674 H393 1 r 2 z rx r 2 y rztx, ryrxtx, r 2 x ryt 1 x , r 2 y rzty, r 2 y rztz, r 2 x ryt 1 z , r 1 y r 1 z t 1 z [ 8, 32, 79, 133, 204, 291, 397, 516, 651, 805 ] 1675 H393 1 r 2 z rx ryrxtx, r 2 x ryt 1 x , r 1 y r 1 z t 1 x , ryrxty, r 2 y rztz, r 2 x ryt 1 z , r 1 y r 1 z t 1 z 30A [ 8, 26, 59, 109, 170, 245, 339, 442, 559, 697 ] 1676*, [ 8, 27, 61, 112, 178, 257, 354, 467, 591, 732 ] 1677*, [ 8, 27, 62, 113, 178, 258, 354, 465, 589, 729 ] 1678*, [ 8, 28, 63, 115, 183, 264, 361, 474, 601, 742 ] 1679*, [ 8, 28, 64, 114, 178, 258, 352, 460, 584, 722 ] 1680*, [ 8, 29, 65, 116, 183, 264, 361, 473, 600, 742 ] 1681*, [ 8, 29, 65, 117, 186, 269, 368, 483, 613, 758 ] 1682*, [ 8, 30, 66, 118, 185, 268, 365, 478, 606, 750 ] 1683*, [ 9, 28, 64, 116, 176, 256, 352, 452, 576, 716 ] 1676 H650 hmxi r 2 y rx r 2 y rxtx,mxtx,mzrxt 1 x ,mxt 1 x ,myt 1 y ,mxrxt 1 y , r 2 y tz,mxr 1 x t 1 z [ 9, 29, 65, 118, 183, 263, 363, 474, 598, 744 ] 1677 H580 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,myt 1 y , rxt 1 y , mztz, r 1 x t 1 z [ 9, 29, 66, 118, 183, 265, 362, 473, 598, 741 ] 1678 H422 1 r 2 z rx r 2 z tx, r 1 x tx, r 2 y t 1 x , rxt 1 x ,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mzt 1 z [ 9, 30, 66, 120, 187, 268, 368, 481, 608, 751 ] 1679 H652 hmxi r 2 y rx itx, r 2 z rxtx, r 2 x t 1 x , r 2 z rxt 1 x ,myt 1 y ,mxrxt 1 y , r 2 y tz,mxr 1 x t 1 z [ 9, 30, 68, 118, 184, 266, 360, 470, 596, 734 ] 1680 H648 hmxi r 2 y rx r 2 y rxtx,mxtx,mzrxt 1 x ,mxt 1 x , r 2 x t 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 z rxt 1 z [ 9, 31, 68, 120, 188, 269, 368, 480, 608, 751 ] 1681 H560 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz,mzr 1 x t 1 z [ 9, 31, 68, 121, 190, 272, 373, 488, 618, 764 ] 1682 H409 1 r 2 z rx r 2 z tx, r 1 x tx, r 2 y t 1 x , rxt 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z [ 9, 32, 68, 122, 189, 273, 370, 485, 613, 758 ] 1683 H649 hmxi r 2 y rx itx, r 2 z rxtx, r 2 x t 1 x , r 2 z rxt 1 x , r 2 x t 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 z rxt 1 z 30B [ 8, 27, 69, 129, 200, 288, 392, 512, 648, 800 ] 1684*, [ 8, 28, 69, 129, 200, 288, 392, 512, 648, 800 ] 1685*, [ 8, 28, 70, 128, 200, 288, 392, 512, 648, 800 ] 1686*, [ 8, 28, 73, 135, 206, 292, 396, 516, 652, 804 ] 1687*, [ 8, 29, 69, 129, 200, 288, 392, 512, 648, 800 ] 1688*, [ 8, 29, 72, 135, 206, 292, 396, 516, 652, 804 ] 1689*, 1690*, [ 8, 30, 71, 135, 206, 292, 396, 516, 652, 804 ] 1691*, [ 9, 29, 75, 130, 202, 290, 394, 514, 650, 802 ] 1684 H650 hmxi r 2 y rx r 2 y rxtx,mxtx,mzrxt 1 x ,mxt 1 x ,myty,myt 1 y , mxrxt 1 y ,mxr 1 x t 1 z [ 9, 30, 74, 131, 201, 291, 393, 515, 649, 803 ] 1685 H580 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z [ 9, 30, 75, 129, 203, 289, 395, 513, 651, 801 ] 1686 H422 1 r 2 z rx r 2 z tx, r 1 x tx, r 2 y t 1 x , rxt 1 x ,myty,mxr 1 x t 1 y , myt 1 y ,mxrxtz [ 9, 30, 79, 132, 206, 290, 398, 514, 654, 802 ] 1687 H648 hmxi r 2 y rx r 2 y rxtx,mxtx,mzrxt 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y , r 2 z rxt 1 y , r 2 z rxt 1 z [ 9, 31, 73, 132, 200, 292, 392, 516, 648, 804 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 107 Nbr. gr Hi L m X 1688 H652 hmxi r 2 y rx itx, r 2 z rxtx, r 2 x t 1 x , r 2 z rxt 1 x ,myty,myt 1 y , mxrxt 1 y ,mxr 1 x t 1 z [ 9, 31, 77, 134, 204, 292, 396, 516, 652, 804 ] 1689 H409 1 r 2 z rx r 2 z tx, r 1 x tx, r 2 y t 1 x , rxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz 1690 H560 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y , mzr 1 x t 1 y ,mzr 1 x t 1 z [ 9, 32, 75, 136, 202, 294, 394, 518, 650, 806 ] 1691 H649 hmxi r 2 y rx itx, r 2 z rxtx, r 2 x t 1 x , r 2 z rxt 1 x , r 2 x ty, r 2 x t 1 y , r 2 z rxt 1 y , r 2 z rxt 1 z 31 [ 7, 22, 50, 96, 160, 240, 336, 448, 576, 720 ] 1692*, [ 7, 23, 58, 117, 192, 281, 387, 508, 644, 796 ] 1693*, [ 7, 23, 59, 119, 195, 285, 391, 512, 648, 800 ] 1694*, [ 7, 24, 60, 124, 197, 284, 388, 508, 644, 796 ] 1695*, [ 7, 24, 61, 126, 198, 283, 393, 517, 647, 795 ] 1696*, [ 7, 24, 64, 124, 198, 288, 392, 512, 648, 800 ] 1697*, [ 7, 25, 60, 114, 186, 274, 378, 498, 634, 786 ] 1698*, [ 7, 25, 62, 120, 195, 284, 388, 508, 644, 796 ] 1699*, 1700*, [ 7, 25, 63, 124, 199, 288, 392, 512, 648, 800 ] 1701*, [ 7, 26, 66, 127, 202, 291, 396, 516, 652, 804 ] 1702*, [ 7, 26, 66, 128, 203, 292, 395, 515, 653, 805 ] 1703*, [ 7, 26, 68, 129, 200, 288, 392, 512, 648, 800 ] 1704*, 1705*, [ 7, 26, 69, 128, 200, 288, 392, 512, 648, 800 ] 1706*, [ 7, 27, 68, 131, 204, 291, 397, 517, 651, 803 ] 1707*, [ 7, 27, 68, 132, 205, 292, 396, 516, 652, 804 ] 1708*, [ 7, 27, 70, 130, 204, 292, 396, 516, 652, 804 ] 1709*, [ 7, 28, 68, 128, 200, 288, 392, 512, 648, 800 ] 1710*, 1711*, 1712*, 1713*, [ 7, 28, 69, 128, 203, 292, 396, 516, 652, 804 ] 1714*, 1715*, 1716*, 1717*, [ 7, 29, 73, 133, 204, 292, 396, 516, 652, 804 ] 1718*, 1719*, 1720*, 1721*, [ 7, 31, 72, 132, 204, 292, 396, 516, 652, 804 ] 1722*, 1723*, 1724*, 1725*, 1726*, 1727*, 1728*, [ 8, 25, 58, 110, 178, 262, 362, 478, 610, 758 ] 1692 H650 hmxi r 2 y rx mxtx,mxt 1 x ,myty,myt 1 y ,mxrxt 1 y ,mxr 1 x t 1 z , r 2 y t 1 z [ 8, 26, 66, 126, 198, 287, 392, 512, 648, 800 ] 1693 H578 1 mzrx mxtx, r 2 y rxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z , mzt 1 z [ 8, 26, 67, 127, 201, 289, 393, 514, 651, 802 ] 1694 H571 1 mzrx mxtx, r 2 y rxt 1 x ,myty, r 2 z t 1 y ,mxrxt 1 y ,mxr 1 x t 1 z , r 2 y t 1 z [ 8, 27, 68, 132, 197, 291, 389, 515, 645, 803 ] 1695 H576 1 mzrx mxtx, r 2 y rxt 1 x , r 2 z ty, r 2 z t 1 y ,mxrxt 1 y ,mxr 1 x t 1 z , r 2 y t 1 z [ 8, 27, 69, 133, 196, 291, 395, 518, 644, 803 ] 1696 H571 1 mzrx mxtx, r 2 y rxt 1 x ,myty, rxty, r 2 z t 1 y , r 1 x tz, mztz [ 8, 27, 72, 128, 202, 290, 394, 514, 650, 802 ] 1697 H650 hmxi r 2 y rx r 2 y rxtx,mzrxt 1 x ,myty,myt 1 y ,mxrxt 1 y ,mxr 1 x t 1 z , r 2 y t 1 z [ 8, 28, 66, 122, 194, 282, 386, 506, 642, 794 ] 1698 H648 hmxi r 2 y rx mxtx,mxt 1 x , r 2 x t 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 x t 1 z , r 2 z rxt 1 z [ 8, 28, 68, 127, 200, 288, 392, 512, 648, 800 ] 1699 H580 1 mzrx itx,mxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z , mzt 1 z 1700 H580 1 mzrx r 2 z rxtx,mxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z , mzt 1 z [ 8, 28, 69, 131, 200, 292, 392, 516, 648, 804 ] 1701 H422 1 r 2 z rx r 2 z tx, r 2 y t 1 x ,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz, mztz [ 8, 29, 72, 131, 203, 292, 396, 516, 652, 804 ] 108 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 1702 H557 1 mzrx mxtx, r 2 y rxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, r 2 x t 1 z , mzr 1 x t 1 z [ 8, 29, 72, 132, 204, 292, 394, 517, 653, 804 ] 1703 H561 1 mzrx mxtx, r 2 y rxt 1 x , it 1 y , r 2 z rxt 1 y , r 2 x tz, it 1 z , r 2 z rxt 1 z [ 8, 29, 74, 131, 201, 291, 393, 515, 649, 803 ] 1704 H580 1 mzrx itx, r 2 y rxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z , mzt 1 z 1705 H580 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z , mzt 1 z [ 8, 29, 75, 129, 203, 289, 395, 513, 651, 801 ] 1706 H422 1 r 2 z rx r 1 x tx, rxt 1 x ,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz, mztz [ 8, 30, 74, 133, 202, 293, 397, 516, 650, 805 ] 1707 H561 1 mzrx mxtx, r 2 y rxt 1 x , r 2 x ty,mzr 1 x ty, it 1 y , r 2 x tz, mzr 1 x tz [ 8, 30, 74, 134, 203, 293, 395, 517, 651, 805 ] 1708 H556 1 mzrx mxtx, r 2 y rxt 1 x , it 1 y , r 2 z rxt 1 y , itz, it 1 z , r 2 z rxt 1 z [ 8, 30, 76, 130, 206, 290, 398, 514, 654, 802 ] 1709 H648 hmxi r 2 y rx r 2 y rxtx,mzrxt 1 x , r 2 x t 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 x t 1 z , r 2 z rxt 1 z [ 8, 31, 72, 132, 200, 292, 392, 516, 648, 804 ] 1710 H652 hmxi r 2 y rx itx, r 2 x t 1 x ,myty,myt 1 y ,mxrxt 1 y ,mxr 1 x t 1 z , r 2 y t 1 z 1711 H652 hmxi r 2 y rx r 2 z rxtx, r 2 z rxt 1 x ,myty,myt 1 y ,mxrxt 1 y ,mxr 1 x t 1 z , r 2 y t 1 z 1712 H572 1 mzrx r 2 z rxtx, it 1 x , r 2 z ty, r 2 z t 1 y ,mxrxt 1 y ,mxr 1 x t 1 z , r 2 y t 1 z 1713 H574 1 mzrx r 2 z rxtx, it 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z , mzt 1 z [ 8, 31, 73, 131, 204, 292, 396, 516, 652, 804 ] 1714 H409 1 r 2 z rx r 2 z tx, r 2 y t 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 x t 1 z 1715 H560 1 mzrx itx,mxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, r 2 x t 1 z , mzr 1 x t 1 z 1716 H560 1 mzrx r 2 z rxtx,mxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, r 2 x t 1 z , mzr 1 x t 1 z 1717 H560 1 mzrx r 2 z rxtx,mxt 1 x , r 2 x ty,mzr 1 x ty, r 2 x t 1 y , r 2 x tz, mzr 1 x tz [ 8, 32, 77, 132, 204, 292, 396, 516, 652, 804 ] 1718 H409 1 r 2 z rx r 1 x tx, rxt 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 x t 1 z 1719 H560 1 mzrx itx, r 2 y rxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, r 2 x t 1 z , mzr 1 x t 1 z 1720 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, r 2 x t 1 z , mzr 1 x t 1 z 1721 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x , r 2 x ty,mzr 1 x ty, r 2 x t 1 y , r 2 x tz, mzr 1 x tz [ 8, 34, 74, 134, 202, 294, 394, 518, 650, 806 ] 1722 H649 hmxi r 2 y rx itx, r 2 x t 1 x , r 2 x ty, r 2 z rxty, r 2 x t 1 y , r 2 x tz, r 2 z rxtz 1723 H649 hmxi r 2 y rx r 2 z rxtx, r 2 z rxt 1 x , r 2 x t 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 x t 1 z , r 2 z rxt 1 z 1724 H649 hmxi r 2 y rx itx, r 2 x t 1 x , r 2 x t 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 x t 1 z , r 2 z rxt 1 z 1725 H551 1 mzrx r 2 z rxtx, it 1 x , it 1 y , r 2 z rxt 1 y , itz, it 1 z , r 2 z rxt 1 z 1726 H551 1 mzrx r 2 z rxtx, it 1 x , ity, r 2 z rxty, it 1 y , itz, r 2 z rxtz 1727 H555 1 mzrx r 2 z rxtx, it 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, r 2 x t 1 z , mzr 1 x t 1 z 1728 H555 1 mzrx itx, r 2 z rxt 1 x , r 2 x ty,mzr 1 x ty, r 2 x t 1 y , r 2 x tz, mzr 1 x tz 32 [ 8, 26, 61, 119, 195, 284, 388, 508, 644, 796 ] 1729*, [ 8, 26, 62, 121, 198, 288, 392, 512, 648, 800 ] 1730*, [ 8, 26, 62, 124, 196, 284, 388, 508, 644, 796 ] 1731*, [ 8, 26, 63, 126, 197, 283, 393, 517, 647, 795 ] 1732*, [ 8, 27, 63, 120, 195, 284, 388, 508, 644, 796 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 109 Nbr. gr Hi L m X 1733*, [ 8, 27, 69, 129, 200, 288, 392, 512, 648, 800 ] 1734*, 1735*, [ 8, 28, 69, 128, 200, 288, 392, 512, 648, 800 ] 1736*, 1737*, [ 8, 28, 69, 129, 200, 288, 392, 512, 648, 800 ] 1738*, [ 8, 29, 69, 128, 200, 288, 392, 512, 648, 800 ] 1739*, 1740*, [ 8, 29, 69, 129, 204, 292, 396, 516, 652, 804 ] 1741*, [ 8, 29, 69, 130, 205, 293, 395, 515, 653, 805 ] 1742*, [ 8, 29, 70, 131, 203, 291, 397, 517, 651, 803 ] 1743*, [ 8, 29, 70, 132, 204, 292, 396, 516, 652, 804 ] 1744*, [ 8, 30, 69, 128, 200, 288, 392, 512, 648, 800 ] 1745*, 1746*, 1747*, 1748*, [ 8, 30, 70, 128, 203, 292, 396, 516, 652, 804 ] 1749*, [ 8, 30, 75, 133, 204, 292, 396, 516, 652, 804 ] 1750*, 1751*, [ 8, 31, 74, 133, 204, 292, 396, 516, 652, 804 ] 1752*, [ 8, 31, 75, 132, 204, 292, 396, 516, 652, 804 ] 1753*, 1754*, [ 8, 32, 74, 132, 204, 292, 396, 516, 652, 804 ] 1755*, 1756*, 1757*, [ 8, 33, 73, 132, 204, 292, 396, 516, 652, 804 ] 1758*, 1759*, 1760*, 1761*, 1762*, 1763*, [ 9, 28, 67, 127, 200, 288, 392, 512, 648, 800 ] 1729 H578 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z ,mzt 1 z [ 9, 28, 68, 128, 203, 290, 393, 514, 651, 802 ] 1730 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, r 2 z t 1 y ,mxrxt 1 y , mxr 1 x t 1 z , r 2 y t 1 z [ 9, 28, 69, 131, 197, 291, 389, 515, 645, 803 ] 1731 H576 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z ty, r 2 z t 1 y ,mxrxt 1 y , mxr 1 x t 1 z , r 2 y t 1 z [ 9, 28, 70, 132, 196, 291, 395, 518, 644, 803 ] 1732 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, rxty, r 2 z t 1 y , r 1 x tz,mztz [ 9, 29, 68, 127, 200, 288, 392, 512, 648, 800 ] 1733 H580 1 mzrx itx, r 2 z rxtx,mxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z ,mzt 1 z [ 9, 29, 75, 130, 202, 290, 394, 514, 650, 802 ] 1734 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, r 2 z t 1 y ,mxrxt 1 y , mxr 1 x t 1 z , r 2 y t 1 z 1735 H578 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z ,mzt 1 z [ 9, 30, 74, 130, 202, 290, 394, 514, 650, 802 ] 1736 H576 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z ty, r 2 z t 1 y ,mxrxt 1 y , mxr 1 x t 1 z , r 2 y t 1 z 1737 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, rxty, r 2 z t 1 y , r 1 x tz,mztz [ 9, 30, 74, 131, 201, 291, 393, 515, 649, 803 ] 1738 H580 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z ,mzt 1 z [ 9, 31, 73, 131, 201, 291, 393, 515, 649, 803 ] 1739 H580 1 mzrx itx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z ,mzt 1 z 1740 H580 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z ,mzt 1 z [ 9, 31, 73, 132, 204, 292, 396, 516, 652, 804 ] 1741 H557 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, r 2 x t 1 z ,mzr 1 x t 1 z [ 9, 31, 73, 133, 205, 292, 394, 517, 653, 804 ] 1742 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , it 1 y , r 2 z rxt 1 y , r 2 x tz, it 1 z , r 2 z rxt 1 z [ 9, 31, 75, 132, 202, 293, 397, 516, 650, 805 ] 1743 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty,mzr 1 x ty, it 1 y , r 2 x tz,mzr 1 x tz [ 9, 31, 75, 133, 203, 293, 395, 517, 651, 805 ] 1744 H556 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , it 1 y , r 2 z rxt 1 y , itz, it 1 z , r 2 z rxt 1 z [ 9, 32, 72, 132, 200, 292, 392, 516, 648, 804 ] 1745 H572 1 mzrx itx, it 1 x , r 2 z rxt 1 x , r 2 z ty, r 2 z t 1 y ,mxrxt 1 y , mxr 1 x t 1 z , r 2 y t 1 z 110 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 1746 H572 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , r 2 z ty, r 2 z t 1 y ,mxrxt 1 y , mxr 1 x t 1 z , r 2 y t 1 z 1747 H574 1 mzrx itx, it 1 x , r 2 z rxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z ,mzt 1 z 1748 H574 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z ,mzt 1 z [ 9, 32, 73, 131, 204, 292, 396, 516, 652, 804 ] 1749 H560 1 mzrx itx, r 2 z rxtx,mxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, r 2 x t 1 z ,mzr 1 x t 1 z [ 9, 32, 79, 130, 206, 290, 398, 514, 654, 802 ] 1750 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , it 1 y , r 2 z rxt 1 y , r 2 x tz, it 1 z , r 2 z rxt 1 z 1751 H557 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, r 2 x t 1 z ,mzr 1 x t 1 z [ 9, 33, 77, 132, 204, 292, 396, 516, 652, 804 ] 1752 H560 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, r 2 x t 1 z ,mzr 1 x t 1 z [ 9, 33, 78, 130, 206, 290, 398, 514, 654, 802 ] 1753 H556 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , it 1 y , r 2 z rxt 1 y , itz, it 1 z , r 2 z rxt 1 z 1754 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty,mzr 1 x ty, it 1 y , r 2 x tz,mzr 1 x tz [ 9, 34, 76, 132, 204, 292, 396, 516, 652, 804 ] 1755 H560 1 mzrx itx, r 2 y rxt 1 x ,mxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, r 2 x t 1 z ,mzr 1 x t 1 z 1756 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, r 2 x t 1 z ,mzr 1 x t 1 z 1757 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty,mzr 1 x ty, r 2 x t 1 y , r 2 x tz,mzr 1 x tz [ 9, 35, 74, 134, 202, 294, 394, 518, 650, 806 ] 1758 H551 1 mzrx itx, it 1 x , r 2 z rxt 1 x , it 1 y , r 2 z rxt 1 y , itz, it 1 z , r 2 z rxt 1 z 1759 H551 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , it 1 y , r 2 z rxt 1 y , itz, it 1 z , r 2 z rxt 1 z 1760 H551 1 mzrx itx, r 2 z rxtx, r 2 z rxt 1 x , ity, r 2 z rxty, it 1 y , itz, r 2 z rxtz 1761 H555 1 mzrx itx, it 1 x , r 2 z rxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, r 2 x t 1 z ,mzr 1 x t 1 z 1762 H555 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, r 2 x t 1 z ,mzr 1 x t 1 z 1763 H555 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , r 2 x ty,mzr 1 x ty, r 2 x t 1 y , r 2 x tz,mzr 1 x tz 33 [ 8, 25, 58, 110, 182, 273, 376, 493, 630, 785 ] 1764*, [ 8, 25, 59, 113, 184, 272, 376, 496, 632, 784 ] 1766*, [ 8, 25, 59, 115, 190, 281, 384, 503, 639, 792 ] 1765*, [ 8, 25, 60, 118, 192, 282, 385, 505, 641, 793 ] 1767*, [ 8, 26, 66, 126, 199, 288, 392, 512, 648, 800 ] 1768*, 1769*, [ 8, 27, 68, 128, 200, 288, 392, 512, 648, 800 ] 1770*, 1771*, [ 8, 27, 69, 128, 200, 288, 392, 512, 648, 800 ] 1772*, [ 8, 28, 66, 123, 196, 284, 387, 507, 644, 796 ] 1774*, [ 8, 28, 66, 124, 196, 284, 388, 508, 644, 796 ] 1776*, [ 8, 28, 67, 123, 195, 283, 387, 507, 643, 795 ] 1773*, [ 8, 28, 67, 124, 195, 283, 388, 508, 643, 795 ] 1775*, [ 8, 28, 69, 127, 200, 288, 392, 512, 648, 800 ] 1777*, [ 8, 29, 68, 128, 200, 288, 392, 512, 648, 800 ] 1778*, 1779*, 1780*, 1781*, [ 8, 29, 72, 131, 204, 292, 396, 516, 652, 804 ] 1782*, 1783*, [ 8, 30, 73, 133, 204, 292, 396, 516, 652, 804 ] 1784*, [ 8, 30, 74, 132, 204, 292, 396, 516, 652, 804 ] 1785*, 1786*, [ 8, 31, 73, 132, 204, 292, 396, 516, 652, 804 ] 1787*, [ 8, 32, 72, 132, 204, 292, 396, 516, 652, 804 ] 1788*, 1789*, 1790*, 1791*, [ 9, 27, 66, 120, 193, 282, 385, 504, 641, 794 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 111 Nbr. gr Hi L m X 1764 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,myty, r 2 z t 1 y , r 1 x t 1 y , r 2 y tz, rxtz [ 9, 27, 66, 124, 198, 287, 388, 510, 645, 799 ] 1765 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , mxrxt 1 z ,mzt 1 z [ 9, 27, 67, 122, 194, 282, 386, 506, 642, 794 ] 1766 H402 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,myty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mztz [ 9, 27, 67, 126, 197, 288, 388, 512, 644, 800 ] 1767 H407 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 z ty, r 1 x ty, r 2 z t 1 y , r 2 y t 1 z , rxt 1 z [ 9, 28, 73, 129, 202, 290, 394, 514, 650, 802 ] 1768 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,myty, r 2 z t 1 y , r 1 x t 1 y , r 2 y tz, rxtz 1769 H419 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,myty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mztz [ 9, 29, 74, 130, 202, 290, 394, 514, 650, 802 ] 1770 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , mxrxt 1 z ,mzt 1 z 1771 H418 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 z ty, r 1 x ty, r 2 z t 1 y , r 2 y t 1 z , rxt 1 z [ 9, 29, 75, 129, 203, 289, 395, 513, 651, 801 ] 1772 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,myty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mztz [ 9, 30, 72, 126, 200, 286, 392, 510, 648, 798 ] 1773 H388 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , it 1 y ,mzrxt 1 y , itz, mzrxtz, it 1 z [ 9, 30, 72, 127, 200, 287, 392, 511, 648, 799 ] 1774 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , it 1 y ,mzrxt 1 y , itz, mzrxtz, r 2 x t 1 z 1775 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 y rxty, r 2 x ty, it 1 y , r 2 y rxt 1 z , r 2 x t 1 z [ 9, 30, 72, 128, 200, 288, 392, 512, 648, 800 ] 1776 H391 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 x t 1 z [ 9, 30, 74, 129, 203, 289, 395, 513, 651, 801 ] 1777 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,mxr 1 x t 1 y ,myt 1 y ,mxrxtz, mztz,mzt 1 z [ 9, 31, 72, 132, 200, 292, 392, 516, 648, 804 ] 1778 H414 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , r 2 z ty, r 1 x ty, r 2 z t 1 y , r 2 y t 1 z , rxt 1 z 1779 H415 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x ,myty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mztz 1780 H410 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , r 2 z ty, r 1 x ty, r 2 z t 1 y , r 2 y t 1 z , rxt 1 z 1781 H404 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x ,myty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mztz [ 9, 31, 77, 130, 206, 290, 398, 514, 654, 802 ] 1782 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , it 1 y ,mzrxt 1 y , itz, mzrxtz, r 2 x t 1 z 1783 H401 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 x t 1 z [ 9, 32, 77, 132, 204, 292, 396, 516, 652, 804 ] 1784 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz [ 9, 32, 78, 130, 206, 290, 398, 514, 654, 802 ] 1785 H400 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , it 1 y ,mzrxt 1 y , itz, mzrxtz, it 1 z 1786 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 y rxty, r 2 x ty, it 1 y , r 2 y rxt 1 z , r 2 x t 1 z [ 9, 33, 76, 132, 204, 292, 396, 516, 652, 804 ] 1787 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 x t 1 z [ 9, 34, 74, 134, 202, 294, 394, 518, 650, 806 ] 1788 H389 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , it 1 y ,mzrxt 1 y , itz, mzrxtz, it 1 z 1789 H396 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 x t 1 z 1790 H398 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , it 1 y ,mzrxt 1 y , itz, mzrxtz, it 1 z 1791 H392 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 x t 1 z 34 112 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X [ 9, 28, 68, 128, 200, 288, 392, 512, 648, 800 ] 1792*, [ 9, 29, 68, 128, 200, 288, 392, 512, 648, 800 ] 1793*, [ 9, 29, 69, 127, 200, 288, 392, 512, 648, 800 ] 1794*, [ 9, 30, 68, 128, 200, 288, 392, 512, 648, 800 ] 1795*, [ 9, 31, 74, 132, 204, 292, 396, 516, 652, 804 ] 1796*, [ 9, 32, 73, 132, 204, 292, 396, 516, 652, 804 ] 1797*, 1798*, [ 9, 33, 72, 132, 204, 292, 396, 516, 652, 804 ] 1799*, [ 9, 33, 73, 131, 206, 291, 395, 518, 651, 803 ] 1800*, [ 10, 29, 74, 130, 202, 290, 394, 514, 650, 802 ] 1792 H650 hmxi r 2 y rx r 2 y rxtx,mxtx,mzrxt 1 x ,mxt 1 x ,myty,myt 1 y , mxrxt 1 y ,mxr 1 x t 1 z , r 2 y t 1 z [ 10, 30, 73, 131, 201, 291, 393, 515, 649, 803 ] 1793 H580 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y , rxt 1 y , r 1 x t 1 z ,mzt 1 z [ 10, 30, 74, 129, 203, 289, 395, 513, 651, 801 ] 1794 H422 1 r 2 z rx r 2 z tx, r 1 x tx, r 2 y t 1 x , rxt 1 x ,myty,mxr 1 x t 1 y , myt 1 y ,mxrxtz,mztz [ 10, 31, 72, 132, 200, 292, 392, 516, 648, 804 ] 1795 H652 hmxi r 2 y rx itx, r 2 z rxtx, r 2 x t 1 x , r 2 z rxt 1 x ,myty,myt 1 y , mxrxt 1 y ,mxr 1 x t 1 z , r 2 y t 1 z [ 10, 32, 78, 130, 206, 290, 398, 514, 654, 802 ] 1796 H648 hmxi r 2 y rx r 2 y rxtx,mxtx,mzrxt 1 x ,mxt 1 x , r 2 x t 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 x t 1 z , r 2 z rxt 1 z [ 10, 33, 76, 132, 204, 292, 396, 516, 652, 804 ] 1797 H409 1 r 2 z rx r 2 z tx, r 1 x tx, r 2 y t 1 x , rxt 1 x , r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 x t 1 z 1798 H560 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, r 2 x t 1 z ,mzr 1 x t 1 z [ 10, 34, 74, 134, 202, 294, 394, 518, 650, 806 ] 1799 H649 hmxi r 2 y rx itx, r 2 z rxtx, r 2 x t 1 x , r 2 z rxt 1 x , r 2 x t 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 x t 1 z , r 2 z rxt 1 z [ 10, 34, 75, 133, 204, 291, 397, 516, 651, 805 ] 1800 H393 1 r 2 z rx r 2 y rztx, ryrxtx, r 2 x ryt 1 x , r 1 y r 1 z t 1 x , r 2 y rzty, ryrxty, r 2 y rztz, r 2 x ryt 1 z , r 1 y r 1 z t 1 z 35 [ 7, 24, 56, 104, 168, 248, 344, 456, 584, 728 ] 1801*, [ 7, 25, 60, 112, 180, 264, 364, 480, 612, 760 ] 1802*, [ 7, 25, 60, 113, 184, 273, 379, 500, 636, 788 ] 1803*, [ 7, 25, 61, 116, 188, 276, 380, 500, 636, 788 ] 1804*, [ 7, 26, 62, 114, 182, 266, 366, 482, 614, 762 ] 1805*, [ 7, 26, 64, 120, 192, 280, 384, 504, 640, 792 ] 1806*, 1807*, [ 7, 26, 64, 122, 196, 284, 388, 508, 644, 796 ] 1808*, [ 7, 26, 65, 123, 196, 284, 388, 508, 644, 796 ] 1809*, 1810*, [ 7, 26, 65, 123, 197, 286, 390, 510, 646, 798 ] 1811*, [ 7, 26, 65, 124, 198, 286, 390, 510, 646, 798 ] 1812*, [ 7, 26, 66, 124, 196, 284, 388, 508, 644, 796 ] 1813*, [ 7, 26, 66, 126, 198, 284, 388, 508, 644, 796 ] 1814*, [ 7, 27, 67, 124, 196, 284, 388, 508, 644, 796 ] 1815*, 1816*, 1817*, 1818*, 1819*, 1820*, [ 7, 27, 67, 125, 199, 289, 395, 516, 652, 804 ] 1821*, [ 7, 27, 68, 125, 196, 284, 388, 508, 644, 796 ] 1822*, 1823*, [ 7, 27, 68, 127, 200, 288, 392, 512, 648, 800 ] 1824*, [ 7, 27, 70, 130, 203, 292, 396, 516, 652, 804 ] 1825*, [ 7, 28, 68, 124, 196, 284, 388, 508, 644, 796 ] 1826*, 1827*, 1828*, 1829*, 1830*, 1831*, 1832*, 1833*, [ 7, 28, 70, 128, 200, 288, 392, 512, 648, 800 ] 1834*, 1835*, 1836*, [ 7, 28, 70, 129, 203, 292, 396, 516, 652, 804 ] 1837*, [ 7, 28, 70, 130, 204, 292, 396, 516, 652, 804 ] 1838*, 1839*, K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 113 Nbr. gr Hi L m X [ 7, 28, 71, 131, 204, 292, 396, 516, 652, 804 ] 1840*, [ 7, 28, 72, 132, 204, 292, 396, 516, 652, 804 ] 1841*, [ 7, 28, 73, 133, 204, 292, 396, 516, 652, 804 ] 1842*, [ 7, 29, 71, 128, 200, 288, 392, 512, 648, 800 ] 1843*, [ 7, 29, 71, 129, 203, 292, 396, 516, 652, 804 ] 1844*, [ 7, 29, 73, 132, 204, 292, 396, 516, 652, 804 ] 1845*, 1846*, 1847*, 1848*, 1849*, [ 7, 29, 75, 134, 204, 292, 396, 516, 652, 804 ] 1850*, 1851*, 1852*, [ 7, 30, 72, 128, 200, 288, 392, 512, 648, 800 ] 1853*, 1854*, 1855*, [ 7, 30, 74, 132, 204, 292, 396, 516, 652, 804 ] 1856*, 1857*, 1858*, 1859*, 1860*, 1861*, 1862*, [ 7, 30, 75, 134, 205, 292, 396, 516, 652, 804 ] 1863*, [ 7, 30, 76, 134, 204, 292, 396, 516, 652, 804 ] 1864*, 1865*, 1866*, [ 7, 31, 75, 132, 204, 292, 396, 516, 652, 804 ] 1867*, 1868*, 1869*, 1870*, [ 7, 32, 76, 132, 204, 292, 396, 516, 652, 804 ] 1871*, 1872*, 1873*, 1874*, 1875*, 1876*, 1877*, [ 8, 28, 64, 116, 184, 268, 368, 484, 616, 764 ] 1801 H779 hmz,mzrx, r 2 x i r2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 x ty,mzr 1 x t 1 y ,mzr 1 x tz, r 2 x t 1 z [ 8, 29, 67, 121, 191, 277, 379, 497, 631, 781 ] 1802 H452 1 r 2 z tx, r 2 z t 1 x , t 1 x , ty, t 1 y , r 2 y rztz, mzt 1 z [ 8, 29, 67, 122, 194, 283, 388, 508, 644, 796 ] 1803 H503 1 r 2 z tx, r 2 z t 1 x , t 1 x , ty, t 1 y , r 2 y rztz, r 2 x t 1 z [ 8, 29, 68, 124, 196, 284, 388, 508, 644, 796 ] 1804 H680 hmxi r 2 z mxtx,mxt 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 y tz, r 2 x t 1 z [ 8, 30, 68, 122, 192, 278, 380, 498, 632, 782 ] 1805 H611 hmzi r 2 z tx, r 2 z t 1 x , t 1 x , ty, t 1 y , r 2 y rztz, mxrzt 1 z [ 8, 30, 70, 126, 198, 286, 390, 510, 646, 798 ] 1806 H749 hmy, r 2 z i i r 2 y tx,mzt 1 x , r 2 z ty, r 2 z t 1 y , r 2 z tz,mzt 1 z , r 2 z t 1 z 1807 H682 hmyi mx mztx, r 2 y t 1 x ,mzt 1 x ,myty, t 1 y ,mztz, r 2 y t 1 z [ 8, 30, 70, 128, 200, 288, 392, 512, 648, 800 ] 1808 H598 hr 2 y i r2 z r 2 y tx, r 2 y t 1 x , r 2 y ty, r 2 x t 1 y , r 2 y t 1 y , r 2 x rztz, rzt 1 z [ 8, 30, 71, 128, 200, 288, 392, 512, 648, 800 ] 1809 H650 hmzrxi mx tx,mxt 1 x , t 1 x ,myty, r 2 z t 1 y , r 1 x tz, mxr 1 x t 1 z 1810 H378 1 mz mxtx, r 2 y t 1 x ,myty,myt 1 y ,mxrztz, r 2 y rzt 1 z , mxrzt 1 z [ 8, 30, 71, 128, 201, 289, 393, 513, 649, 801 ] 1811 H503 1 r 2 z tx, r 2 z t 1 x , t 1 x , ty, t 1 y , r 2 y rztz, r 2 y t 1 z [ 8, 30, 71, 129, 201, 289, 393, 513, 649, 801 ] 1812 H452 1 r 2 z tx, r 2 z t 1 x , t 1 x , ty, t 1 y , r 2 y rztz, it 1 z [ 8, 30, 72, 128, 200, 288, 392, 512, 648, 800 ] 1813 H648 hmzrxi mx tx,mxt 1 x , t 1 x , r 2 x ty, it 1 y ,mzr 1 x tz, r 2 z rxt 1 z [ 8, 30, 72, 130, 200, 288, 392, 512, 648, 800 ] 1814 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,myty,myt 1 y ,mztz, it 1 z [ 8, 31, 72, 128, 200, 288, 392, 512, 648, 800 ] 1815 H361 1 mz r 2 z tx, it 1 x , r 2 z ty, it 1 y ,mxtz, r 2 y t 1 z , mxt 1 z 1816 H680 hmxi r 2 z r 2 z tx,myt 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 y tz, r 2 x t 1 z 1817 H368 1 mz mxtx, r 2 y t 1 x ,myty, r 2 x t 1 y ,mxtz, r 2 y t 1 z , mxt 1 z 1818 H527 1 r 2 z mxtx,mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, mzt 1 z 1819 H527 1 r 2 z mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y , r 2 y rztz, mzt 1 z 1820 H529 1 r 2 z mxtx,mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, r 2 y t 1 z 114 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X [ 8, 31, 72, 129, 202, 291, 396, 516, 652, 804 ] 1821 H529 1 r 2 z mytx,myt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, r 2 y t 1 z [ 8, 31, 73, 128, 200, 288, 392, 512, 648, 800 ] 1822 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,myty,mxr 1 x t 1 y ,mxrxtz, mzt 1 z 1823 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 y rxtz, r 2 x t 1 z [ 8, 31, 73, 130, 202, 290, 394, 514, 650, 802 ] 1824 H359 1 mz r 2 z tx,mxt 1 x ,myty, r 2 x t 1 y , tz,mzt 1 z , t 1 z [ 8, 31, 75, 131, 204, 292, 396, 516, 652, 804 ] 1825 H377 1 mz r 2 y tx, r 1 z t 1 x , rzty,myt 1 y , tz,mzt 1 z , t 1 z [ 8, 32, 72, 128, 200, 288, 392, 512, 648, 800 ] 1826 H779 hmz,mzrx, r 2 x i r2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 z ty,mxt 1 y ,mxr 1 x tz, r 2 y rxt 1 z 1827 H750 hmy, r 2 z i i r 2 y tx,mzt 1 x , r 2 z ty, r 2 z t 1 y ,mxrztz, r 2 x rzt 1 z , mxrzt 1 z 1828 H622 hmzi r 2 z mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y , r 2 y rztz, mxrzt 1 z 1829 H649 hmzrxi mx mzr 1 x tx, r 2 z rxt 1 x ,mzr 1 x t 1 x , r 2 x ty, it 1 y ,mzr 1 x tz, r 2 z rxt 1 z 1830 H682 hmyi mx mztx, r 2 y t 1 x ,mzt 1 x , r 2 z ty,mxt 1 y ,mztz, r 2 y t 1 z 1831 H652 hmzrxi mx mzr 1 x tx, r 2 z rxt 1 x ,mzr 1 x t 1 x ,myty, r 2 z t 1 y , r 1 x tz, mxr 1 x t 1 z 1832 H378 1 mz r 2 y tx, r 2 y t 1 x ,myty, r 2 x t 1 y ,mxrztz, r 2 y rzt 1 z , mxrzt 1 z 1833 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,mxty,mxt 1 y ,mztz, it 1 z [ 8, 32, 74, 130, 202, 290, 394, 514, 650, 802 ] 1834 H370 1 mz r 2 z tx,mxt 1 x ,myty, r 2 x t 1 y ,mytz, r 2 x t 1 z , myt 1 z 1835 H527 1 r 2 z mxtx,mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, it 1 z 1836 H529 1 r 2 z mytx,mxt 1 x ,myt 1 x ,myty,myt 1 y , r 2 y rztz, r 2 y t 1 z [ 8, 32, 74, 131, 204, 292, 396, 516, 652, 804 ] 1837 H529 1 r 2 z mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y , r 2 y rztz, r 2 y t 1 z [ 8, 32, 74, 132, 204, 292, 396, 516, 652, 804 ] 1838 H604 hr 2 y i r2 z itx,myt 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 x rztz, rzt 1 z 1839 H527 1 r 2 z mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y , r 2 y rztz, it 1 z [ 8, 32, 75, 132, 204, 292, 396, 516, 652, 804 ] 1840 H359 1 mz itx,mxt 1 x ,myty, r 2 x t 1 y , tz,mzt 1 z , t 1 z [ 8, 32, 76, 132, 204, 292, 396, 516, 652, 804 ] 1841 H359 1 mz r 2 z tx, r 2 y t 1 x ,myty, r 2 x t 1 y , tz,mzt 1 z , t 1 z [ 8, 32, 77, 132, 204, 292, 396, 516, 652, 804 ] 1842 H382 1 mz r 2 y tx, r 1 z t 1 x , rzty,myt 1 y ,mxrztz, r 2 y rzt 1 z , mxrzt 1 z [ 8, 33, 74, 130, 202, 290, 394, 514, 650, 802 ] 1843 H682 hmxi r 2 z r 2 y tx, r 2 y t 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 y tz, r 2 x t 1 z [ 8, 33, 74, 131, 204, 292, 396, 516, 652, 804 ] 1844 H343 1 mz r 2 z tx,mxrzt 1 x , ity,mxrzt 1 y , tz,mzt 1 z , t 1 z [ 8, 33, 76, 132, 204, 292, 396, 516, 652, 804 ] 1845 H370 1 mz itx,mxt 1 x ,myty, r 2 x t 1 y ,mytz, r 2 x t 1 z , myt 1 z 1846 H370 1 mz r 2 z tx, r 2 y t 1 x ,myty, r 2 x t 1 y ,mytz, r 2 x t 1 z , myt 1 z 1847 H359 1 mz itx, r 2 y t 1 x ,myty, r 2 x t 1 y , tz,mzt 1 z , t 1 z 1848 H452 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rztz, mzt 1 z 1849 H503 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rztz, r 2 x t 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 115 Nbr. gr Hi L m X [ 8, 33, 78, 132, 204, 292, 396, 516, 652, 804 ] 1850 H377 1 mz r 2 y tx,mzr 1 z t 1 x ,mzrzty,myt 1 y , tz,mzt 1 z , t 1 z 1851 H541 1 r 2 z r 2 z rxtx, r 2 y rxt 1 x ,mxty,mxt 1 y ,myt 1 y ,mztz, it 1 z 1852 H541 1 r 2 z rxtx, r 1 x t 1 x ,mxty,mxt 1 y ,myt 1 y ,mztz, it 1 z [ 8, 34, 74, 130, 202, 290, 394, 514, 650, 802 ] 1853 H604 hr 2 y i r2 z mxtx,mxt 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 x rztz, rzt 1 z 1854 H622 hmzi r 2 z mxtx,mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, mxrzt 1 z 1855 H683 hmxi r 2 z r 2 y tx, r 2 y t 1 x , r 2 y ty, r 2 x t 1 y , r 2 y t 1 y , r 2 y tz, r 2 x t 1 z [ 8, 34, 76, 132, 204, 292, 396, 516, 652, 804 ] 1856 H344 1 mz r 2 z tx, it 1 x , r 2 z ty, r 2 z t 1 y ,mxrztz, r 2 y rzt 1 z , mxrzt 1 z 1857 H350 1 mz r 2 z tx, it 1 x ,myty, r 2 x t 1 y , tz,mzt 1 z , t 1 z 1858 H350 1 mz r 2 z tx, it 1 x , r 2 x ty,myt 1 y , tz,mzt 1 z , t 1 z 1859 H370 1 mz itx, r 2 y t 1 x ,myty, r 2 x t 1 y ,mytz, r 2 x t 1 z , myt 1 z 1860 H347 1 mz r 2 z tx,mxrzt 1 x , ity,mxrzt 1 y ,mxr 1 z tz, r 2 x rzt 1 z , mxr 1 z t 1 z 1861 H452 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rztz, it 1 z 1862 H503 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rztz, r 2 y t 1 z [ 8, 34, 77, 133, 204, 292, 396, 516, 652, 804 ] 1863 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 x ty, r 2 x t 1 y , r 2 x tz, r 2 y t 1 z [ 8, 34, 78, 132, 204, 292, 396, 516, 652, 804 ] 1864 H382 1 mz r 2 y tx,mzr 1 z t 1 x ,mzrzty,myt 1 y ,mxrztz, r 2 y rzt 1 z , mxrzt 1 z 1865 H511 1 r 2 z r 2 z rxtx, r 2 y rxt 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z 1866 H511 1 r 2 z rxtx, r 1 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z [ 8, 35, 76, 132, 204, 292, 396, 516, 652, 804 ] 1867 H682 hmxi r 2 z itx, r 2 x t 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 y tz, r 2 x t 1 z 1868 H372 1 mz r 2 z tx, it 1 x ,myty, r 2 x t 1 y ,mxtz, r 2 y t 1 z , mxt 1 z 1869 H365 1 mz r 2 z tx, it 1 x ,myty, r 2 x t 1 y ,mytz, r 2 x t 1 z , myt 1 z 1870 H343 1 mz r 2 z tx, r 2 y rzt 1 x , ity, r 2 y rzt 1 y , tz,mzt 1 z , t 1 z [ 8, 36, 76, 132, 204, 292, 396, 516, 652, 804 ] 1871 H598 hr 2 y i r2 z r 2 z tx, r 2 x t 1 x , r 2 y ty, r 2 x t 1 y , r 2 y t 1 y , r 2 x rztz, rzt 1 z 1872 H611 hmzi r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rztz, mxrzt 1 z 1873 H683 hmxi r 2 z itx, r 2 x t 1 x , r 2 y ty, r 2 x t 1 y , r 2 y t 1 y , r 2 y tz, r 2 x t 1 z 1874 H344 1 mz r 2 z tx, it 1 x , ity, it 1 y ,mxrztz, r 2 y rzt 1 z , mxrzt 1 z 1875 H364 1 mz r 2 z tx, it 1 x ,myty, r 2 x t 1 y , r 2 z tz, it 1 z , r 2 z t 1 z 1876 H347 1 mz r 2 z tx, r 2 y rzt 1 x , ity, r 2 y rzt 1 y ,mxr 1 z tz, r 2 x rzt 1 z , mxr 1 z t 1 z 1877 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x tz, r 2 y t 1 z 36 [ 7, 25, 62, 121, 199, 291, 396, 516, 652, 804 ] 1878*, [ 7, 26, 63, 116, 184, 268, 368, 484, 616, 764 ] 1879*, [ 7, 26, 65, 123, 196, 284, 388, 508, 644, 796 ] 1880*, [ 7, 26, 65, 125, 202, 292, 396, 516, 652, 804 ] 1881*, [ 7, 26, 67, 131, 209, 296, 397, 516, 652, 804 ] 116 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 1882*, [ 7, 27, 65, 117, 184, 268, 368, 484, 616, 764 ] 1883*, [ 7, 27, 68, 126, 197, 284, 388, 508, 644, 796 ] 1884*, [ 7, 27, 68, 127, 200, 288, 392, 512, 648, 800 ] 1885*, 1886*, [ 7, 27, 69, 129, 202, 290, 394, 514, 650, 802 ] 1887*, [ 7, 27, 69, 130, 204, 292, 396, 516, 652, 804 ] 1888*, [ 7, 27, 70, 132, 205, 292, 396, 516, 652, 804 ] 1889*, [ 7, 27, 70, 133, 207, 293, 396, 516, 652, 804 ] 1890*, 1891*, [ 7, 27, 70, 134, 209, 294, 396, 516, 652, 804 ] 1892*, [ 7, 27, 71, 132, 204, 292, 396, 516, 652, 804 ] 1893*, [ 7, 27, 71, 139, 212, 293, 396, 516, 652, 804 ] 1894*, [ 7, 28, 65, 116, 184, 268, 368, 484, 616, 764 ] 1895*, [ 7, 28, 69, 125, 196, 284, 388, 508, 644, 796 ] 1896*, [ 7, 28, 70, 130, 202, 288, 392, 512, 648, 800 ] 1897*, 1898*, [ 7, 28, 71, 126, 193, 279, 381, 499, 633, 783 ] 1899*, [ 7, 28, 71, 130, 202, 290, 394, 514, 650, 802 ] 1900*, 1901*, 1902*, [ 7, 28, 71, 131, 204, 292, 396, 516, 652, 804 ] 1903*, 1904*, [ 7, 28, 71, 132, 204, 290, 394, 514, 650, 802 ] 1905*, [ 7, 28, 72, 135, 207, 292, 396, 516, 652, 804 ] 1906*, [ 7, 28, 72, 135, 210, 296, 397, 516, 652, 804 ] 1907*, 1908*, [ 7, 28, 72, 139, 211, 292, 396, 516, 652, 804 ] 1909*, [ 7, 28, 73, 133, 204, 292, 396, 516, 652, 804 ] 1910*, [ 7, 28, 73, 134, 204, 290, 394, 514, 650, 802 ] 1911*, [ 7, 28, 73, 135, 206, 292, 396, 516, 652, 804 ] 1912*, 1913*, 1914*, 1915*, [ 7, 28, 74, 137, 207, 292, 396, 516, 652, 804 ] 1916*, [ 7, 28, 74, 137, 209, 294, 396, 516, 652, 804 ] 1917*, [ 7, 29, 74, 136, 209, 294, 396, 516, 652, 804 ] 1918*, 1919*, 1920*, [ 7, 29, 75, 137, 208, 293, 396, 516, 652, 804 ] 1921*, 1922*, 1923*, [ 7, 29, 76, 137, 207, 293, 396, 516, 652, 804 ] 1924*, [ 7, 30, 76, 136, 207, 293, 396, 516, 652, 804 ] 1925*, [ 7, 30, 77, 137, 206, 292, 396, 516, 652, 804 ] 1926*, 1927*, 1928*, 1929*, [ 7, 31, 75, 131, 202, 290, 394, 514, 650, 802 ] 1930*, [ 7, 31, 76, 135, 207, 293, 396, 516, 652, 804 ] 1931*, 1932*, 1933*, 1934*, [ 7, 31, 77, 134, 204, 292, 396, 516, 652, 804 ] 1935*, [ 7, 31, 77, 136, 206, 292, 396, 516, 652, 804 ] 1936*, [ 7, 32, 75, 130, 202, 290, 394, 514, 650, 802 ] 1937*, 1938*, [ 7, 32, 77, 133, 204, 292, 396, 516, 652, 804 ] 1939*, 1940*, [ 7, 32, 77, 135, 206, 292, 396, 516, 652, 804 ] 1941*, 1942*, 1943*, 1944*, 1945*, 1946*, [ 7, 32, 78, 134, 204, 292, 396, 516, 652, 804 ] 1947*, [ 8, 29, 69, 128, 203, 292, 396, 516, 652, 804 ] 1878 H650 hmxi r 2 y rx mxtx,mxt 1 x ,myty,myt 1 y ,mxrxt 1 y , r 2 y tz, mxr 1 x t 1 z [ 8, 30, 69, 123, 193, 279, 381, 499, 633, 783 ] 1879 H648 hmxi r 2 y rx mxtx,mxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 z rxt 1 y , r 2 z rxtz, r 2 z rxt 1 z [ 8, 30, 71, 128, 200, 288, 392, 512, 648, 800 ] 1880 H650 hmxi r 2 y rx mxtx,mxt 1 x ,mxrxty,myt 1 y ,mxrxt 1 y ,mxr 1 x tz, mxr 1 x t 1 z [ 8, 30, 71, 130, 204, 292, 396, 516, 652, 804 ] 1881 H648 hmxi r 2 y rx mxtx,mxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 x tz, r 2 z rxtz, r 2 x t 1 z [ 8, 30, 73, 134, 207, 293, 396, 516, 652, 804 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 117 Nbr. gr Hi L m X 1882 H578 1 mzrx mxtx, r 2 y rxt 1 x ,myty,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z [ 8, 31, 70, 123, 193, 279, 381, 499, 633, 783 ] 1883 H560 1 mzrx r 2 z rxtx,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y ,mzr 1 x tz, mzr 1 x t 1 z [ 8, 31, 73, 129, 200, 288, 392, 512, 648, 800 ] 1884 H580 1 mzrx r 2 z rxtx,mxt 1 x , rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z [ 8, 31, 73, 130, 202, 290, 394, 514, 650, 802 ] 1885 H560 1 mzrx itx,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y ,mzr 1 x tz, mzr 1 x t 1 z 1886 H557 1 mzrx mxtx, r 2 y rxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y ,mzr 1 x tz, mzr 1 x t 1 z [ 8, 31, 74, 131, 203, 291, 395, 515, 651, 803 ] 1887 H409 1 r 2 z rx r 2 z tx, r 2 y t 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 y rxt 1 z [ 8, 31, 74, 132, 204, 292, 396, 516, 652, 804 ] 1888 H578 1 mzrx mxtx, r 2 y rxt 1 x , rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z [ 8, 31, 75, 133, 204, 292, 396, 516, 652, 804 ] 1889 H580 1 mzrx itx,mxt 1 x , rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z [ 8, 31, 75, 134, 205, 292, 396, 516, 652, 804 ] 1890 H650 hmxi r 2 y rx r 2 y rxtx,mzrxt 1 x ,myty,myt 1 y ,mxrxt 1 y , r 2 y tz, mxr 1 x t 1 z 1891 H571 1 mzrx mxtx, r 2 y rxt 1 x ,myty, r 2 z t 1 y ,mxrxt 1 y ,mztz, mxr 1 x t 1 z [ 8, 31, 75, 135, 206, 292, 396, 516, 652, 804 ] 1892 H557 1 mzrx mxtx, r 2 y rxt 1 x ,mzr 1 x ty, r 2 x t 1 y , r 2 x tz,mzr 1 x tz, r 2 x t 1 z [ 8, 31, 76, 132, 204, 292, 396, 516, 652, 804 ] 1893 H422 1 r 2 z rx r 2 z tx, r 2 y t 1 x ,mxr 1 x ty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz, mxrxt 1 z [ 8, 31, 76, 139, 205, 292, 396, 516, 652, 804 ] 1894 H572 1 mzrx r 2 z rxtx, it 1 x , r 2 z ty, r 2 z t 1 y ,mxrxt 1 y , r 2 y tz, mxr 1 x t 1 z [ 8, 32, 69, 123, 193, 279, 381, 499, 633, 783 ] 1895 H649 hmxi r 2 y rx r 2 z rxtx, r 2 z rxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 z rxt 1 y , r 2 z rxtz, r 2 z rxt 1 z [ 8, 32, 73, 128, 200, 288, 392, 512, 648, 800 ] 1896 H652 hmxi r 2 y rx r 2 z rxtx, r 2 z rxt 1 x ,mxrxty,myt 1 y ,mxrxt 1 y ,mxr 1 x tz, mxr 1 x t 1 z [ 8, 32, 74, 132, 202, 290, 394, 514, 650, 802 ] 1897 H555 1 mzrx r 2 z rxtx, it 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y ,mzr 1 x tz, mzr 1 x t 1 z 1898 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y ,mzr 1 x tz, mzr 1 x t 1 z [ 8, 32, 75, 127, 198, 285, 388, 507, 642, 793 ] 1899 H393 1 r 2 z rx ryrxtx, r 1 y r 1 z t 1 x , ryrxty, r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz, r 1 y r 1 z t 1 z [ 8, 32, 75, 131, 203, 291, 395, 515, 651, 803 ] 1900 H649 hmxi r 2 y rx itx, r 2 x t 1 x , r 2 z rxty, r 2 x t 1 y , r 2 z rxt 1 y , r 2 z rxtz, r 2 z rxt 1 z 1901 H648 hmxi r 2 y rx r 2 y rxtx,mzrxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 z rxt 1 y , r 2 z rxtz, r 2 z rxt 1 z 1902 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mzr 1 x ty, it 1 y , r 2 z rxt 1 y ,mzr 1 x tz, r 2 z rxt 1 z [ 8, 32, 75, 132, 204, 292, 396, 516, 652, 804 ] 1903 H650 hmxi r 2 y rx r 2 y rxtx,mzrxt 1 x ,mxrxty,myt 1 y ,mxrxt 1 y ,mxr 1 x tz, mxr 1 x t 1 z 1904 H571 1 mzrx mxtx, r 2 y rxt 1 x , rxty, r 2 z t 1 y ,mxrxt 1 y , r 1 x tz, mxr 1 x t 1 z [ 8, 32, 75, 133, 203, 291, 395, 515, 651, 803 ] 1905 H409 1 r 2 z rx r 1 x tx, rxt 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 y rxt 1 z [ 8, 32, 76, 135, 204, 292, 396, 516, 652, 804 ] 1906 H580 1 mzrx r 2 z rxtx, r 2 y rxt 1 x , rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z [ 8, 32, 76, 135, 207, 293, 396, 516, 652, 804 ] 1907 H580 1 mzrx itx,mxt 1 x ,myty,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z 1908 H580 1 mzrx r 2 z rxtx,mxt 1 x ,myty,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z [ 8, 32, 76, 139, 204, 292, 396, 516, 652, 804 ] 118 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 1909 H551 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxty, it 1 y , itz, r 2 z rxtz, it 1 z [ 8, 32, 77, 132, 204, 292, 396, 516, 652, 804 ] 1910 H652 hmxi r 2 y rx itx, r 2 x t 1 x ,mxrxty,myt 1 y ,mxrxt 1 y ,mxr 1 x tz, mxr 1 x t 1 z [ 8, 32, 77, 133, 203, 291, 395, 515, 651, 803 ] 1911 H560 1 mzrx itx, r 2 y rxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y ,mzr 1 x tz, mzr 1 x t 1 z [ 8, 32, 77, 134, 204, 292, 396, 516, 652, 804 ] 1912 H422 1 r 2 z rx r 1 x tx, rxt 1 x ,mxr 1 x ty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz, mxrxt 1 z 1913 H648 hmxi r 2 y rx r 2 y rxtx,mzrxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 x tz, r 2 z rxtz, r 2 x t 1 z 1914 H574 1 mzrx r 2 z rxtx, it 1 x , rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z 1915 H561 1 mzrx mxtx, r 2 y rxt 1 x , r 2 x ty, it 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 z rxt 1 z [ 8, 32, 78, 135, 204, 292, 396, 516, 652, 804 ] 1916 H580 1 mzrx itx, r 2 y rxt 1 x , rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z [ 8, 32, 78, 135, 206, 292, 396, 516, 652, 804 ] 1917 H422 1 r 2 z rx r 2 z tx, r 2 y t 1 x ,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz, mzt 1 z [ 8, 33, 77, 135, 206, 292, 396, 516, 652, 804 ] 1918 H560 1 mzrx itx,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y , r 2 x tz,mzr 1 x tz, r 2 x t 1 z 1919 H560 1 mzrx r 2 z rxtx,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y , r 2 x tz,mzr 1 x tz, r 2 x t 1 z 1920 H560 1 mzrx r 2 z rxtx,mxt 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, mzr 1 x t 1 z [ 8, 33, 78, 135, 205, 292, 396, 516, 652, 804 ] 1921 H409 1 r 2 z rx r 2 z tx, r 2 y t 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z 1922 H580 1 mzrx itx, r 2 y rxt 1 x ,myty,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z 1923 H580 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,myty,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z [ 8, 33, 79, 134, 205, 292, 396, 516, 652, 804 ] 1924 H422 1 r 2 z rx r 1 x tx, rxt 1 x ,myty,mxr 1 x t 1 y ,myt 1 y ,mxrxtz, mzt 1 z [ 8, 34, 78, 134, 205, 292, 396, 516, 652, 804 ] 1925 H571 1 mzrx mxtx, r 2 y rxt 1 x ,myty, rxty, r 2 z t 1 y , r 1 x tz, r 2 y t 1 z [ 8, 34, 79, 134, 204, 292, 396, 516, 652, 804 ] 1926 H409 1 r 2 z rx r 1 x tx, rxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z 1927 H560 1 mzrx itx, r 2 y rxt 1 x ,mzr 1 x ty, r 2 x t 1 y , r 2 x tz,mzr 1 x tz, r 2 x t 1 z 1928 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mzr 1 x ty, r 2 x t 1 y , r 2 x tz,mzr 1 x tz, r 2 x t 1 z 1929 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, mzr 1 x t 1 z [ 8, 35, 76, 131, 203, 291, 395, 515, 651, 803 ] 1930 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mzr 1 x ty, r 2 z rxt 1 y , r 2 x tz,mzr 1 x tz, r 2 z rxt 1 z [ 8, 35, 77, 134, 205, 292, 396, 516, 652, 804 ] 1931 H652 hmxi r 2 y rx itx, r 2 x t 1 x ,myty,myt 1 y ,mxrxt 1 y , r 2 y tz, mxr 1 x t 1 z 1932 H652 hmxi r 2 y rx r 2 z rxtx, r 2 z rxt 1 x ,myty,myt 1 y ,mxrxt 1 y , r 2 y tz, mxr 1 x t 1 z 1933 H574 1 mzrx r 2 z rxtx, it 1 x ,myty,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z 1934 H576 1 mzrx mxtx, r 2 y rxt 1 x , r 2 z ty, r 2 z t 1 y ,mxrxt 1 y , r 2 y tz, mxr 1 x t 1 z [ 8, 35, 78, 132, 204, 292, 396, 516, 652, 804 ] 1935 H571 1 mzrx mxtx, r 2 y rxt 1 x ,myty, rxty,mxrxt 1 y , r 1 x tz, mxr 1 x t 1 z [ 8, 35, 78, 134, 204, 292, 396, 516, 652, 804 ] 1936 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mzr 1 x ty, it 1 y , r 2 x tz,mzr 1 x tz, it 1 z [ 8, 36, 75, 131, 203, 291, 395, 515, 651, 803 ] 1937 H551 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxty, it 1 y , r 2 z rxt 1 y , r 2 z rxtz, r 2 z rxt 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 119 Nbr. gr Hi L m X 1938 H556 1 mzrx mxtx, r 2 y rxt 1 x , r 2 z rxty, it 1 y , r 2 z rxt 1 y , r 2 z rxtz, r 2 z rxt 1 z [ 8, 36, 77, 132, 204, 292, 396, 516, 652, 804 ] 1939 H572 1 mzrx r 2 z rxtx, it 1 x ,mxrxty, r 2 z t 1 y ,mxrxt 1 y ,mxr 1 x tz, mxr 1 x t 1 z 1940 H576 1 mzrx mxtx, r 2 y rxt 1 x ,mxrxty, r 2 z t 1 y ,mxrxt 1 y ,mxr 1 x tz, mxr 1 x t 1 z [ 8, 36, 77, 134, 204, 292, 396, 516, 652, 804 ] 1941 H649 hmxi r 2 y rx itx, r 2 x t 1 x , r 2 x ty, r 2 x t 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 z rxt 1 z 1942 H649 hmxi r 2 y rx r 2 z rxtx, r 2 z rxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 x tz, r 2 z rxtz, r 2 x t 1 z 1943 H649 hmxi r 2 y rx itx, r 2 x t 1 x , r 2 z rxty, r 2 x t 1 y , r 2 x tz, r 2 z rxtz, r 2 x t 1 z 1944 H555 1 mzrx r 2 z rxtx, it 1 x ,mzr 1 x ty, r 2 x t 1 y , r 2 x tz,mzr 1 x tz, r 2 x t 1 z 1945 H555 1 mzrx itx, r 2 z rxt 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, mzr 1 x t 1 z 1946 H556 1 mzrx mxtx, r 2 y rxt 1 x , r 2 z rxty, it 1 y , itz, r 2 z rxtz, it 1 z [ 8, 36, 78, 132, 204, 292, 396, 516, 652, 804 ] 1947 H393 1 r 2 z rx r 2 y rztx, r 2 x ryt 1 x , r 2 y rzty, r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz, r 2 x ryt 1 z 37 [ 8, 29, 68, 124, 196, 284, 388, 508, 644, 796 ] 1948*, 1949*, [ 8, 30, 70, 126, 198, 286, 390, 510, 646, 798 ] 1950*, 1951*, 1952*, 1953*, [ 8, 30, 71, 128, 200, 288, 392, 512, 648, 800 ] 1954*, 1955*, 1956*, 1957*, 1958*, [ 8, 30, 72, 131, 204, 292, 396, 516, 652, 804 ] 1959*, 1960*, 1961*, 1962*, 1963*, 1964*, [ 8, 30, 73, 134, 206, 292, 396, 516, 652, 804 ] 1965*, [ 8, 31, 72, 128, 200, 288, 392, 512, 648, 800 ] 1966*, [ 8, 31, 73, 129, 200, 288, 392, 512, 648, 800 ] 1967*, 1968*, 1969*, [ 8, 31, 73, 131, 204, 292, 396, 516, 652, 804 ] 1970*, [ 8, 31, 74, 132, 204, 292, 396, 516, 652, 804 ] 1971*, 1972*, 1973*, 1974*, 1975*, 1976*, 1977*, 1978*, 1979*, [ 8, 31, 75, 133, 204, 292, 396, 516, 652, 804 ] 1980*, [ 8, 31, 75, 134, 205, 292, 396, 516, 652, 804 ] 1981*, 1982*, 1983*, [ 8, 32, 73, 128, 200, 288, 392, 512, 648, 800 ] 1984*, 1985*, 1986*, 1987*, 1988*, 1989*, 1990*, 1991*, 1992*, 1993*, 1994*, 1995*, [ 8, 32, 75, 132, 204, 292, 396, 516, 652, 804 ] 1996*, 1997*, 1998*, 1999*, 2000*, 2001*, 2002*, 2003*, 2004*, 2005*, 2006*, 2007*, 2008*, 2009*, 2010*, 2011*, 2012*, 2013*, 2014*, 2015*, 2016*, 2017*, 2018*, 2019*, 2020*, 2021*, 2022*, 2023*, 2024*, 2025*, [ 8, 32, 76, 133, 204, 292, 396, 516, 652, 804 ] 2026*, 2027*, 2028*, 2029*, 2030*, 2031*, 2032*, [ 8, 32, 76, 134, 205, 292, 396, 516, 652, 804 ] 2033*, 2034*, 2035*, 2036*, [ 8, 32, 77, 134, 204, 292, 396, 516, 652, 804 ] 2037*, 2038*, 2039*, 2040*, [ 8, 32, 77, 135, 205, 292, 396, 516, 652, 804 ] 2041*, [ 8, 33, 76, 132, 204, 292, 396, 516, 652, 804 ] 2042*, 2043*, 2044*, 2045*, 2046*, 2047*, 2048*, 2049*, 2050*, 2051*, 2052*, 2053*, 2054*, 2055*, 2056*, 2057*, 2058*, 2059*, 2060*, 2061*, 2062*, 2063*, [ 8, 33, 77, 133, 204, 292, 396, 516, 652, 804 ] 2064*, 2065*, 2066*, [ 8, 33, 77, 134, 205, 292, 396, 516, 652, 804 ] 2067*, [ 8, 33, 78, 135, 205, 292, 396, 516, 652, 804 ] 2068*, 2069*, 2070*, [ 8, 34, 77, 132, 204, 292, 396, 516, 652, 804 ] 2071*, 2072*, 2073*, 2074*, 2075*, 2076*, 2077*, 2078*, 2079*, 2080*, 2081*, 2082*, 2083*, 2084*, 2085*, 2086*, 2087*, 2088*, 2089*, 2090*, 2091*, 2092*, 2093*, 2094*, 2095*, 2096*, 2097*, 2098*, 2099*, 2100*, 2101*, 2102*, 2103*, 2104*, 2105*, 2106*, 2107*, 2108*, 2109*, 2110*, [ 8, 34, 78, 134, 205, 292, 396, 516, 652, 804 ] 2111*, 2112*, 2113*, [ 9, 32, 72, 128, 200, 288, 392, 512, 648, 800 ] 1948 H680 hmxi r 2 z mxtx,mxt 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z 1949 H690 hmxi r 2 z mxtx,mxt 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 y rztz, mzr 1 z tz, r 2 y t 1 z [ 9, 33, 73, 129, 201, 289, 393, 513, 649, 801 ] 120 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 1950 H452 1 r 2 z tx, r 2 z t 1 x , t 1 x , ty, t 1 y , r 2 y rztz, it 1 z ,mzt 1 z 1951 H503 1 r 2 z tx, r 2 z t 1 x , t 1 x , ty, t 1 y , r 2 y rztz, r 2 y t 1 z , r 2 x t 1 z 1952 H438 1 r 2 z tx, r 2 z t 1 x , t 1 x , ty, t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 1953 H495 1 r 2 z tx, r 2 z t 1 x , t 1 x , ty, t 1 y ,mzr 1 z tz, mzrztz, r 2 y rzt 1 z [ 9, 33, 74, 130, 202, 290, 394, 514, 650, 802 ] 1954 H682 hmyi mx mztx, r 2 y t 1 x ,mzt 1 x ,myty, t 1 y , r 2 y tz, r 2 y t 1 z ,mzt 1 z 1955 H682 hmyi mx mztx, r 2 y t 1 x ,mzt 1 x ,myty, t 1 y ,mztz, r 2 y t 1 z ,mzt 1 z 1956 H680 hmxi r 2 z mxtx,mxt 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z 1957 H359 1 mz r 2 z tx,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , tz, mzt 1 z , t 1 z 1958 H690 hmxi r 2 z mxtx,mxt 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 y rztz, mzr 1 z tz, r 2 x t 1 z [ 9, 33, 75, 132, 204, 292, 396, 516, 652, 804 ] 1959 H359 1 mz itx, r 2 z tx,mxt 1 x ,myty, r 2 x t 1 y , tz, mzt 1 z , t 1 z 1960 H378 1 mz mxtx, r 2 y t 1 x ,mxt 1 x , r 2 x ty,myt 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 1961 H378 1 mz mxtx, r 2 y t 1 x ,mxt 1 x ,myty,myt 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 1962 H353 1 mz mxtx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , tz, mzt 1 z , t 1 z 1963 H353 1 mz mxtx, r 2 y t 1 x ,mxt 1 x , r 2 x ty,myt 1 y , tz, mzt 1 z , t 1 z 1964 H377 1 mz r 2 y tx,mxtx, r 1 z t 1 x , rzty,myt 1 y , tz, mzt 1 z , t 1 z [ 9, 33, 76, 134, 204, 292, 396, 516, 652, 804 ] 1965 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,myty,myt 1 y ,mztz, it 1 z ,mzt 1 z [ 9, 34, 74, 130, 202, 290, 394, 514, 650, 802 ] 1966 H682 hmxi r 2 z r 2 y tx, r 2 y t 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z [ 9, 34, 75, 130, 202, 290, 394, 514, 650, 802 ] 1967 H370 1 mz r 2 z tx,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y ,mytz, r 2 x t 1 z ,myt 1 z 1968 H370 1 mz r 2 z tx,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y ,mytz, r 2 x t 1 z ,myt 1 z 1969 H359 1 mz r 2 z tx,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , tz, mzt 1 z , t 1 z [ 9, 34, 75, 132, 204, 292, 396, 516, 652, 804 ] 1970 H343 1 mz r 2 z tx,mxrzt 1 x , ity, r 2 z ty,mxrzt 1 y , tz, mzt 1 z , t 1 z [ 9, 34, 76, 132, 204, 292, 396, 516, 652, 804 ] 1971 H680 hmxi r 2 z r 2 z tx,myt 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z 1972 H370 1 mz itx, r 2 z tx,mxt 1 x ,myty, r 2 x t 1 y ,mytz, r 2 x t 1 z ,myt 1 z 1973 H359 1 mz itx, r 2 z tx, r 2 y t 1 x ,myty, r 2 x t 1 y , tz, mzt 1 z , t 1 z 1974 H359 1 mz itx,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , tz, mzt 1 z , t 1 z 1975 H368 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,mxtz, r 2 y t 1 z ,mxt 1 z 1976 H368 1 mz mxtx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,mxtz, r 2 y t 1 z ,mxt 1 z 1977 H368 1 mz mxtx, r 2 y t 1 x ,myty, r 2 x t 1 y ,myt 1 y ,mxtz, r 2 y t 1 z ,mxt 1 z 1978 H353 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , tz, mzt 1 z , t 1 z 1979 H690 hmxi r 2 z r 2 z tx,myt 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 y rztz, mzr 1 z tz, r 2 y t 1 z [ 9, 34, 77, 132, 204, 292, 396, 516, 652, 804 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 121 Nbr. gr Hi L m X 1980 H359 1 mz r 2 z tx, r 2 y t 1 x ,myty, r 2 x t 1 y ,myt 1 y , tz, mzt 1 z , t 1 z [ 9, 34, 77, 133, 204, 292, 396, 516, 652, 804 ] 1981 H382 1 mz r 2 y tx,mxtx, r 1 z t 1 x , rzty, r 2 x t 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 1982 H382 1 mz r 2 y tx,mxtx, r 1 z t 1 x , rzty,myt 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 1983 H377 1 mz r 2 y tx,mxtx, r 1 z t 1 x , rzty, r 2 x t 1 y , tz, mzt 1 z , t 1 z [ 9, 35, 74, 130, 202, 290, 394, 514, 650, 802 ] 1984 H683 hmxi r 2 z r 2 y tx, r 2 y t 1 x , r 2 y ty, r 2 x t 1 y , r 2 y t 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z 1985 H683 hmxi r 2 z r 2 y tx, r 2 y t 1 x , r 2 y ty, r 2 x t 1 y , r 2 y t 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z 1986 H682 hmxi r 2 z r 2 y tx, r 2 y t 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z 1987 H453 1 r 2 z mztx, it 1 x ,mzt 1 x ,mzty,mzt 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 1988 H539 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 z tz, mzrztz, r 2 y rzt 1 z 1989 H539 1 r 2 z mztx,mzt 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y ,mzr 1 z tz, mzrztz, r 2 y rzt 1 z 1990 H505 1 r 2 z r 2 y tx, r 2 y t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 1991 H505 1 r 2 z r 2 y tx, r 2 y t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y rztz, r 2 x rztz, r 2 y rzt 1 z 1992 H527 1 r 2 z mxtx,mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, it 1 z ,mzt 1 z 1993 H529 1 r 2 z mxtx,mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, r 2 y t 1 z , r 2 x t 1 z 1994 H496 1 r 2 z mxtx,mxt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 1995 H502 1 r 2 z mxtx,mxt 1 x ,mxty,mxt 1 y ,myt 1 y ,mzr 1 z tz, mzrztz, r 2 y rzt 1 z [ 9, 35, 76, 132, 204, 292, 396, 516, 652, 804 ] 1996 H344 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , ity, r 2 z t 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 1997 H344 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 z ty, r 2 z t 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 1998 H330 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 z ty, it 1 y , tz, mzt 1 z , t 1 z 1999 H330 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , ity, r 2 z t 1 y , tz, mzt 1 z , t 1 z 2000 H682 hmyi mx mztx, r 2 y t 1 x ,mzt 1 x , r 2 z ty,mxt 1 y , r 2 y tz, r 2 y t 1 z ,mzt 1 z 2001 H682 hmyi mx mztx, r 2 y t 1 x ,mzt 1 x , r 2 z ty,mxt 1 y ,mztz, r 2 y t 1 z ,mzt 1 z 2002 H680 hmxi r 2 z r 2 z tx,myt 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z 2003 H350 1 mz r 2 z tx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y , tz, mzt 1 z , t 1 z 2004 H350 1 mz r 2 z tx, it 1 x ,myty, r 2 x t 1 y ,myt 1 y , tz, mzt 1 z , t 1 z 2005 H350 1 mz r 2 z tx, it 1 x , r 2 x ty,myty,myt 1 y , tz, mzt 1 z , t 1 z 2006 H680 hmxi r 2 z r 2 z tx,myt 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 x tz, r 2 y tz, r 2 x t 1 z 2007 H370 1 mz itx, r 2 z tx, r 2 y t 1 x ,myty, r 2 x t 1 y ,mytz, r 2 x t 1 z ,myt 1 z 2008 H359 1 mz r 2 z tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , tz, mzt 1 z , t 1 z 2009 H373 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , r 2 z tz, it 1 z , r 2 z t 1 z 2010 H373 1 mz mxtx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , r 2 z tz, it 1 z , r 2 z t 1 z 2011 H368 1 mz mxtx, r 2 y t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y ,mxtz, r 2 y t 1 z ,mxt 1 z 122 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 2012 H378 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2013 H378 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2014 H378 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x , r 2 x ty,myt 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2015 H378 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x ,myty,myt 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2016 H378 1 mz r 2 y tx, r 2 y t 1 x ,myty, r 2 x t 1 y ,myt 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2017 H378 1 mz mxtx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2018 H690 hmxi r 2 z r 2 z tx,myt 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 y rztz, mzr 1 z tz, r 2 x t 1 z 2019 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,mxty,mxt 1 y , itz, it 1 z ,mzt 1 z 2020 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,myty,myt 1 y , itz, it 1 z ,mzt 1 z 2021 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,mxty,mxt 1 y ,mztz, it 1 z ,mzt 1 z 2022 H527 1 r 2 z mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y , r 2 y rztz, it 1 z ,mzt 1 z 2023 H529 1 r 2 z mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y , r 2 y rztz, r 2 y t 1 z , r 2 x t 1 z 2024 H496 1 r 2 z mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 2025 H502 1 r 2 z mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y ,mzr 1 z tz, mzrztz, r 2 y rzt 1 z [ 9, 35, 77, 132, 204, 292, 396, 516, 652, 804 ] 2026 H370 1 mz itx,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y ,mytz, r 2 x t 1 z ,myt 1 z 2027 H370 1 mz itx,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y ,mytz, r 2 x t 1 z ,myt 1 z 2028 H370 1 mz r 2 z tx, r 2 y t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y ,mytz, r 2 x t 1 z ,myt 1 z 2029 H370 1 mz r 2 z tx, r 2 y t 1 x ,myty, r 2 x t 1 y ,myt 1 y ,mytz, r 2 x t 1 z ,myt 1 z 2030 H359 1 mz itx, r 2 y t 1 x ,myty, r 2 x t 1 y ,myt 1 y , tz, mzt 1 z , t 1 z 2031 H359 1 mz itx,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , tz, mzt 1 z , t 1 z 2032 H359 1 mz r 2 z tx, r 2 y t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , tz, mzt 1 z , t 1 z [ 9, 35, 77, 133, 204, 292, 396, 516, 652, 804 ] 2033 H347 1 mz itx,mxrzt 1 x , ity, r 2 z ty,mxrzt 1 y ,mxr 1 z tz, r 2 x rzt 1 z ,mxr 1 z t 1 z 2034 H347 1 mz r 2 z tx,mxrzt 1 x , ity, r 2 z ty,mxrzt 1 y ,mxr 1 z tz, r 2 x rzt 1 z ,mxr 1 z t 1 z 2035 H343 1 mz itx,mxrzt 1 x , ity, r 2 z ty,mxrzt 1 y , tz, mzt 1 z , t 1 z 2036 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 x ty, r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z [ 9, 35, 78, 132, 204, 292, 396, 516, 652, 804 ] 2037 H511 1 r 2 z r 2 z rxtx, r 2 y rxt 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2038 H511 1 r 2 z rxtx, r 1 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2039 H541 1 r 2 z r 2 z rxtx, r 2 y rxt 1 x ,mxty,mxt 1 y ,myt 1 y , itz, it 1 z ,mzt 1 z 2040 H541 1 r 2 z rxtx, r 1 x t 1 x ,mxty,mxt 1 y ,myt 1 y , itz, it 1 z ,mzt 1 z [ 9, 35, 78, 133, 204, 292, 396, 516, 652, 804 ] 2041 H377 1 mz r 2 y tx,mxtx,mzr 1 z t 1 x ,mzrzty,myt 1 y , tz, mzt 1 z , t 1 z [ 9, 36, 76, 132, 204, 292, 396, 516, 652, 804 ] 2042 H361 1 mz itx, it 1 x , r 2 z t 1 x , r 2 z ty, it 1 y ,mxtz, r 2 y t 1 z ,mxt 1 z 2043 H682 hmxi r 2 z itx, r 2 x t 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 123 Nbr. gr Hi L m X 2044 H361 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 z ty, it 1 y ,mxtz, r 2 y t 1 z ,mxt 1 z 2045 H361 1 mz r 2 z tx, it 1 x , r 2 z ty, it 1 y , r 2 z t 1 y ,mxtz, r 2 y t 1 z ,mxt 1 z 2046 H330 1 mz r 2 z tx, it 1 x , ity, it 1 y , r 2 z t 1 y , tz, mzt 1 z , t 1 z 2047 H682 hmxi r 2 z itx, r 2 x t 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 x tz, r 2 y tz, r 2 y t 1 z 2048 H372 1 mz itx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,mxtz, r 2 y t 1 z ,mxt 1 z 2049 H372 1 mz r 2 z tx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,mxtz, r 2 y t 1 z ,mxt 1 z 2050 H372 1 mz r 2 z tx, it 1 x ,myty, r 2 x t 1 y ,myt 1 y ,mxtz, r 2 y t 1 z ,mxt 1 z 2051 H372 1 mz r 2 z tx, it 1 x , r 2 x ty,myty,myt 1 y ,mxtz, r 2 y t 1 z ,mxt 1 z 2052 H365 1 mz r 2 z tx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,mytz, r 2 x t 1 z ,myt 1 z 2053 H365 1 mz r 2 z tx, it 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y ,mytz, r 2 x t 1 z ,myt 1 z 2054 H365 1 mz r 2 z tx, it 1 x , r 2 x ty,myty, r 2 x t 1 y ,mytz, r 2 x t 1 z ,myt 1 z 2055 H365 1 mz r 2 z tx, it 1 x ,myty, r 2 x t 1 y ,myt 1 y ,mytz, r 2 x t 1 z ,myt 1 z 2056 H365 1 mz r 2 z tx, it 1 x , r 2 x ty,myty,myt 1 y ,mytz, r 2 x t 1 z ,myt 1 z 2057 H350 1 mz itx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y , tz, mzt 1 z , t 1 z 2058 H350 1 mz r 2 z tx, it 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , tz, mzt 1 z , t 1 z 2059 H350 1 mz r 2 z tx, it 1 x , r 2 x ty,myty, r 2 x t 1 y , tz, mzt 1 z , t 1 z 2060 H370 1 mz r 2 z tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,mytz, r 2 x t 1 z ,myt 1 z 2061 H359 1 mz itx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , tz, mzt 1 z , t 1 z 2062 H541 1 r 2 z r 2 z rxtx, r 2 y rxt 1 x ,mxty,mxt 1 y ,myt 1 y ,mztz, it 1 z ,mzt 1 z 2063 H541 1 r 2 z rxtx, r 1 x t 1 x ,mxty,mxt 1 y ,myt 1 y ,mztz, it 1 z ,mzt 1 z [ 9, 36, 77, 132, 204, 292, 396, 516, 652, 804 ] 2064 H370 1 mz itx, r 2 y t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y ,mytz, r 2 x t 1 z ,myt 1 z 2065 H370 1 mz itx, r 2 y t 1 x ,myty, r 2 x t 1 y ,myt 1 y ,mytz, r 2 x t 1 z ,myt 1 z 2066 H359 1 mz itx, r 2 y t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , tz, mzt 1 z , t 1 z [ 9, 36, 77, 133, 204, 292, 396, 516, 652, 804 ] 2067 H343 1 mz r 2 z tx, r 2 y rzt 1 x , ity, r 2 z ty, r 2 y rzt 1 y , tz, mzt 1 z , t 1 z [ 9, 36, 78, 133, 204, 292, 396, 516, 652, 804 ] 2068 H382 1 mz r 2 y tx,mxtx,mzr 1 z t 1 x ,mzrzty, r 2 x t 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2069 H382 1 mz r 2 y tx,mxtx,mzr 1 z t 1 x ,mzrzty,myt 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2070 H377 1 mz r 2 y tx,mxtx,mzr 1 z t 1 x ,mzrzty, r 2 x t 1 y , tz, mzt 1 z , t 1 z [ 9, 37, 76, 132, 204, 292, 396, 516, 652, 804 ] 2071 H683 hmxi r 2 z itx, r 2 x t 1 x , r 2 y ty, r 2 x t 1 y , r 2 y t 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z 2072 H683 hmxi r 2 z itx, r 2 x t 1 x , r 2 y ty, r 2 x t 1 y , r 2 y t 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z 2073 H332 1 mz r 2 z tx, it 1 x , ity, it 1 y , r 2 z t 1 y , r 2 z tz, it 1 z , r 2 z t 1 z 2074 H332 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 z ty, it 1 y , r 2 z tz, it 1 z , r 2 z t 1 z 2075 H682 hmxi r 2 z itx, r 2 x t 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 x tz, r 2 y tz, r 2 x t 1 z 124 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 2076 H361 1 mz r 2 z tx, it 1 x , ity, it 1 y , r 2 z t 1 y ,mxtz, r 2 y t 1 z ,mxt 1 z 2077 H344 1 mz itx, it 1 x , r 2 z t 1 x , ity, it 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2078 H344 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , ity, it 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2079 H344 1 mz r 2 z tx, it 1 x , ity, it 1 y , r 2 z t 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2080 H344 1 mz r 2 z tx, it 1 x , ity, r 2 z ty, it 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2081 H344 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 z ty, it 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2082 H344 1 mz itx, it 1 x , r 2 z t 1 x , ity, r 2 z t 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2083 H344 1 mz itx, it 1 x , r 2 z t 1 x , r 2 z ty, r 2 z t 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2084 H682 hmxi r 2 z itx, r 2 x t 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z 2085 H683 hmxi r 2 z itx, r 2 x t 1 x , r 2 y ty, r 2 x t 1 y , r 2 y t 1 y , r 2 x tz, r 2 y tz, r 2 x t 1 z 2086 H683 hmxi r 2 z itx, r 2 x t 1 x , r 2 y ty, r 2 x t 1 y , r 2 y t 1 y , r 2 x tz, r 2 y tz, r 2 y t 1 z 2087 H364 1 mz itx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y , r 2 z tz, it 1 z , r 2 z t 1 z 2088 H364 1 mz r 2 z tx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y , r 2 z tz, it 1 z , r 2 z t 1 z 2089 H364 1 mz r 2 z tx, it 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 z tz, it 1 z , r 2 z t 1 z 2090 H364 1 mz r 2 z tx, it 1 x , r 2 x ty,myty, r 2 x t 1 y , r 2 z tz, it 1 z , r 2 z t 1 z 2091 H364 1 mz r 2 z tx, it 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 z tz, it 1 z , r 2 z t 1 z 2092 H364 1 mz r 2 z tx, it 1 x , r 2 x ty,myty,myt 1 y , r 2 z tz, it 1 z , r 2 z t 1 z 2093 H372 1 mz r 2 z tx, it 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y ,mxtz, r 2 y t 1 z ,mxt 1 z 2094 H372 1 mz r 2 z tx, it 1 x , r 2 x ty,myty, r 2 x t 1 y ,mxtz, r 2 y t 1 z ,mxt 1 z 2095 H365 1 mz itx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,mytz, r 2 x t 1 z ,myt 1 z 2096 H370 1 mz itx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,mytz, r 2 x t 1 z ,myt 1 z 2097 H453 1 r 2 z mztx, it 1 x ,mzt 1 x , ity, it 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 2098 H539 1 r 2 z itx, it 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y ,mzr 1 z tz, mzrztz, r 2 y rzt 1 z 2099 H539 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 y ty, r 2 y t 1 y ,mzr 1 z tz, mzrztz, r 2 y rzt 1 z 2100 H452 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rztz, it 1 z ,mzt 1 z 2101 H503 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rztz, r 2 y t 1 z , r 2 x t 1 z 2102 H438 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z ty, r 2 z t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 2103 H495 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z ty, r 2 z t 1 y ,mzr 1 z tz, mzrztz, r 2 y rzt 1 z 2104 H511 1 r 2 z r 2 z rxtx, r 2 y rxt 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 2105 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2106 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 2107 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 x ty, r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2108 H511 1 r 2 z rxtx, r 1 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 125 Nbr. gr Hi L m X 2109 H505 1 r 2 z r 2 x tx, r 2 y t 1 x , r 2 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 2110 H505 1 r 2 z r 2 x tx, r 2 y t 1 x , r 2 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 y rztz, r 2 x rztz, r 2 y rzt 1 z [ 9, 37, 77, 133, 204, 292, 396, 516, 652, 804 ] 2111 H347 1 mz itx, r 2 y rzt 1 x , ity, r 2 z ty, r 2 y rzt 1 y ,mxr 1 z tz, r 2 x rzt 1 z ,mxr 1 z t 1 z 2112 H347 1 mz r 2 z tx, r 2 y rzt 1 x , ity, r 2 z ty, r 2 y rzt 1 y ,mxr 1 z tz, r 2 x rzt 1 z ,mxr 1 z t 1 z 2113 H343 1 mz itx, r 2 y rzt 1 x , ity, r 2 z ty, r 2 y rzt 1 y , tz, mzt 1 z , t 1 z 38A [ 6, 18, 38, 66, 102, 146, 198, 258, 326, 402 ] 2114*, [ 6, 20, 46, 82, 128, 186, 254, 332, 422, 522 ] 2115*, [ 6, 20, 46, 84, 134, 194, 263, 343, 434, 535 ] 2116*, [ 6, 20, 47, 85, 132, 191, 261, 340, 431, 533 ] 2117*, [ 6, 20, 47, 87, 137, 195, 263, 343, 434, 535 ] 2118*, [ 6, 22, 50, 90, 142, 206, 282, 370, 470, 582 ] 2119*, [ 6, 22, 52, 94, 148, 214, 292, 382, 484, 598 ] 2120*, [ 6, 22, 54, 98, 152, 218, 296, 386, 488, 602 ] 2123*, [ 6, 22, 54, 99, 156, 227, 311, 408, 519, 643 ] 2121*, 2122*, [ 6, 22, 56, 100, 152, 218, 296, 386, 488, 602 ] 2124*, [ 6, 24, 58, 105, 166, 240, 328, 430, 545, 675 ] 2125*, 2126*, [ 6, 24, 59, 106, 166, 240, 328, 430, 545, 675 ] 2127*, 2128*, [ 6, 26, 62, 109, 172, 249, 338, 444, 564, 695 ] 2129*, [ 7, 23, 50, 87, 135, 194, 263, 343, 434, 535 ] 2114 H751 hmy, r 2 z i i r2 z tx, r 2 z t 1 x , r 2 y rxty,mxrxt 1 y ,mxrxt 1 z , r 2 y rxt 1 z [ 7, 25, 56, 98, 154, 222, 301, 395, 500, 616 ] 2115 H459 1 r 2 z mxrzt 1 x ,mxr 1 z t 1 x ,mxr 1 z ty,mxrzt 1 y , r 2 y rztz,mzt 1 z [ 7, 25, 56, 100, 158, 226, 306, 401, 506, 623 ] 2116 H533 1 r 2 z mxr 1 z tx,mxrzt 1 x ,mxrzt 1 y ,mxr 1 z t 1 y , r 2 y rztz, r 2 y t 1 z [ 7, 25, 57, 100, 156, 225, 305, 399, 505, 622 ] 2117 H536 1 r 2 z r 1 z t 1 x , rzt 1 x , rzty, r 1 z t 1 y , r 2 y rztz,mzt 1 z [ 7, 25, 57, 102, 159, 226, 306, 401, 506, 623 ] 2118 H512 1 r 2 z rztx, r 1 z t 1 x , r 1 z ty, rzty, r 2 y rztz, r 2 y t 1 z [ 7, 27, 58, 105, 163, 236, 321, 421, 532, 659 ] 2119 H613 hmzi r 2 z mxrzt 1 x ,mxr 1 z t 1 x ,mxr 1 z ty,mxrzt 1 y , r 2 y rztz,mxrzt 1 z [ 7, 27, 60, 107, 167, 240, 327, 427, 540, 667 ] 2120 H627 hmzi r 2 z r 1 z t 1 x , rzt 1 x , rzty, r 1 z t 1 y , r 2 y rztz,mxrzt 1 z [ 7, 27, 62, 108, 168, 242, 328, 428, 542, 668 ] 2121 H693 hmxi r 2 z r 2 x rztx,mzrzt 1 x ,mzrzt 1 y , r 2 x rzt 1 y , r 2 y tz, r 2 x t 1 z 2122 H695 hmxi r 2 z mxr 1 z tx, rzt 1 x , rzt 1 y ,mxr 1 z t 1 y , r 2 y tz, r 2 x t 1 z [ 7, 27, 62, 109, 169, 242, 329, 429, 542, 669 ] 2123 H608 hr 2 y i r2 z r 2 y rxtx, r 2 z rxt 1 x ,mxr 1 x t 1 y ,mzrxt 1 y ,mzrxtz,mxr 1 x t 1 z [ 7, 27, 64, 109, 169, 242, 329, 429, 542, 669 ] 2124 H600 hr 2 y i r2 z r 2 y rxtx, r 2 z rxt 1 x , r 1 x t 1 y , r 2 y rxt 1 y , rxtz, r 2 z rxt 1 z [ 7, 29, 64, 113, 176, 251, 342, 446, 563, 696 ] 2125 H459 1 r 2 z mxrzt 1 x ,mxr 1 z t 1 x ,mxr 1 z ty,mxrzt 1 y , r 2 y rztz, it 1 z 2126 H533 1 r 2 z mxrzt 1 x ,mxr 1 z t 1 x ,mxr 1 z ty,mxrzt 1 y , r 2 y rztz, r 2 y t 1 z [ 7, 29, 65, 113, 176, 251, 342, 446, 563, 696 ] 2127 H536 1 r 2 z r 1 z t 1 x , rzt 1 x , rzty, r 1 z t 1 y , r 2 y rztz, it 1 z 2128 H512 1 r 2 z r 1 z t 1 x , rzt 1 x , rzty, r 1 z t 1 y , r 2 y rztz, r 2 y t 1 z [ 7, 31, 66, 115, 181, 258, 349, 459, 578, 711 ] 2129 H751 hmy, r 2 z i i r2 y tx,mzt 1 x , r 2 y rxty,mxrxt 1 y ,mxrxt 1 z , r 2 y rxt 1 z 38B [ 6, 19, 46, 92, 156, 236, 332, 444, 572, 716 ] 2130*, [ 6, 20, 48, 92, 152, 228, 320, 428, 552, 692 ] 2131*, [ 6, 21, 56, 114, 188, 276, 380, 500, 636, 788 ] 2132*, [ 6, 21, 57, 117, 193, 283, 388, 508, 644, 796 ] 2133*, [ 6, 21, 57, 119, 197, 287, 392, 512, 648, 800 ] 2134*, [ 6, 22, 56, 108, 176, 260, 360, 476, 608, 756 ] 2135*, [ 6, 22, 57, 111, 182, 271, 376, 496, 632, 784 ] 126 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 2136*, [ 6, 22, 61, 121, 195, 285, 390, 510, 646, 798 ] 2137*, [ 6, 23, 62, 120, 192, 280, 384, 504, 640, 792 ] 2138*, [ 6, 23, 62, 122, 196, 284, 388, 508, 644, 796 ] 2139*, [ 6, 23, 62, 124, 196, 284, 388, 508, 644, 796 ] 2140*, [ 6, 23, 64, 126, 200, 288, 392, 512, 648, 800 ] 2141*, [ 6, 23, 66, 128, 201, 290, 393, 512, 648, 800 ] 2142*, [ 6, 23, 67, 129, 200, 288, 392, 512, 648, 800 ] 2143*, [ 6, 24, 60, 112, 180, 264, 364, 480, 612, 760 ] 2144*, [ 6, 24, 62, 120, 194, 282, 386, 506, 642, 794 ] 2145*, [ 6, 24, 63, 120, 192, 280, 384, 504, 640, 792 ] 2146*, [ 6, 24, 65, 123, 194, 282, 386, 506, 642, 794 ] 2147*, [ 6, 24, 65, 124, 195, 282, 386, 506, 642, 794 ] 2148*, [ 6, 24, 65, 124, 196, 284, 388, 508, 644, 796 ] 2149*, [ 6, 25, 68, 128, 200, 288, 392, 512, 648, 800 ] 2150*, 2151*, [ 6, 25, 69, 129, 200, 288, 392, 512, 648, 800 ] 2152*, [ 6, 26, 67, 124, 196, 284, 388, 508, 644, 796 ] 2153*, [ 6, 26, 68, 125, 196, 284, 388, 508, 644, 796 ] 2154*, [ 6, 26, 68, 126, 198, 286, 390, 510, 646, 798 ] 2155*, [ 6, 27, 70, 128, 200, 288, 392, 512, 648, 800 ] 2156*, [ 6, 28, 70, 126, 198, 286, 390, 510, 646, 798 ] 2157*, [ 7, 24, 58, 110, 178, 262, 362, 478, 610, 758 ] 2130 H651 hmxi r 2 y rx mxtx,mxt 1 x ,mxr 1 x t 1 y ,mytz,mxrxtz,myt 1 z [ 7, 25, 59, 109, 175, 257, 355, 469, 599, 745 ] 2131 H646 hmxi r 2 y rx mxtx,mxt 1 x ,mxty,mzrxt 1 y ,mxt 1 y ,mzrxtz [ 7, 26, 66, 124, 196, 284, 388, 508, 644, 796 ] 2132 H581 1 mzrx r 2 z rxtx,mxt 1 x , r 1 x t 1 y ,mytz, rxtz,myt 1 z [ 7, 26, 67, 126, 199, 288, 392, 512, 648, 800 ] 2133 H581 1 mzrx itx,mxt 1 x , r 1 x t 1 y ,mytz, rxtz,myt 1 z [ 7, 26, 67, 128, 201, 290, 394, 514, 650, 802 ] 2134 H423 1 r 2 z rx r 2 z tx, r 2 y t 1 x ,mxrxt 1 y ,mytz,mxr 1 x t 1 z ,myt 1 z [ 7, 27, 65, 119, 189, 275, 377, 495, 629, 779 ] 2135 H552 1 mzrx r 2 z rxtx,mxt 1 x , ty,mzrxt 1 y , t 1 y ,mzrxtz [ 7, 27, 66, 121, 193, 282, 386, 506, 642, 794 ] 2136 H552 1 mzrx itx,mxt 1 x , ty,mzrxt 1 y , t 1 y ,mzrxtz [ 7, 27, 70, 127, 200, 289, 393, 513, 649, 801 ] 2137 H406 1 r 2 z rx r 2 z tx, r 2 y t 1 x , ty, r 2 z rxt 1 y , t 1 y , r 2 z rxt 1 z [ 7, 28, 70, 126, 198, 286, 390, 510, 646, 798 ] 2138 H653 hmxi r 2 y rx r 2 z rxtx, r 2 z rxt 1 x ,mxr 1 x t 1 y ,mytz,mxrxtz,myt 1 z [ 7, 28, 70, 128, 200, 288, 392, 512, 648, 800 ] 2139 H579 1 mzrx mxtx, r 2 y rxt 1 x , r 1 x t 1 y ,mytz, rxtz,myt 1 z [ 7, 28, 70, 130, 198, 290, 390, 514, 646, 802 ] 2140 H577 1 mzrx mxtx, r 2 y rxt 1 x ,mxr 1 x t 1 y , r 2 z tz,mxrxtz, r 2 z t 1 z [ 7, 28, 72, 130, 202, 290, 394, 514, 650, 802 ] 2141 H653 hmxi r 2 y rx itx, r 2 x t 1 x ,mxr 1 x t 1 y ,mytz,mxrxtz,myt 1 z [ 7, 28, 74, 130, 203, 291, 394, 514, 650, 802 ] 2142 H573 1 mzrx r 2 z rxtx, it 1 x ,mxr 1 x t 1 y , r 2 z tz,mxrxtz, r 2 z t 1 z [ 7, 28, 75, 130, 202, 290, 394, 514, 650, 802 ] 2143 H575 1 mzrx r 2 z rxtx, it 1 x , r 1 x t 1 y ,mytz, rxtz,myt 1 z [ 7, 29, 67, 121, 191, 277, 379, 497, 631, 781 ] 2144 H647 hmxi r 2 y rx r 2 z rxtx, r 2 z rxt 1 x ,mxty,mzrxt 1 y ,mxt 1 y ,mzrxtz [ 7, 29, 69, 127, 199, 287, 391, 511, 647, 799 ] 2145 H647 hmxi r 2 y rx itx, r 2 x t 1 x ,mxty,mzrxt 1 y ,mxt 1 y ,mzrxtz [ 7, 29, 70, 126, 198, 286, 390, 510, 646, 798 ] 2146 H546 1 mzrx mxtx, r 2 y rxt 1 x , ty,mzrxt 1 y , t 1 y ,mzrxtz [ 7, 29, 72, 127, 199, 287, 391, 511, 647, 799 ] 2147 H549 1 mzrx r 2 z rxtx, it 1 x , ty,mzrxt 1 y , t 1 y ,mzrxtz [ 7, 29, 72, 128, 199, 287, 391, 511, 647, 799 ] 2148 H554 1 mzrx r 2 z rxtx, it 1 x ,mxty, r 2 y rxt 1 y ,mxt 1 y , r 2 y rxtz [ 7, 29, 72, 128, 200, 288, 392, 512, 648, 800 ] 2149 H548 1 mzrx mxtx, r 2 y rxt 1 x ,mxty, r 2 y rxt 1 y ,mxt 1 y , r 2 y rxtz [ 7, 30, 74, 130, 202, 290, 394, 514, 650, 802 ] 2150 H423 1 r 2 z rx r 1 x tx, rxt 1 x ,mxrxt 1 y ,mytz,mxr 1 x t 1 z ,myt 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 127 Nbr. gr Hi L m X 2151 H581 1 mzrx r 2 z rxtx, r 2 y rxt 1 x , r 1 x t 1 y ,mytz, rxtz,myt 1 z [ 7, 30, 75, 130, 202, 290, 394, 514, 650, 802 ] 2152 H581 1 mzrx itx, r 2 y rxt 1 x , r 1 x t 1 y ,mytz, rxtz,myt 1 z [ 7, 31, 72, 128, 200, 288, 392, 512, 648, 800 ] 2153 H552 1 mzrx r 2 z rxtx, r 2 y rxt 1 x , ty,mzrxt 1 y , t 1 y ,mzrxtz [ 7, 31, 73, 128, 200, 288, 392, 512, 648, 800 ] 2154 H552 1 mzrx itx, r 2 y rxt 1 x , ty,mzrxt 1 y , t 1 y ,mzrxtz [ 7, 31, 73, 129, 201, 289, 393, 513, 649, 801 ] 2155 H406 1 r 2 z rx r 1 x tx, rxt 1 x , ty, r 2 z rxt 1 y , t 1 y , r 2 z rxt 1 z [ 7, 32, 74, 130, 202, 290, 394, 514, 650, 802 ] 2156 H651 hmxi r 2 y rx r 2 y rxtx,mzrxt 1 x ,mxr 1 x t 1 y ,mytz,mxrxtz,myt 1 z [ 7, 33, 73, 129, 201, 289, 393, 513, 649, 801 ] 2157 H646 hmxi r 2 y rx r 2 y rxtx,mzrxt 1 x ,mxty,mzrxt 1 y ,mxt 1 y ,mzrxtz 39A [ 8, 28, 64, 116, 184, 268, 368, 484, 616, 764 ] 2158*, [ 8, 29, 67, 121, 191, 277, 379, 497, 631, 781 ] 2159*, [ 8, 29, 68, 124, 196, 284, 388, 508, 644, 796 ] 2160*, 2161*, 2162*, [ 8, 29, 68, 125, 198, 286, 390, 510, 646, 798 ] 2163*, [ 8, 29, 68, 125, 199, 288, 392, 512, 648, 800 ] 2164*, [ 8, 30, 70, 126, 198, 286, 390, 510, 646, 798 ] 2165*, [ 8, 30, 71, 128, 200, 288, 392, 512, 648, 800 ] 2166*, 2167*, 2168*, 2169*, 2170*, 2171*, [ 8, 30, 71, 129, 203, 292, 396, 516, 652, 804 ] 2172*, [ 8, 30, 72, 130, 202, 290, 394, 514, 650, 802 ] 2173*, [ 8, 31, 72, 128, 200, 288, 392, 512, 648, 800 ] 2174*, 2175*, 2176*, 2177*, 2178*, 2179*, 2180*, 2181*, 2182*, 2183*, [ 8, 31, 73, 130, 202, 290, 394, 514, 650, 802 ] 2184*, 2185*, 2186*, 2187*, [ 8, 31, 73, 131, 204, 292, 396, 516, 652, 804 ] 2188*, [ 8, 31, 74, 131, 202, 290, 394, 514, 650, 802 ] 2189*, [ 8, 31, 74, 132, 204, 292, 396, 516, 652, 804 ] 2190*, 2191*, 2192*, 2193*, 2194*, 2195*, [ 8, 32, 74, 130, 202, 290, 394, 514, 650, 802 ] 2196*, 2197*, 2198*, 2199*, 2200*, 2201*, [ 8, 32, 75, 132, 204, 292, 396, 516, 652, 804 ] 2202*, 2203*, 2204*, 2205*, 2206*, 2207*, 2208*, 2209*, 2210*, 2211*, 2212*, 2213*, [ 8, 32, 76, 133, 204, 292, 396, 516, 652, 804 ] 2214*, [ 8, 33, 76, 132, 204, 292, 396, 516, 652, 804 ] 2215*, 2216*, 2217*, 2218*, 2219*, 2220*, 2221*, 2222*, 2223*, 2224*, 2225*, 2226*, 2227*, 2228*, 2229*, 2230*, 2231*, [ 9, 31, 69, 123, 193, 279, 381, 499, 633, 783 ] 2158 H745 hmzr 1 x , r 2 z rxi i r 2 z rxtx, r 2 z rxt 1 x ,mxty,mzrxt 1 y ,mxt 1 y , r 2 x tz, r 2 z rxtz, r 2 z rxt 1 z [ 9, 32, 71, 126, 197, 284, 387, 506, 641, 792 ] 2159 H305 1 i tx, it 1 x , t 1 x , ty, t 1 y ,mxr 1 z tz, r 2 y rzt 1 z ,mxr 1 z t 1 z [ 9, 32, 72, 128, 200, 288, 392, 512, 648, 800 ] 2160 H615 hmzi r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , mztz, t 1 z 2161 H747 hmzr 1 x , r 2 z rxi i r 2 z rxtx, r 2 z rxt 1 x , r 2 y ty,mxr 1 x t 1 y , r 2 y t 1 y ,mxr 1 x tz, r 2 y tz,mxr 1 x t 1 z 2162 H460 1 r 2 z tx, r 2 z t 1 x , t 1 x , ty, r 2 z t 1 y , t 1 y , r 2 y tz,mzt 1 z [ 9, 32, 72, 129, 201, 289, 393, 513, 649, 801 ] 2163 H504 1 r 2 z tx, r 2 z t 1 x , t 1 x , ryty, r 2 x ryt 1 y , r 1 y t 1 y , tz, t 1 z [ 9, 32, 72, 129, 202, 290, 394, 514, 650, 802 ] 2164 H503 1 r 2 z tx, r 2 z t 1 x , t 1 x , ty, r 2 z t 1 y , t 1 y , r 2 x rztz, r 2 y t 1 z [ 9, 33, 73, 129, 201, 289, 393, 513, 649, 801 ] 2165 H614 hmzi r 2 z tx, r 2 z t 1 x , t 1 x , ty, r 2 z t 1 y , t 1 y , r 2 y tz,mxt 1 z [ 9, 33, 74, 130, 202, 290, 394, 514, 650, 802 ] 2166 H629 hmzr 1 x i r2 x r 2 z rxtx, r 2 y rxt 1 x , ty,mzrxt 1 y , t 1 y , r 2 x tz, mzr 1 x tz,mzr 1 x t 1 z 2167 H461 1 r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz,mzt 1 z 2168 H461 1 r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz,mzt 1 z 128 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 2169 H483 1 r 2 z mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz,mzt 1 z 2170 H531 1 r 2 z mxrztx,mxrzt 1 x ,mxr 1 z t 1 x ,mxrzty,mxr 1 z ty,mxrzt 1 y , r 2 y tz,mzt 1 z 2171 H533 1 r 2 z mxrztx,mxrzt 1 x ,mxr 1 z t 1 x ,mxrzty,mxr 1 z ty,mxrzt 1 y , r 2 y rztz, r 2 y t 1 z [ 9, 33, 74, 131, 204, 292, 396, 516, 652, 804 ] 2172 H529 1 r 2 z mytx,mxt 1 x ,myt 1 x ,mxty,myty,mxt 1 y , r 2 y rztz, r 2 y t 1 z [ 9, 33, 75, 131, 203, 291, 395, 515, 651, 803 ] 2173 H669 hr 2 y rxi r2zrx r 2 y tx, r 2 z t 1 x , ty, r 2 x t 1 y , t 1 y , r 2 y rxtz, r 2 x t 1 z , r 2 y rxt 1 z [ 9, 34, 74, 130, 202, 290, 394, 514, 650, 802 ] 2174 H616 hmzi r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz,mxt 1 z 2175 H616 hmzi r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz,myt 1 z 2176 H618 hmzi r 2 z mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz,mxt 1 z 2177 H308 1 i mztx, r 2 z t 1 x ,mzt 1 x ,mzty,mzt 1 y ,mxr 1 z tz, r 2 y rzt 1 z ,mxr 1 z t 1 z 2178 H323 1 i r 2 z tx, r 2 z t 1 x ,mxryty, r 2 x ryt 1 y ,mxryt 1 y ,mxtz, mxt 1 z , r 2 x t 1 z 2179 H323 1 i mztx, r 2 z t 1 x ,mzt 1 x ,mxryty, r 2 x ryt 1 y ,mxryt 1 y , mxtz,mxt 1 z 2180 H326 1 i mzrxtx, r 2 z rxt 1 x ,mzrxt 1 x ,myty,myt 1 y , r 2 z tz, r 2 z t 1 z ,mzt 1 z 2181 H540 1 r 2 z mztx, it 1 x ,mzt 1 x , ryty, r 2 x ryt 1 y , r 1 y t 1 y , mytz,myt 1 z 2182 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z 2183 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,mxty,mxt 1 y ,myt 1 y , mztz,mzt 1 z [ 9, 34, 75, 131, 203, 291, 395, 515, 651, 803 ] 2184 H433 1 r 2 z mztx, it 1 x ,mzt 1 x , ty, r 2 z t 1 y , t 1 y , mztz, it 1 z 2185 H462 1 r 2 z mztx, it 1 x ,mzt 1 x , ty, r 2 z t 1 y , t 1 y , r 2 x tz, r 2 y t 1 z 2186 H460 1 r 2 z tx, r 2 z t 1 x , t 1 x , ty, r 2 z t 1 y , t 1 y , r 2 y tz, it 1 z 2187 H503 1 r 2 z tx, r 2 z t 1 x , t 1 x , ty, r 2 z t 1 y , t 1 y , r 2 y rztz, r 2 y t 1 z [ 9, 34, 75, 132, 204, 292, 396, 516, 652, 804 ] 2188 H540 1 r 2 z mztx, it 1 x ,mzt 1 x , ryty, r 2 x ryt 1 y , r 1 y t 1 y , mxtz,mxt 1 z [ 9, 34, 76, 131, 203, 291, 395, 515, 651, 803 ] 2189 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mzr 1 x ty, it 1 y , r 2 z rxt 1 y , r 2 x tz, mzr 1 x tz, r 2 z rxt 1 z [ 9, 34, 76, 132, 204, 292, 396, 516, 652, 804 ] 2190 H675 hr 2 y rxi r2zrx r 2 y tx, r 2 z t 1 x ,mxr 1 x ty,mzt 1 y ,myt 1 y ,mytz, mxr 1 x t 1 z ,mxrxt 1 z 2191 H468 1 r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , mztz, it 1 z 2192 H469 1 r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz, r 2 y t 1 z 2193 H640 hmzr 1 x i r2 x r 2 z rxtx, r 2 y rxt 1 x ,mzty, r 1 x t 1 y ,mzt 1 y , r 1 x tz, mztz, r 1 x t 1 z 2194 H535 1 r 2 z rztx, r 1 z t 1 x , rzt 1 x , r 1 z ty, rzty, r 1 z t 1 y , r 2 y tz,mzt 1 z 2195 H512 1 r 2 z rztx, r 1 z t 1 x , rzt 1 x , r 1 z ty, rzty, r 1 z t 1 y , r 2 y rztz, r 2 y t 1 z [ 9, 35, 75, 131, 203, 291, 395, 515, 651, 803 ] 2196 H745 hmzr 1 x , r 2 z rxi i r 2 y rxtx,mzrxt 1 x ,mxty,mzrxt 1 y ,mxt 1 y , r 2 x tz, r 2 z rxtz, r 2 z rxt 1 z 2197 H631 hmzr 1 x i r2 x r 2 z rxtx, r 2 y rxt 1 x ,mxty, r 2 y rxt 1 y ,mxt 1 y , itz, r 2 z rxtz, r 2 z rxt 1 z 2198 H674 hr 2 y rxi r2zrx r 2 y tx, r 2 z t 1 x ,mzrxty, it 1 y ,mxt 1 y ,mxtz, mzr 1 x t 1 z ,mzrxt 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 129 Nbr. gr Hi L m X 2199 H624 hmzi r 2 z mxrztx,mxrzt 1 x ,mxr 1 z t 1 x ,mxrzty,mxr 1 z ty,mxrzt 1 y , r 2 y tz,mxt 1 z 2200 H531 1 r 2 z mxrztx,mxrzt 1 x ,mxr 1 z t 1 x ,mxrzty,mxr 1 z ty,mxrzt 1 y , r 2 y tz, it 1 z 2201 H533 1 r 2 z mxr 1 z tx,mxrzt 1 x ,mxr 1 z t 1 x ,mxr 1 z ty,mxrzt 1 y ,mxr 1 z t 1 y , r 2 y rztz, r 2 y t 1 z [ 9, 35, 76, 132, 204, 292, 396, 516, 652, 804 ] 2202 H475 1 r 2 z mztx, it 1 x ,mzt 1 x ,mxty,mxt 1 y ,myt 1 y , mztz, it 1 z 2203 H485 1 r 2 z mztx, it 1 x ,mzt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz, r 2 y t 1 z 2204 H466 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , mztz, it 1 z 2205 H427 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z 2206 H461 1 r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, it 1 z 2207 H461 1 r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz, it 1 z 2208 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,mxty,mxt 1 y ,myt 1 y , mztz, it 1 z 2209 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,mxty,myty,mxt 1 y , mztz, it 1 z 2210 H486 1 r 2 z r 2 x tx, r 2 y t 1 x , r 2 x t 1 x ,mxty,mxt 1 y ,myt 1 y , mztz, it 1 z 2211 H479 1 r 2 z r 2 x tx, r 2 y t 1 x , r 2 x t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz, r 2 y t 1 z 2212 H483 1 r 2 z mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, it 1 z 2213 H529 1 r 2 z mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, r 2 y t 1 z [ 9, 35, 77, 132, 204, 292, 396, 516, 652, 804 ] 2214 H571 1 mzrx mxtx, r 2 y rxt 1 x , rxty, r 2 z t 1 y ,mxrxt 1 y , r 1 x tz, mztz,mxr 1 x t 1 z [ 9, 36, 76, 132, 204, 292, 396, 516, 652, 804 ] 2215 H615 hmzi r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , itz, r 2 z t 1 z 2216 H305 1 i tx, it 1 x , t 1 x , ity, it 1 y ,mxr 1 z tz, r 2 y rzt 1 z ,mxr 1 z t 1 z 2217 H747 hmzr 1 x , r 2 z rxi i r 2 y rxtx,mzrxt 1 x , r 2 y ty,mxr 1 x t 1 y , r 2 y t 1 y ,mxr 1 x tz, r 2 y tz,mxr 1 x t 1 z 2218 H308 1 i mztx, r 2 z t 1 x ,mzt 1 x , r 2 z ty, r 2 z t 1 y ,mxr 1 z tz, r 2 y rzt 1 z ,mxr 1 z t 1 z 2219 H326 1 i mzrxtx, r 2 z rxt 1 x ,mzrxt 1 x , r 2 y ty, r 2 y t 1 y ,myt 1 y , r 2 z tz, r 2 z t 1 z 2220 H473 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , mztz, it 1 z 2221 H474 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z 2222 H504 1 r 2 z tx, r 2 z t 1 x , t 1 x , ryty, r 2 x ryt 1 y , r 1 y t 1 y , r 2 z tz, r 2 z t 1 z 2223 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z 2224 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 y ty, r 2 x ty, r 2 y t 1 y , r 2 x tz, r 2 y t 1 z 2225 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 x t 1 z 2226 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,mxty,mxt 1 y ,myt 1 y , itz, it 1 z 2227 H639 hmzr 1 x i r2 x r 2 z rxtx, r 2 y rxt 1 x , r 2 y ty,mxr 1 x t 1 y , r 2 y t 1 y ,mxr 1 x tz, r 2 y tz,mxr 1 x t 1 z 2228 H671 hr 2 y rxi r2zrx r 2 y tx, r 2 z t 1 x , r 1 x ty, r 2 z t 1 y , r 2 y t 1 y , r 2 z tz, rxt 1 z , r 1 x t 1 z 2229 H626 hmzi r 2 z rztx, r 1 z t 1 x , rzt 1 x , r 1 z ty, rzty, r 1 z t 1 y , r 2 y tz,mxt 1 z 2230 H535 1 r 2 z rztx, r 1 z t 1 x , rzt 1 x , r 1 z ty, rzty, r 1 z t 1 y , r 2 y tz, it 1 z 130 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 2231 H512 1 r 2 z r 1 z tx, r 1 z t 1 x , rzt 1 x , rzty, r 1 z t 1 y , rzt 1 y , r 2 y rztz, r 2 y t 1 z 39B [ 8, 31, 74, 132, 204, 292, 396, 516, 652, 804 ] 2232*, [ 8, 33, 78, 134, 204, 292, 396, 516, 652, 804 ] 2233*, [ 8, 34, 78, 133, 204, 292, 396, 516, 652, 804 ] 2234*, 2235*, [ 8, 35, 78, 132, 204, 292, 396, 516, 652, 804 ] 2236*, 2237*, [ 9, 34, 76, 132, 204, 292, 396, 516, 652, 804 ] 2232 H779 hi, r 2 x , r 2 z i r2 z rx mxtx, t 1 x ,myty,mxr 1 x ty, t 1 y , r 2 z rxt 1 y , r 2 z rxtz,mxrxt 1 z [ 9, 36, 78, 132, 204, 292, 396, 516, 652, 804 ] 2233 H766 hmy, r 2 x i mzr1x r 2 z rxtx,mxt 1 x ,myty, r 1 x ty, t 1 y ,mzr 1 x t 1 y , mzr 1 x tz, rxt 1 z [ 9, 37, 77, 132, 204, 292, 396, 516, 652, 804 ] 2234 H561 1 mzrx mxtx, r 2 y rxt 1 x , r 2 x ty,mzr 1 x ty, it 1 y , r 2 z rxt 1 y , mzr 1 x tz, r 2 z rxt 1 z 2235 H571 1 mzrx mxtx, r 2 y rxt 1 x ,myty, rxty, r 2 z t 1 y ,mxrxt 1 y , r 1 x tz,mxr 1 x t 1 z [ 9, 38, 76, 132, 204, 292, 396, 516, 652, 804 ] 2236 H779 hi, r 2 x , r 2 z i r2 z rx mzr 1 x tx, r 2 z rxt 1 x ,myty,mxr 1 x ty, t 1 y , r 2 z rxt 1 y , r 2 z rxtz,mxrxt 1 z 2237 H768 hmy, r 2 x i mzr1x r 2 z rxtx,mxt 1 x , r 2 z ty,mxr 1 x ty,mxt 1 y , r 2 z rxt 1 y , r 2 z rxtz,mxrxt 1 z 40 [ 9, 32, 72, 128, 200, 288, 392, 512, 648, 800 ] 2238*, 2239*, 2240*, [ 9, 33, 75, 132, 204, 292, 396, 516, 652, 804 ] 2241*, 2242*, 2243*, 2244*, 2245*, 2246*, 2247*, 2248*, [ 9, 34, 76, 132, 204, 292, 396, 516, 652, 804 ] 2249*, 2250*, 2251*, 2252*, 2253*, 2254*, 2255*, 2256*, 2257*, 2258*, 2259*, 2260*, 2261*, 2262*, 2263*, 2264*, 2265*, 2266*, 2267*, 2268*, 2269*, [ 10, 34, 74, 130, 202, 290, 394, 514, 650, 802 ] 2238 H749 hmy, r 2 z i i r 2 y tx, r 2 z tx,mzt 1 x , r 2 z t 1 x , r 2 z ty, r 2 z t 1 y , r 2 z tz,mzt 1 z , r 2 z t 1 z 2239 H750 hmy,mzi r 2 z mytx,mxt 1 x ,myt 1 x ,myty,myt 1 y , r 2 y rztz, mzrztz,mxrzt 1 z , rzt 1 z 2240 H690 hmxi r 2 z mxtx,mxt 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 y rztz, mzr 1 z tz, r 2 x t 1 z , r 2 y t 1 z [ 10, 35, 76, 132, 204, 292, 396, 516, 652, 804 ] 2241 H682 hmxi r 2 z itx, r 2 y tx, r 2 x t 1 x , r 2 y t 1 x ,mxty,myt 1 y , mxt 1 y , r 2 y tz, r 2 x t 1 z 2242 H680 hmxi r 2 z r 2 z tx,mxtx,myt 1 x ,mxt 1 x ,mxty,myt 1 y , mxt 1 y , r 2 y tz, r 2 x t 1 z 2243 H359 1 mz itx, r 2 z tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , tz,mzt 1 z , t 1 z 2244 H452 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z ty, ty, r 2 z t 1 y , t 1 y , r 2 y rztz,mzt 1 z 2245 H503 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z ty, ty, r 2 z t 1 y , t 1 y , r 2 y rztz, r 2 x t 1 z 2246 H541 1 r 2 z r 2 z rxtx, rxtx, r 2 y rxt 1 x , r 1 x t 1 x ,mxty,mxt 1 y , myt 1 y ,mztz, it 1 z 2247 H527 1 r 2 z mxtx,mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y , myt 1 y , r 2 y rztz,mzt 1 z 2248 H529 1 r 2 z mxtx,mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y , myt 1 y , r 2 y rztz, r 2 y t 1 z [ 10, 36, 76, 132, 204, 292, 396, 516, 652, 804 ] 2249 H749 hmy, r 2 z i i r 2 y tx, r 2 z tx,mzt 1 x , r 2 z t 1 x , r 2 x ty,mzt 1 y , r 2 z tz,mzt 1 z , r 2 z t 1 z 2250 H604 hr 2 y i r2 z itx,mxtx,myt 1 x ,mxt 1 x ,mxty,myt 1 y , mxt 1 y , r 2 x rztz, rzt 1 z 2251 H750 hmy,mzi r 2 z mytx,mxt 1 x ,myt 1 x , r 2 z ty,mxt 1 y , r 2 y rztz, mzrztz,mxrzt 1 z , rzt 1 z 2252 H750 hmy, r 2 z i i r 2 y tx, r 2 z tx,mzt 1 x , r 2 z t 1 x , r 2 x ty,mzt 1 y , mxrztz, r 2 x rzt 1 z ,mxrzt 1 z 2253 H750 hmy, r 2 z i i r 2 y tx, r 2 z tx,mzt 1 x , r 2 z t 1 x , r 2 z ty, r 2 z t 1 y , mxrztz, r 2 x rzt 1 z ,mxrzt 1 z 2254 H598 hr 2 y i r2 z r 2 z tx, r 2 y tx, r 2 x t 1 x , r 2 y t 1 x , r 2 y ty, r 2 x t 1 y , r 2 y t 1 y , r 2 x rztz, rzt 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 131 Nbr. gr Hi L m X 2255 H611 hmzi r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z ty, ty, r 2 z t 1 y , t 1 y , r 2 y rztz,mxrzt 1 z 2256 H622 hmzi r 2 z mxtx,mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y , myt 1 y , r 2 y rztz,mxrzt 1 z 2257 H683 hmxi r 2 z itx, r 2 y tx, r 2 x t 1 x , r 2 y t 1 x , r 2 y ty, r 2 x t 1 y , r 2 y t 1 y , r 2 y tz, r 2 x t 1 z 2258 H344 1 mz r 2 z tx, it 1 x , ity, r 2 z ty, it 1 y , r 2 z t 1 y , mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2259 H682 hmyi mx mztx, r 2 y t 1 x ,mzt 1 x , r 2 z ty,myty,mxt 1 y , t 1 y ,mztz, r 2 y t 1 z 2260 H370 1 mz itx, r 2 z tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , mytz, r 2 x t 1 z ,myt 1 z 2261 H378 1 mz r 2 y tx,mxtx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2262 H690 hmxi r 2 z r 2 z tx,myt 1 x ,mxty,myt 1 y ,mxt 1 y , r 2 y rztz, mzr 1 z tz, r 2 x t 1 z , r 2 y t 1 z 2263 H452 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z ty, ty, r 2 z t 1 y , t 1 y , r 2 y rztz, it 1 z 2264 H503 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z ty, ty, r 2 z t 1 y , t 1 y , r 2 y rztz, r 2 y t 1 z 2265 H511 1 r 2 z r 2 z rxtx, rxtx, r 2 y rxt 1 x , r 1 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z 2266 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 y ty, r 2 x ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z 2267 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,mxty,myty,mxt 1 y , myt 1 y ,mztz, it 1 z 2268 H527 1 r 2 z mxtx,mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y , myt 1 y , r 2 y rztz, it 1 z 2269 H529 1 r 2 z mytx,mxt 1 x ,myt 1 x ,mxty,myty,mxt 1 y , myt 1 y , r 2 y rztz, r 2 y t 1 z 41 [ 9, 33, 75, 132, 204, 292, 396, 516, 652, 804 ] 2270*, 2271*, [ 9, 34, 77, 133, 204, 292, 396, 516, 652, 804 ] 2272*, 2273*, 2274*, 2275*, 2276*, 2277*, 2278*, 2279*, 2280*, 2281*, [ 9, 35, 77, 132, 204, 292, 396, 516, 652, 804 ] 2282*, 2283*, 2284*, 2285*, 2286*, 2287*, 2288*, 2289*, 2290*, 2291*, 2292*, 2293*, 2294*, 2295*, 2296*, 2297*, 2298*, 2299*, 2300*, 2301*, [ 10, 35, 76, 132, 204, 292, 396, 516, 652, 804 ] 2270 H648 hmxi r 2 y rx mxtx,mxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 z rxtz, r 2 x t 1 z , r 2 z rxt 1 z 2271 H650 hmxi r 2 y rx mxtx,mxt 1 x ,myty,mxrxty,myt 1 y ,mxrxt 1 y , mxr 1 x tz,mxr 1 x t 1 z , r 2 y t 1 z [ 10, 36, 77, 132, 204, 292, 396, 516, 652, 804 ] 2272 H422 1 r 2 z rx r 2 z tx, r 2 y t 1 x ,mxr 1 x ty,myty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mxrxt 1 z ,mzt 1 z 2273 H409 1 r 2 z rx r 2 z tx, r 2 y t 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 y rxt 1 z , r 2 x t 1 z 2274 H560 1 mzrx itx,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, mzr 1 x tz, r 2 x t 1 z ,mzr 1 x t 1 z 2275 H560 1 mzrx r 2 z rxtx,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, mzr 1 x tz, r 2 x t 1 z ,mzr 1 x t 1 z 2276 H580 1 mzrx itx,mxt 1 x ,myty, rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z ,mzt 1 z 2277 H580 1 mzrx r 2 z rxtx,mxt 1 x ,myty, rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z ,mzt 1 z 2278 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mzr 1 x ty, it 1 y , r 2 z rxt 1 y , r 2 x tz, mzr 1 x tz, it 1 z , r 2 z rxt 1 z 2279 H571 1 mzrx mxtx, r 2 y rxt 1 x ,myty, rxty, r 2 z t 1 y ,mxrxt 1 y , r 1 x tz,mxr 1 x t 1 z , r 2 y t 1 z 2280 H557 1 mzrx mxtx, r 2 y rxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, mzr 1 x tz, r 2 x t 1 z ,mzr 1 x t 1 z 2281 H578 1 mzrx mxtx, r 2 y rxt 1 x ,myty, rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z ,mzt 1 z [ 10, 37, 76, 132, 204, 292, 396, 516, 652, 804 ] 2282 H649 hmxi r 2 y rx itx, r 2 x t 1 x , r 2 z rxty, r 2 x t 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 z rxtz, r 2 x t 1 z , r 2 z rxt 1 z 2283 H649 hmxi r 2 y rx r 2 z rxtx, r 2 z rxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 z rxtz, r 2 x t 1 z , r 2 z rxt 1 z 132 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 2284 H652 hmxi r 2 y rx itx, r 2 x t 1 x ,myty,mxrxty,myt 1 y ,mxrxt 1 y , mxr 1 x tz,mxr 1 x t 1 z , r 2 y t 1 z 2285 H652 hmxi r 2 y rx r 2 z rxtx, r 2 z rxt 1 x ,myty,mxrxty,myt 1 y ,mxrxt 1 y , mxr 1 x tz,mxr 1 x t 1 z , r 2 y t 1 z 2286 H422 1 r 2 z rx r 1 x tx, rxt 1 x ,mxr 1 x ty,myty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mxrxt 1 z ,mzt 1 z 2287 H409 1 r 2 z rx r 1 x tx, rxt 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 y rxt 1 z , r 2 x t 1 z 2288 H648 hmxi r 2 y rx r 2 y rxtx,mzrxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 z rxtz, r 2 x t 1 z , r 2 z rxt 1 z 2289 H650 hmxi r 2 y rx r 2 y rxtx,mzrxt 1 x ,myty,mxrxty,myt 1 y ,mxrxt 1 y , mxr 1 x tz,mxr 1 x t 1 z , r 2 y t 1 z 2290 H551 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxty, it 1 y , r 2 z rxt 1 y , itz, r 2 z rxtz, it 1 z , r 2 z rxt 1 z 2291 H572 1 mzrx r 2 z rxtx, it 1 x , r 2 z ty,mxrxty, r 2 z t 1 y ,mxrxt 1 y , mxr 1 x tz,mxr 1 x t 1 z , r 2 y t 1 z 2292 H555 1 mzrx r 2 z rxtx, it 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, mzr 1 x tz, r 2 x t 1 z ,mzr 1 x t 1 z 2293 H574 1 mzrx r 2 z rxtx, it 1 x ,myty, rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z ,mzt 1 z 2294 H560 1 mzrx itx, r 2 y rxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, mzr 1 x tz, r 2 x t 1 z ,mzr 1 x t 1 z 2295 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz, mzr 1 x tz, r 2 x t 1 z ,mzr 1 x t 1 z 2296 H580 1 mzrx itx, r 2 y rxt 1 x ,myty, rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z ,mzt 1 z 2297 H580 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,myty, rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z ,mzt 1 z 2298 H556 1 mzrx mxtx, r 2 y rxt 1 x , r 2 z rxty, it 1 y , r 2 z rxt 1 y , itz, r 2 z rxtz, it 1 z , r 2 z rxt 1 z 2299 H561 1 mzrx mxtx, r 2 y rxt 1 x , r 2 x ty,mzr 1 x ty, it 1 y , r 2 z rxt 1 y , r 2 x tz,mzr 1 x tz, r 2 z rxt 1 z 2300 H576 1 mzrx mxtx, r 2 y rxt 1 x , r 2 z ty,mxrxty, r 2 z t 1 y ,mxrxt 1 y , mxr 1 x tz,mxr 1 x t 1 z , r 2 y t 1 z 2301 H571 1 mzrx mxtx, r 2 y rxt 1 x ,myty, rxty, r 2 z t 1 y ,mxrxt 1 y , r 1 x tz,mztz,mxr 1 x t 1 z 42A [ 6, 24, 64, 130, 207, 289, 396, 519, 649, 804 ] 2302*, [ 6, 24, 65, 116, 179, 260, 354, 467, 593, 734 ] 2303*, [ 7, 29, 70, 134, 205, 289, 398, 517, 649, 806 ] 2302 H393 1 r 2 z rx r 2 y rztx, r 1 y r 1 z t 1 x , r 2 y rzty, ryrxty, r 2 x ryt 1 y , r 2 y rztz [ 7, 29, 71, 119, 188, 269, 368, 481, 609, 752 ] 2303 H393 1 r 2 z rx r 2 y rztx, r 1 y r 1 z t 1 x , r 2 y rzty, ryrxty, r 1 y r 1 z t 1 y , ryrxtz 42B [ 6, 24, 56, 98, 153, 222, 303, 396, 503, 622 ] 2304*, [ 6, 24, 57, 101, 158, 228, 310, 405, 513, 633 ] 2305*, [ 7, 29, 63, 110, 172, 249, 338, 441, 560, 691 ] 2304 H393 1 r 2 z rx r 2 y rztx, ryrxtx, r 2 y rzty, r 1 y r 1 z t 1 y , ryrxtz, r 1 y r 1 z t 1 z [ 7, 29, 64, 112, 175, 252, 342, 446, 565, 697 ] 2305 H393 1 r 2 z rx r 2 y rztx, ryrxtx, r 2 y rzty, r 2 x ryt 1 y , r 2 y rztz, r 1 y r 1 z t 1 z 43 [ 8, 30, 73, 135, 210, 296, 397, 516, 652, 804 ] 2306*, 2307*, [ 8, 31, 73, 129, 200, 288, 392, 512, 648, 800 ] 2308*, 2309*, [ 8, 31, 75, 134, 205, 292, 396, 516, 652, 804 ] 2310*, 2311*, [ 8, 31, 75, 136, 209, 294, 396, 516, 652, 804 ] 2312*, 2313*, [ 8, 31, 76, 137, 208, 293, 396, 516, 652, 804 ] 2314*, 2315*, 2316*, 2317*, [ 8, 32, 73, 128, 200, 288, 392, 512, 648, 800 ] 2318*, 2319*, [ 8, 32, 76, 132, 202, 290, 394, 514, 650, 802 ] 2320*, 2321*, 2322*, 2323*, 2324*, 2325*, [ 8, 32, 76, 133, 204, 292, 396, 516, 652, 804 ] 2326*, 2327*, [ 8, 32, 77, 135, 205, 292, 396, 516, 652, 804 ] 2328*, 2329*, 2330*, 2331*, [ 8, 32, 77, 136, 207, 293, 396, 516, 652, 804 ] 2332*, [ 8, 32, 78, 135, 204, 292, 396, 516, 652, 804 ] 2333*, 2334*, K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 133 Nbr. gr Hi L m X [ 8, 32, 78, 137, 206, 292, 396, 516, 652, 804 ] 2335*, 2336*, 2337*, 2338*, [ 8, 33, 76, 131, 202, 290, 394, 514, 650, 802 ] 2339*, [ 8, 33, 77, 135, 207, 293, 396, 516, 652, 804 ] 2340*, 2341*, 2342*, 2343*, 2344*, 2345*, 2346*, 2347*, 2348*, 2349*, 2350*, [ 8, 33, 78, 134, 204, 292, 396, 516, 652, 804 ] 2351*, [ 8, 33, 78, 136, 206, 292, 396, 516, 652, 804 ] 2352*, [ 8, 34, 76, 130, 202, 290, 394, 514, 650, 802 ] 2353*, 2354*, 2355*, 2356*, 2357*, 2358*, 2359*, [ 8, 34, 78, 133, 204, 292, 396, 516, 652, 804 ] 2360*, 2361*, 2362*, 2363*, 2364*, 2365*, 2366*, [ 8, 34, 78, 135, 206, 292, 396, 516, 652, 804 ] 2367*, 2368*, 2369*, 2370*, 2371*, 2372*, 2373*, 2374*, 2375*, 2376*, 2377*, 2378*, 2379*, [ 9, 33, 76, 135, 207, 293, 396, 516, 652, 804 ] 2306 H580 1 mzrx itx, r 2 z rxtx,mxt 1 x ,myty,myt 1 y , rxt 1 y , mztz, r 1 x t 1 z 2307 H578 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y , rxt 1 y , mztz, r 1 x t 1 z [ 9, 34, 75, 130, 202, 290, 394, 514, 650, 802 ] 2308 H560 1 mzrx itx, r 2 z rxtx,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , mzr 1 x tz,mzr 1 x t 1 z 2309 H557 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , mzr 1 x tz,mzr 1 x t 1 z [ 9, 34, 77, 133, 204, 292, 396, 516, 652, 804 ] 2310 H580 1 mzrx itx, r 2 z rxtx,mxt 1 x , rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z 2311 H578 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z [ 9, 34, 77, 135, 206, 292, 396, 516, 652, 804 ] 2312 H560 1 mzrx itx, r 2 z rxtx,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y , r 2 x tz, mzr 1 x tz, r 2 x t 1 z 2313 H557 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y , r 2 x tz, mzr 1 x tz, r 2 x t 1 z [ 9, 34, 78, 135, 205, 292, 396, 516, 652, 804 ] 2314 H580 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,myty,myt 1 y , rxt 1 y , mztz, r 1 x t 1 z 2315 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, r 2 z t 1 y ,mxrxt 1 y , mztz,mxr 1 x t 1 z 2316 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, r 2 z t 1 y ,mxrxt 1 y , mztz,mxr 1 x t 1 z 2317 H578 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y , rxt 1 y , mztz, r 1 x t 1 z [ 9, 35, 74, 130, 202, 290, 394, 514, 650, 802 ] 2318 H555 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , mzr 1 x tz,mzr 1 x t 1 z 2319 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , mzr 1 x tz,mzr 1 x t 1 z [ 9, 35, 77, 131, 203, 291, 395, 515, 651, 803 ] 2320 H555 1 mzrx itx, it 1 x , r 2 z rxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , mzr 1 x tz,mzr 1 x t 1 z 2321 H560 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , mzr 1 x tz,mzr 1 x t 1 z 2322 H560 1 mzrx itx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , mzr 1 x tz,mzr 1 x t 1 z 2323 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, it 1 y , r 2 z rxt 1 y , mzr 1 x tz, r 2 z rxt 1 z 2324 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, it 1 y , r 2 z rxt 1 y , mzr 1 x tz, r 2 z rxt 1 z 2325 H557 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , mzr 1 x tz,mzr 1 x t 1 z [ 9, 35, 77, 132, 204, 292, 396, 516, 652, 804 ] 2326 H574 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z 2327 H580 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x , rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z [ 9, 35, 78, 133, 204, 292, 396, 516, 652, 804 ] 2328 H580 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x , rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z 2329 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , rxty, r 2 z t 1 y ,mxrxt 1 y , r 1 x tz,mxr 1 x t 1 z 2330 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , rxty, r 2 z t 1 y ,mxrxt 1 y , r 1 x tz,mxr 1 x t 1 z 134 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 2331 H578 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z [ 9, 35, 78, 134, 205, 292, 396, 516, 652, 804 ] 2332 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, rxty, r 2 z t 1 y , r 1 x tz, r 2 y t 1 z [ 9, 35, 79, 132, 204, 292, 396, 516, 652, 804 ] 2333 H574 1 mzrx itx, it 1 x , r 2 z rxt 1 x , rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z 2334 H580 1 mzrx itx, r 2 y rxt 1 x ,mxt 1 x , rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z [ 9, 35, 79, 134, 204, 292, 396, 516, 652, 804 ] 2335 H560 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mzr 1 x ty, r 2 x t 1 y , r 2 x tz, mzr 1 x tz, r 2 x t 1 z 2336 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty, it 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 z rxt 1 z 2337 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty, it 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 z rxt 1 z 2338 H557 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y , r 2 x tz, mzr 1 x tz, r 2 x t 1 z [ 9, 36, 76, 131, 203, 291, 395, 515, 651, 803 ] 2339 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 z rxt 1 y , r 2 x tz, mzr 1 x tz, r 2 z rxt 1 z [ 9, 36, 77, 134, 205, 292, 396, 516, 652, 804 ] 2340 H572 1 mzrx itx, it 1 x , r 2 z rxt 1 x , r 2 z ty, r 2 z t 1 y ,mxrxt 1 y , r 2 y tz,mxr 1 x t 1 z 2341 H572 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , r 2 z ty, r 2 z t 1 y ,mxrxt 1 y , r 2 y tz,mxr 1 x t 1 z 2342 H574 1 mzrx itx, it 1 x , r 2 z rxt 1 x ,myty,myt 1 y , rxt 1 y , mztz, r 1 x t 1 z 2343 H574 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x ,myty,myt 1 y , rxt 1 y , mztz, r 1 x t 1 z 2344 H580 1 mzrx itx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y , rxt 1 y , mztz, r 1 x t 1 z 2345 H580 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y , rxt 1 y , mztz, r 1 x t 1 z 2346 H576 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z ty, r 2 z t 1 y ,mxrxt 1 y , r 2 y tz,mxr 1 x t 1 z 2347 H576 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z ty, r 2 z t 1 y ,mxrxt 1 y , r 2 y tz,mxr 1 x t 1 z 2348 H576 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z ty,mxrxty, r 2 z t 1 y , mxr 1 x tz, r 2 y t 1 z 2349 H576 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z ty,mxrxt 1 y , r 2 y tz, mxr 1 x t 1 z , r 2 y t 1 z 2350 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, rxty, r 2 z t 1 y , r 1 x tz, r 2 y t 1 z [ 9, 36, 78, 132, 204, 292, 396, 516, 652, 804 ] 2351 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, rxty,mxrxt 1 y , r 1 x tz,mxr 1 x t 1 z [ 9, 36, 78, 134, 204, 292, 396, 516, 652, 804 ] 2352 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, it 1 y , r 2 x tz, mzr 1 x tz, it 1 z [ 9, 37, 75, 131, 203, 291, 395, 515, 651, 803 ] 2353 H551 1 mzrx itx, it 1 x , r 2 z rxt 1 x , r 2 z rxty, it 1 y , r 2 z rxt 1 y , r 2 z rxtz, r 2 z rxt 1 z 2354 H551 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , r 2 z rxty, it 1 y , r 2 z rxt 1 y , r 2 z rxtz, r 2 z rxt 1 z 2355 H551 1 mzrx itx, it 1 x , r 2 z rxt 1 x , r 2 z rxty, r 2 z rxt 1 y , itz, r 2 z rxtz, r 2 z rxt 1 z 2356 H556 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z rxty, it 1 y , r 2 z rxt 1 y , r 2 z rxtz, r 2 z rxt 1 z 2357 H556 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z rxty, it 1 y , r 2 z rxt 1 y , r 2 z rxtz, r 2 z rxt 1 z 2358 H556 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z rxty, r 2 z rxt 1 y , itz, r 2 z rxtz, r 2 z rxt 1 z 2359 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 z rxt 1 y , r 2 x tz, mzr 1 x tz, r 2 z rxt 1 z [ 9, 37, 77, 132, 204, 292, 396, 516, 652, 804 ] 2360 H572 1 mzrx itx, it 1 x , r 2 z rxt 1 x ,mxrxty, r 2 z t 1 y ,mxrxt 1 y , mxr 1 x tz,mxr 1 x t 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 135 Nbr. gr Hi L m X 2361 H572 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x ,mxrxty, r 2 z t 1 y ,mxrxt 1 y , mxr 1 x tz,mxr 1 x t 1 z 2362 H576 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,mxrxty, r 2 z t 1 y ,mxrxt 1 y , mxr 1 x tz,mxr 1 x t 1 z 2363 H576 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,mxrxty, r 2 z t 1 y ,mxrxt 1 y , mxr 1 x tz,mxr 1 x t 1 z 2364 H576 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z ty,mxrxty,mxrxt 1 y , mxr 1 x tz,mxr 1 x t 1 z 2365 H576 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,mxrxty,mxrxt 1 y ,mxr 1 x tz, mxr 1 x t 1 z , r 2 y t 1 z 2366 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, rxty,mxrxt 1 y , r 1 x tz,mxr 1 x t 1 z [ 9, 37, 77, 134, 204, 292, 396, 516, 652, 804 ] 2367 H551 1 mzrx itx, it 1 x , r 2 z rxt 1 x , r 2 z rxty, it 1 y , itz, r 2 z rxtz, it 1 z 2368 H551 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , r 2 z rxty, it 1 y , itz, r 2 z rxtz, it 1 z 2369 H551 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , ity, it 1 y , r 2 z rxt 1 y , itz, r 2 z rxt 1 z 2370 H555 1 mzrx itx, it 1 x , r 2 z rxt 1 x ,mzr 1 x ty, r 2 x t 1 y , r 2 x tz, mzr 1 x tz, r 2 x t 1 z 2371 H555 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x ,mzr 1 x ty, r 2 x t 1 y , r 2 x tz, mzr 1 x tz, r 2 x t 1 z 2372 H555 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz,mzr 1 x t 1 z 2373 H560 1 mzrx itx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y , r 2 x tz, mzr 1 x tz, r 2 x t 1 z 2374 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y , r 2 x tz, mzr 1 x tz, r 2 x t 1 z 2375 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz,mzr 1 x t 1 z 2376 H556 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z rxty, it 1 y , itz, r 2 z rxtz, it 1 z 2377 H556 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z rxty, it 1 y , itz, r 2 z rxtz, it 1 z 2378 H556 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , ity, it 1 y , r 2 z rxt 1 y , itz, r 2 z rxt 1 z 2379 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, it 1 y , r 2 x tz, mzr 1 x tz, it 1 z 44 [ 8, 29, 69, 128, 203, 292, 396, 516, 652, 804 ] 2380*, [ 8, 30, 69, 123, 193, 279, 381, 499, 633, 783 ] 2381*, [ 8, 30, 71, 128, 200, 288, 392, 512, 648, 800 ] 2382*, [ 8, 30, 71, 130, 204, 292, 396, 516, 652, 804 ] 2383*, [ 8, 30, 73, 134, 207, 293, 396, 516, 652, 804 ] 2384*, 2385*, 2386*, [ 8, 31, 70, 123, 193, 279, 381, 499, 633, 783 ] 2387*, 2388*, [ 8, 31, 73, 129, 200, 288, 392, 512, 648, 800 ] 2389*, [ 8, 31, 73, 130, 202, 290, 394, 514, 650, 802 ] 2390*, 2391*, 2392*, 2393*, [ 8, 31, 74, 132, 204, 292, 396, 516, 652, 804 ] 2394*, 2395*, 2396*, [ 8, 31, 75, 133, 204, 292, 396, 516, 652, 804 ] 2397*, [ 8, 31, 75, 135, 206, 292, 396, 516, 652, 804 ] 2398*, 2399*, 2400*, [ 8, 31, 75, 135, 207, 293, 396, 516, 652, 804 ] 2401*, [ 8, 31, 76, 136, 207, 293, 396, 516, 652, 804 ] 2402*, 2403*, [ 8, 31, 76, 137, 208, 293, 396, 516, 652, 804 ] 2404*, [ 8, 32, 74, 130, 202, 290, 394, 514, 650, 802 ] 2405*, [ 8, 32, 76, 133, 204, 292, 396, 516, 652, 804 ] 2406*, [ 8, 32, 76, 135, 206, 292, 396, 516, 652, 804 ] 2407*, [ 8, 32, 76, 135, 207, 293, 396, 516, 652, 804 ] 2408*, 2409*, 2410*, 2411*, 2412*, 2413*, 2414*, [ 8, 32, 77, 136, 206, 292, 396, 516, 652, 804 ] 2415*, 2416*, [ 8, 33, 75, 130, 202, 290, 394, 514, 650, 802 ] 136 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 2417*, 2418*, 2419*, 2420*, 2421*, 2422*, 2423*, [ 8, 33, 77, 133, 204, 292, 396, 516, 652, 804 ] 2424*, 2425*, 2426*, 2427*, 2428*, 2429*, 2430*, 2431*, [ 8, 33, 77, 135, 206, 292, 396, 516, 652, 804 ] 2432*, 2433*, 2434*, 2435*, 2436*, 2437*, 2438*, [ 8, 33, 78, 135, 206, 293, 396, 516, 652, 804 ] 2439*, [ 9, 32, 73, 131, 204, 292, 396, 516, 652, 804 ] 2380 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,myty, r 2 z t 1 y , r 1 x t 1 y , rxtz,mzt 1 z [ 9, 33, 72, 127, 198, 285, 388, 507, 642, 793 ] 2381 H388 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mzrxty, it 1 y ,mzrxt 1 y , mzrxtz,mzrxt 1 z [ 9, 33, 74, 130, 202, 290, 394, 514, 650, 802 ] 2382 H407 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, rxt 1 z [ 9, 33, 74, 132, 204, 292, 396, 516, 652, 804 ] 2383 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 x ty, it 1 y ,mzrxt 1 y , mzrxtz, r 2 x t 1 z [ 9, 33, 76, 134, 205, 292, 396, 516, 652, 804 ] 2384 H402 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,myty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mzt 1 z 2385 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,myty, r 2 z t 1 y , r 1 x t 1 y , rxtz,mzt 1 z 2386 H419 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,myty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mzt 1 z [ 9, 34, 72, 127, 198, 285, 388, 507, 642, 793 ] 2387 H389 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x ,mzrxty, it 1 y ,mzrxt 1 y , mzrxtz,mzrxt 1 z 2388 H393 1 r 2 z rx r 2 y rztx, ryrxtx, r 1 y r 1 z t 1 x , r 2 y rzty, ryrxty, r 1 y r 1 z t 1 y , ryrxtz, r 1 y r 1 z t 1 z [ 9, 34, 75, 130, 202, 290, 394, 514, 650, 802 ] 2389 H414 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , r 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, rxt 1 z [ 9, 34, 75, 131, 203, 291, 395, 515, 651, 803 ] 2390 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 y rxty,mzrxt 1 y ,mzrxtz, r 2 y rxt 1 z , r 2 x t 1 z 2391 H398 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x ,mzrxty, it 1 y ,mzrxt 1 y , mzrxtz,mzrxt 1 z 2392 H400 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mzrxty, it 1 y ,mzrxt 1 y , mzrxtz,mzrxt 1 z 2393 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 y rxty,mzrxt 1 y ,mzrxtz, r 2 y rxt 1 z , r 2 x t 1 z [ 9, 34, 76, 132, 204, 292, 396, 516, 652, 804 ] 2394 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mxr 1 x ty,myty, r 1 x t 1 y , rxtz,mxrxt 1 z 2395 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mxr 1 x ty,myty, r 1 x t 1 y , rxtz,mxrxt 1 z 2396 H418 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, rxt 1 z [ 9, 34, 77, 132, 204, 292, 396, 516, 652, 804 ] 2397 H410 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , r 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, rxt 1 z [ 9, 34, 77, 134, 204, 292, 396, 516, 652, 804 ] 2398 H391 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z 2399 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 x ty, it 1 y ,mzrxt 1 y , mzrxtz, r 2 x t 1 z 2400 H401 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z [ 9, 34, 77, 134, 205, 292, 396, 516, 652, 804 ] 2401 H407 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 z ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z [ 9, 34, 78, 134, 205, 292, 396, 516, 652, 804 ] 2402 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,myty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mzt 1 z 2403 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,myty,mxr 1 x t 1 y ,mxrxtz, mztz,mzt 1 z [ 9, 34, 78, 135, 205, 292, 396, 516, 652, 804 ] 2404 H393 1 r 2 z rx r 2 y rztx, r 2 x ryt 1 x , r 1 y r 1 z t 1 x , r 2 y rzty, ryrxty, r 2 x ryt 1 y , r 2 y rztz, r 2 x ryt 1 z [ 9, 35, 75, 131, 203, 291, 395, 515, 651, 803 ] 2405 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 y rxty, it 1 y ,mzrxt 1 y , mzrxtz, r 2 y rxt 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 137 Nbr. gr Hi L m X [ 9, 35, 77, 132, 204, 292, 396, 516, 652, 804 ] 2406 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mxr 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz,mxrxt 1 z [ 9, 35, 77, 134, 204, 292, 396, 516, 652, 804 ] 2407 H388 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mzrxty, it 1 y , itz, it 1 z ,mzrxt 1 z [ 9, 35, 77, 134, 205, 292, 396, 516, 652, 804 ] 2408 H414 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , r 2 z ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z 2409 H415 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x ,myty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mzt 1 z 2410 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , r 2 y tz,mxrxt 1 z 2411 H410 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , r 2 z ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z 2412 H404 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x ,myty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mzt 1 z 2413 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , r 2 y tz,mxrxt 1 z 2414 H418 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 z ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z [ 9, 35, 78, 134, 204, 292, 396, 516, 652, 804 ] 2415 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z 2416 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 y rxtz, r 2 x tz, r 2 x t 1 z [ 9, 36, 75, 131, 203, 291, 395, 515, 651, 803 ] 2417 H396 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 y rxt 1 z 2418 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 y rxt 1 z 2419 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 y rxtz, r 2 y rxt 1 z , r 2 x t 1 z 2420 H391 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 y rxt 1 z 2421 H392 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 y rxt 1 z 2422 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 y rxty, it 1 y ,mzrxt 1 y , mzrxtz, r 2 y rxt 1 z 2423 H401 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 y rxt 1 z [ 9, 36, 77, 132, 204, 292, 396, 516, 652, 804 ] 2424 H415 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x ,mxr 1 x ty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mxrxt 1 z 2425 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,mxr 1 x ty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mxrxt 1 z 2426 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,mxr 1 x ty,mxr 1 x t 1 y ,mxrxtz, mxrxt 1 z ,mzt 1 z 2427 H402 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mxr 1 x ty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mxrxt 1 z 2428 H404 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x ,mxr 1 x ty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mxrxt 1 z 2429 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mxr 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz,mxrxt 1 z 2430 H419 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mxr 1 x ty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mxrxt 1 z 2431 H393 1 r 2 z rx r 2 y rztx, r 2 x ryt 1 x , r 1 y r 1 z t 1 x , r 2 y rzty, ryrxty, r 1 y r 1 z t 1 y , ryrxtz, r 2 x ryt 1 z [ 9, 36, 77, 134, 204, 292, 396, 516, 652, 804 ] 2432 H389 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x ,mzrxty, it 1 y , itz, it 1 z ,mzrxt 1 z 2433 H396 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z 2434 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 y rxty, it 1 y , itz, r 2 y rxt 1 z , r 2 x t 1 z 2435 H398 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x ,mzrxty, it 1 y , itz, it 1 z ,mzrxt 1 z 2436 H392 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z 138 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 2437 H400 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mzrxty, it 1 y , itz, it 1 z ,mzrxt 1 z 2438 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 y rxty, it 1 y , itz, r 2 y rxt 1 z , r 2 x t 1 z [ 9, 36, 78, 133, 205, 292, 396, 516, 652, 804 ] 2439 H393 1 r 2 z rx r 2 y rztx, ryrxtx, r 1 y r 1 z t 1 x , r 2 y rzty, ryrxty, r 2 x ryt 1 y , r 2 y rztz, r 1 y r 1 z t 1 z 45 [ 9, 33, 76, 135, 207, 293, 396, 516, 652, 804 ] 2440*, 2441*, 2442*, 2443*, [ 9, 34, 75, 130, 202, 290, 394, 514, 650, 802 ] 2444*, 2445*, 2446*, 2447*, [ 9, 34, 77, 133, 204, 292, 396, 516, 652, 804 ] 2448*, 2449*, 2450*, 2451*, [ 9, 34, 77, 135, 206, 292, 396, 516, 652, 804 ] 2452*, 2453*, 2454*, 2455*, [ 10, 35, 77, 134, 205, 292, 396, 516, 652, 804 ] 2440 H652 hmxi r 2 y rx itx, r 2 z rxtx, r 2 x t 1 x , r 2 z rxt 1 x ,myty,myt 1 y , mxrxt 1 y , r 2 y tz,mxr 1 x t 1 z 2441 H422 1 r 2 z rx r 2 z tx, r 1 x tx, r 2 y t 1 x , rxt 1 x ,myty,mxr 1 x t 1 y , myt 1 y ,mxrxtz,mzt 1 z 2442 H650 hmxi r 2 y rx r 2 y rxtx,mxtx,mzrxt 1 x ,mxt 1 x ,myty,myt 1 y , mxrxt 1 y , r 2 y tz,mxr 1 x t 1 z 2443 H580 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty,myt 1 y , rxt 1 y ,mztz, r 1 x t 1 z [ 10, 36, 75, 131, 203, 291, 395, 515, 651, 803 ] 2444 H649 hmxi r 2 y rx itx, r 2 z rxtx, r 2 x t 1 x , r 2 z rxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 z rxt 1 y , r 2 z rxtz, r 2 z rxt 1 z 2445 H409 1 r 2 z rx r 2 z tx, r 1 x tx, r 2 y t 1 x , rxt 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 y rxt 1 z 2446 H648 hmxi r 2 y rx r 2 y rxtx,mxtx,mzrxt 1 x ,mxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 z rxt 1 y , r 2 z rxtz, r 2 z rxt 1 z 2447 H560 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y , mzr 1 x t 1 y ,mzr 1 x tz,mzr 1 x t 1 z [ 10, 36, 77, 132, 204, 292, 396, 516, 652, 804 ] 2448 H652 hmxi r 2 y rx itx, r 2 z rxtx, r 2 x t 1 x , r 2 z rxt 1 x ,mxrxty,myt 1 y , mxrxt 1 y ,mxr 1 x tz,mxr 1 x t 1 z 2449 H422 1 r 2 z rx r 2 z tx, r 1 x tx, r 2 y t 1 x , rxt 1 x ,mxr 1 x ty,mxr 1 x t 1 y , myt 1 y ,mxrxtz,mxrxt 1 z 2450 H650 hmxi r 2 y rx r 2 y rxtx,mxtx,mzrxt 1 x ,mxt 1 x ,mxrxty,myt 1 y , mxrxt 1 y ,mxr 1 x tz,mxr 1 x t 1 z 2451 H580 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x , rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z [ 10, 36, 77, 134, 204, 292, 396, 516, 652, 804 ] 2452 H649 hmxi r 2 y rx itx, r 2 z rxtx, r 2 x t 1 x , r 2 z rxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 x tz, r 2 z rxtz, r 2 x t 1 z 2453 H409 1 r 2 z rx r 2 z tx, r 1 x tx, r 2 y t 1 x , rxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x t 1 z 2454 H648 hmxi r 2 y rx r 2 y rxtx,mxtx,mzrxt 1 x ,mxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 x tz, r 2 z rxtz, r 2 x t 1 z 2455 H560 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y , r 2 x tz,mzr 1 x tz, r 2 x t 1 z 46 [ 9, 33, 75, 132, 204, 292, 396, 516, 652, 804 ] 2456*, 2457*, [ 9, 34, 76, 132, 204, 292, 396, 516, 652, 804 ] 2458*, 2459*, 2460*, [ 9, 34, 77, 133, 204, 292, 396, 516, 652, 804 ] 2461*, 2462*, 2463*, 2464*, 2465*, 2466*, 2467*, 2468*, [ 9, 34, 78, 134, 204, 292, 396, 516, 652, 804 ] 2469*, [ 9, 35, 77, 132, 204, 292, 396, 516, 652, 804 ] 2470*, 2471*, 2472*, 2473*, 2474*, 2475*, 2476*, 2477*, 2478*, 2479*, 2480*, 2481*, 2482*, 2483*, [ 9, 35, 78, 133, 204, 292, 396, 516, 652, 804 ] 2484*, 2485*, 2486*, 2487*, 2488*, 2489*, 2490*, 2491*, 2492*, 2493*, 2494*, 2495*, 2496*, 2497*, 2498*, [ 9, 36, 78, 132, 204, 292, 396, 516, 652, 804 ] 2499*, 2500*, 2501*, 2502*, 2503*, 2504*, 2505*, 2506*, 2507*, 2508*, 2509*, 2510*, 2511*, 2512*, 2513*, 2514*, 2515*, 2516*, 2517*, 2518*, 2519*, 2520*, [ 10, 35, 76, 132, 204, 292, 396, 516, 652, 804 ] 2456 H359 1 mz itx, r 2 z tx,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , tz,mzt 1 z , t 1 z 2457 H650 hmzrxi mx tx,mxt 1 x , t 1 x ,myty, r 2 z t 1 y ,myt 1 y , r 1 x tz,mxr 1 x t 1 z , r 1 x t 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 139 Nbr. gr Hi L m X [ 10, 36, 76, 132, 204, 292, 396, 516, 652, 804 ] 2458 H648 hmzrxi mx tx,mxt 1 x , t 1 x , r 2 x ty, it 1 y , r 2 x t 1 y , mzr 1 x tz, r 2 z rxt 1 z ,mzr 1 x t 1 z 2459 H350 1 mz r 2 z tx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,myt 1 y , tz,mzt 1 z , t 1 z 2460 H359 1 mz r 2 z tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , tz,mzt 1 z , t 1 z [ 10, 36, 77, 132, 204, 292, 396, 516, 652, 804 ] 2461 H370 1 mz itx, r 2 z tx,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , mytz, r 2 x t 1 z ,myt 1 z 2462 H370 1 mz itx, r 2 z tx,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , mytz, r 2 x t 1 z ,myt 1 z 2463 H359 1 mz itx, r 2 z tx, r 2 y t 1 x ,myty, r 2 x t 1 y ,myt 1 y , tz,mzt 1 z , t 1 z 2464 H359 1 mz itx, r 2 z tx,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , tz,mzt 1 z , t 1 z 2465 H368 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , mxtz, r 2 y t 1 z ,mxt 1 z 2466 H368 1 mz mxtx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , mxtz, r 2 y t 1 z ,mxt 1 z 2467 H353 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , tz,mzt 1 z , t 1 z 2468 H377 1 mz r 2 y tx,mzr 1 z t 1 x , r 1 z t 1 x ,mzrzty, rzty,myt 1 y , tz,mzt 1 z , t 1 z [ 10, 36, 78, 132, 204, 292, 396, 516, 652, 804 ] 2469 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,myty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mztz,mzt 1 z [ 10, 37, 76, 132, 204, 292, 396, 516, 652, 804 ] 2470 H361 1 mz itx, it 1 x , r 2 z t 1 x , r 2 z ty, it 1 y , r 2 z t 1 y , mxtz, r 2 y t 1 z ,mxt 1 z 2471 H361 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 z ty, it 1 y , r 2 z t 1 y , mxtz, r 2 y t 1 z ,mxt 1 z 2472 H330 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , ity, it 1 y , r 2 z t 1 y , tz,mzt 1 z , t 1 z 2473 H372 1 mz itx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,myt 1 y , mxtz, r 2 y t 1 z ,mxt 1 z 2474 H372 1 mz r 2 z tx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,myt 1 y , mxtz, r 2 y t 1 z ,mxt 1 z 2475 H365 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , mytz, r 2 x t 1 z ,myt 1 z 2476 H365 1 mz r 2 z tx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,myt 1 y , mytz, r 2 x t 1 z ,myt 1 z 2477 H350 1 mz itx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,myt 1 y , tz,mzt 1 z , t 1 z 2478 H350 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , tz,mzt 1 z , t 1 z 2479 H370 1 mz r 2 z tx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , mytz, r 2 x t 1 z ,myt 1 z 2480 H370 1 mz r 2 z tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , mytz, r 2 x t 1 z ,myt 1 z 2481 H359 1 mz itx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , tz,mzt 1 z , t 1 z 2482 H359 1 mz r 2 z tx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , tz,mzt 1 z , t 1 z 2483 H343 1 mz r 2 z tx, r 2 y rzt 1 x ,mxrzt 1 x , ity, r 2 y rzt 1 y ,mxrzt 1 y , tz,mzt 1 z , t 1 z [ 10, 37, 77, 132, 204, 292, 396, 516, 652, 804 ] 2484 H370 1 mz itx, r 2 z tx, r 2 y t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , mytz, r 2 x t 1 z ,myt 1 z 2485 H370 1 mz itx, r 2 z tx, r 2 y t 1 x ,myty, r 2 x t 1 y ,myt 1 y , mytz, r 2 x t 1 z ,myt 1 z 2486 H359 1 mz itx, r 2 z tx, r 2 y t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , tz,mzt 1 z , t 1 z 2487 H652 hmzrxi mx mzr 1 x tx, r 2 z rxt 1 x ,mzr 1 x t 1 x , r 2 z ty, r 2 z t 1 y ,myt 1 y , mxr 1 x tz,mxr 1 x t 1 z , r 1 x t 1 z 2488 H373 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 z tz, it 1 z , r 2 z t 1 z 2489 H652 hmzrxi mx mzr 1 x tx, r 2 z rxt 1 x ,mzr 1 x t 1 x ,myty, r 2 z t 1 y ,myt 1 y , r 1 x tz,mxr 1 x t 1 z , r 1 x t 1 z 140 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 2490 H368 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , mxtz, r 2 y t 1 z ,mxt 1 z 2491 H368 1 mz mxtx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , mxtz, r 2 y t 1 z ,mxt 1 z 2492 H650 hmzrxi mx tx,mxt 1 x , t 1 x , r 2 z ty, r 2 z t 1 y ,myt 1 y , mxr 1 x tz,mxr 1 x t 1 z , r 1 x t 1 z 2493 H378 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2494 H378 1 mz r 2 y tx, r 2 y t 1 x ,mxt 1 x , r 2 x ty,myty,myt 1 y , mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2495 H650 hmzrxi mx tx,mxt 1 x , t 1 x , r 2 z ty,myty, r 2 z t 1 y , mxr 1 x tz, r 1 x tz,mxr 1 x t 1 z 2496 H382 1 mz r 2 y tx,mzr 1 z t 1 x , r 1 z t 1 x ,mzrzty, rzty,myt 1 y , mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2497 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,mxr 1 x ty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mztz,mxrxt 1 z 2498 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 x t 1 z [ 10, 38, 76, 132, 204, 292, 396, 516, 652, 804 ] 2499 H649 hmzrxi mx mzr 1 x tx, r 2 z rxt 1 x ,mzr 1 x t 1 x , ity, it 1 y , r 2 x t 1 y , r 2 z rxtz, r 2 z rxt 1 z ,mzr 1 x t 1 z 2500 H332 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , ity, it 1 y , r 2 z t 1 y , r 2 z tz, it 1 z , r 2 z t 1 z 2501 H649 hmzrxi mx mzr 1 x tx, r 2 z rxt 1 x ,mzr 1 x t 1 x , r 2 x ty, it 1 y , r 2 x t 1 y , mzr 1 x tz, r 2 z rxt 1 z ,mzr 1 x t 1 z 2502 H361 1 mz itx, it 1 x , r 2 z t 1 x , ity, it 1 y , r 2 z t 1 y , mxtz, r 2 y t 1 z ,mxt 1 z 2503 H361 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , ity, it 1 y , r 2 z t 1 y , mxtz, r 2 y t 1 z ,mxt 1 z 2504 H648 hmzrxi mx tx,mxt 1 x , t 1 x , ity, it 1 y , r 2 x t 1 y , r 2 z rxtz, r 2 z rxt 1 z ,mzr 1 x t 1 z 2505 H344 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , ity, it 1 y , r 2 z t 1 y , mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2506 H344 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , ity, r 2 z ty, it 1 y , mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2507 H364 1 mz itx, it 1 x , r 2 z t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 z tz, it 1 z , r 2 z t 1 z 2508 H364 1 mz itx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 z tz, it 1 z , r 2 z t 1 z 2509 H364 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , r 2 z tz, it 1 z , r 2 z t 1 z 2510 H364 1 mz r 2 z tx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,myt 1 y , r 2 z tz, it 1 z , r 2 z t 1 z 2511 H372 1 mz itx, it 1 x , r 2 z t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , mxtz, r 2 y t 1 z ,mxt 1 z 2512 H372 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , mxtz, r 2 y t 1 z ,mxt 1 z 2513 H365 1 mz itx, it 1 x , r 2 z t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , mytz, r 2 x t 1 z ,myt 1 z 2514 H365 1 mz itx, it 1 x , r 2 z t 1 x ,myty, r 2 x t 1 y ,myt 1 y , mytz, r 2 x t 1 z ,myt 1 z 2515 H350 1 mz itx, it 1 x , r 2 z t 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , tz,mzt 1 z , t 1 z 2516 H370 1 mz itx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , mytz, r 2 x t 1 z ,myt 1 z 2517 H370 1 mz itx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y ,myt 1 y , mytz, r 2 x t 1 z ,myt 1 z 2518 H359 1 mz itx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y ,myt 1 y , tz,mzt 1 z , t 1 z 2519 H347 1 mz r 2 z tx, r 2 y rzt 1 x ,mxrzt 1 x , ity, r 2 y rzt 1 y ,mxrzt 1 y , mxr 1 z tz, r 2 x rzt 1 z ,mxr 1 z t 1 z 2520 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 y rxt 1 z 47A [ 7, 23, 50, 87, 135, 194, 263, 343, 434, 535 ] 2521*, [ 7, 24, 55, 99, 156, 227, 311, 408, 519, 643 ] 2522*, 2523*, [ 7, 25, 56, 98, 152, 218, 296, 386, 488, 602 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 141 Nbr. gr Hi L m X 2524*, 2525*, [ 7, 25, 57, 99, 152, 218, 296, 386, 488, 602 ] 2526*, 2527*, [ 7, 26, 57, 98, 152, 218, 296, 386, 488, 602 ] 2528*, 2529*, [ 7, 26, 59, 105, 166, 240, 328, 430, 545, 675 ] 2530*, 2531*, [ 7, 26, 60, 106, 166, 240, 328, 430, 545, 675 ] 2532*, 2533*, [ 7, 27, 60, 105, 166, 240, 328, 430, 545, 675 ] 2534*, [ 7, 28, 63, 109, 172, 249, 338, 444, 564, 695 ] 2535*, 2536*, [ 8, 27, 58, 101, 158, 226, 306, 401, 506, 623 ] 2521 H697 hmxi r 2 z mxr 1 z tx, rzt 1 x , rzt 1 y ,mxr 1 z t 1 y , r 2 y rztz,mzr 1 z tz, r 2 y t 1 z [ 8, 28, 62, 108, 168, 242, 328, 428, 542, 668 ] 2522 H693 hmxi r 2 z r 2 x rztx,mzrzt 1 x ,mzrzt 1 y , r 2 x rzt 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z 2523 H695 hmxi r 2 z mxr 1 z tx, rzt 1 x , rzt 1 y ,mxr 1 z t 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z [ 8, 29, 62, 109, 169, 242, 329, 429, 542, 669 ] 2524 H458 1 r 2 z mxrzt 1 x ,mxr 1 z t 1 x ,mxr 1 z ty,mxrzt 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 2525 H534 1 r 2 z mxrzt 1 x ,mxr 1 z t 1 x ,mxr 1 z ty,mxrzt 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z [ 8, 29, 63, 109, 169, 242, 329, 429, 542, 669 ] 2526 H510 1 r 2 z r 1 z t 1 x , rzt 1 x , rzty, r 1 z t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 2527 H538 1 r 2 z r 1 z t 1 x , rzt 1 x , rzty, r 1 z t 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z [ 8, 30, 62, 109, 169, 242, 329, 429, 542, 669 ] 2528 H440 1 r 2 z r 2 y rzt 1 x , r 2 x rzt 1 x , r 2 x rzty, r 2 y rzt 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 2529 H501 1 r 2 z mzr 1 z t 1 x ,mzrzt 1 x ,mzrzty,mzr 1 z t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z [ 8, 30, 64, 113, 176, 251, 342, 446, 563, 696 ] 2530 H459 1 r 2 z mxrzt 1 x ,mxr 1 z t 1 x ,mxr 1 z ty,mxrzt 1 y , r 2 y rztz, it 1 z , mzt 1 z 2531 H533 1 r 2 z mxrzt 1 x ,mxr 1 z t 1 x ,mxr 1 z ty,mxrzt 1 y , r 2 y rztz, r 2 y t 1 z , r 2 x t 1 z [ 8, 30, 65, 113, 176, 251, 342, 446, 563, 696 ] 2532 H536 1 r 2 z r 1 z t 1 x , rzt 1 x , rzty, r 1 z t 1 y , r 2 y rztz, it 1 z , mzt 1 z 2533 H512 1 r 2 z r 1 z t 1 x , rzt 1 x , rzty, r 1 z t 1 y , r 2 y rztz, r 2 y t 1 z , r 2 x t 1 z [ 8, 31, 64, 113, 176, 251, 342, 446, 563, 696 ] 2534 H697 hmxi r 2 z mxr 1 z tx, rzt 1 x , rzt 1 y ,mxr 1 z t 1 y , r 2 y rztz,mzr 1 z tz, r 2 x t 1 z [ 8, 32, 66, 115, 181, 258, 349, 459, 578, 711 ] 2535 H693 hmxi r 2 z r 2 x rztx,mzrzt 1 x ,mzrzt 1 y , r 2 x rzt 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z 2536 H695 hmxi r 2 z mxr 1 z tx, rzt 1 x , rzt 1 y ,mxr 1 z t 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z 47B [ 7, 24, 60, 119, 195, 284, 388, 508, 644, 796 ] 2537*, [ 7, 25, 60, 113, 184, 272, 376, 496, 632, 784 ] 2538*, [ 7, 25, 63, 122, 196, 284, 388, 508, 644, 796 ] 2539*, [ 7, 25, 63, 124, 196, 284, 388, 508, 644, 796 ] 2540*, [ 7, 26, 64, 120, 192, 280, 384, 504, 640, 792 ] 2541*, [ 7, 26, 66, 124, 196, 284, 388, 508, 644, 796 ] 2542*, [ 7, 26, 68, 128, 200, 288, 392, 512, 648, 800 ] 2543*, [ 7, 26, 69, 129, 200, 288, 392, 512, 648, 800 ] 2544*, [ 7, 27, 66, 122, 194, 282, 386, 506, 642, 794 ] 2545*, [ 7, 27, 67, 123, 194, 282, 386, 506, 642, 794 ] 2546*, [ 7, 27, 69, 128, 200, 288, 392, 512, 648, 800 ] 2547*, 2548*, 2549*, [ 7, 27, 70, 129, 200, 288, 392, 512, 648, 800 ] 2550*, [ 7, 28, 67, 122, 194, 282, 386, 506, 642, 794 ] 2551*, 2552*, 2553*, 142 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X [ 7, 28, 68, 124, 196, 284, 388, 508, 644, 796 ] 2554*, [ 7, 28, 69, 125, 196, 284, 388, 508, 644, 796 ] 2555*, [ 7, 28, 70, 128, 200, 288, 392, 512, 648, 800 ] 2556*, [ 7, 29, 69, 124, 196, 284, 388, 508, 644, 796 ] 2557*, [ 7, 29, 71, 128, 200, 288, 392, 512, 648, 800 ] 2558*, 2559*, [ 7, 30, 71, 126, 198, 286, 390, 510, 646, 798 ] 2560*, 2561*, [ 8, 28, 68, 127, 200, 288, 392, 512, 648, 800 ] 2537 H581 1 mzrx itx, r 2 z rxtx,mxt 1 x , r 1 x t 1 y ,mytz, rxtz, myt 1 z [ 8, 29, 67, 122, 194, 282, 386, 506, 642, 794 ] 2538 H552 1 mzrx itx, r 2 z rxtx,mxt 1 x , ty,mzrxt 1 y , t 1 y , mzrxtz [ 8, 29, 70, 128, 200, 288, 392, 512, 648, 800 ] 2539 H579 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 1 x t 1 y ,mytz, rxtz, myt 1 z [ 8, 29, 70, 130, 198, 290, 390, 514, 646, 802 ] 2540 H577 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,mxr 1 x t 1 y , r 2 z tz,mxrxtz, r 2 z t 1 z [ 8, 30, 70, 126, 198, 286, 390, 510, 646, 798 ] 2541 H546 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , ty,mzrxt 1 y , t 1 y , mzrxtz [ 8, 30, 72, 128, 200, 288, 392, 512, 648, 800 ] 2542 H548 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,mxty, r 2 y rxt 1 y ,mxt 1 y , r 2 y rxtz [ 8, 30, 74, 130, 202, 290, 394, 514, 650, 802 ] 2543 H575 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , r 1 x t 1 y ,mytz, rxtz, myt 1 z [ 8, 30, 75, 130, 202, 290, 394, 514, 650, 802 ] 2544 H575 1 mzrx itx, it 1 x , r 2 z rxt 1 x , r 1 x t 1 y ,mytz, rxtz, myt 1 z [ 8, 31, 71, 127, 199, 287, 391, 511, 647, 799 ] 2545 H549 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , ty,mzrxt 1 y , t 1 y , mzrxtz [ 8, 31, 72, 127, 199, 287, 391, 511, 647, 799 ] 2546 H549 1 mzrx itx, it 1 x , r 2 z rxt 1 x , ty,mzrxt 1 y , t 1 y , mzrxtz [ 8, 31, 74, 130, 202, 290, 394, 514, 650, 802 ] 2547 H573 1 mzrx itx, it 1 x , r 2 z rxt 1 x ,mxr 1 x t 1 y , r 2 z tz,mxrxtz, r 2 z t 1 z 2548 H573 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x ,mxr 1 x t 1 y , r 2 z tz,mxrxtz, r 2 z t 1 z 2549 H581 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x , r 1 x t 1 y ,mytz, rxtz, myt 1 z [ 8, 31, 75, 130, 202, 290, 394, 514, 650, 802 ] 2550 H581 1 mzrx itx, r 2 y rxt 1 x ,mxt 1 x , r 1 x t 1 y ,mytz, rxtz, myt 1 z [ 8, 32, 71, 127, 199, 287, 391, 511, 647, 799 ] 2551 H554 1 mzrx itx, it 1 x , r 2 z rxt 1 x ,mxty, r 2 y rxt 1 y ,mxt 1 y , r 2 y rxtz 2552 H554 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x ,mxty, r 2 y rxt 1 y ,mxt 1 y , r 2 y rxtz 2553 H554 1 mzrx itx, r 2 z rxtx, r 2 z rxt 1 x , r 2 y rxt 1 y , r 2 y rxtz,mxtz, mxt 1 z [ 8, 32, 72, 128, 200, 288, 392, 512, 648, 800 ] 2554 H552 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x , ty,mzrxt 1 y , t 1 y , mzrxtz [ 8, 32, 73, 128, 200, 288, 392, 512, 648, 800 ] 2555 H552 1 mzrx itx, r 2 y rxt 1 x ,mxt 1 x , ty,mzrxt 1 y , t 1 y , mzrxtz [ 8, 32, 74, 130, 202, 290, 394, 514, 650, 802 ] 2556 H581 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x , r 1 x t 1 y ,mytz, rxtz, myt 1 z [ 8, 33, 72, 128, 200, 288, 392, 512, 648, 800 ] 2557 H552 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x , ty,mzrxt 1 y , t 1 y , mzrxtz [ 8, 33, 74, 130, 202, 290, 394, 514, 650, 802 ] 2558 H577 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,mxr 1 x t 1 y , r 2 z tz,mxrxtz, r 2 z t 1 z 2559 H579 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 1 x t 1 y ,mytz, rxtz, myt 1 z [ 8, 34, 73, 129, 201, 289, 393, 513, 649, 801 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 143 Nbr. gr Hi L m X 2560 H548 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,mxty, r 2 y rxt 1 y ,mxt 1 y , r 2 y rxtz 2561 H546 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , ty,mzrxt 1 y , t 1 y , mzrxtz 48A [ 9, 32, 72, 128, 200, 288, 392, 512, 648, 800 ] 2562*, 2563*, [ 9, 33, 74, 130, 202, 290, 394, 514, 650, 802 ] 2564*, 2565*, 2566*, 2567*, 2568*, 2569*, [ 9, 33, 75, 131, 202, 290, 394, 514, 650, 802 ] 2570*, [ 9, 33, 75, 132, 204, 292, 396, 516, 652, 804 ] 2571*, 2572*, 2573*, 2574*, [ 9, 34, 75, 130, 202, 290, 394, 514, 650, 802 ] 2575*, 2576*, 2577*, 2578*, 2579*, 2580*, 2581*, 2582*, 2583*, 2584*, 2585*, 2586*, 2587*, [ 9, 34, 76, 132, 204, 292, 396, 516, 652, 804 ] 2588*, 2589*, 2590*, 2591*, 2592*, 2593*, 2594*, 2595*, 2596*, 2597*, 2598*, 2599*, 2600*, 2601*, 2602*, 2603*, [ 9, 34, 77, 133, 204, 292, 396, 516, 652, 804 ] 2604*, [ 9, 35, 77, 132, 204, 292, 396, 516, 652, 804 ] 2605*, 2606*, 2607*, 2608*, 2609*, 2610*, 2611*, 2612*, 2613*, 2614*, 2615*, 2616*, 2617*, 2618*, 2619*, 2620*, 2621*, 2622*, 2623*, 2624*, 2625*, 2626*, 2627*, 2628*, 2629*, 2630*, 2631*, 2632*, 2633*, 2634*, 2635*, [ 10, 34, 74, 130, 202, 290, 394, 514, 650, 802 ] 2562 H629 hmzr 1 x i r2 x r 2 z rxtx, r 2 y rxt 1 x , r 2 z rxt 1 x , ty,mzrxt 1 y , t 1 y , r 2 x tz,mzr 1 x tz,mzr 1 x t 1 z 2563 H638 hmzr 1 x i r2 x r 2 z rxtx, r 2 z t 1 x ,mxrxt 1 x , ty,mzrxt 1 y , t 1 y , r 2 x tz,mzr 1 x tz,mzr 1 x t 1 z [ 10, 35, 75, 131, 203, 291, 395, 515, 651, 803 ] 2564 H433 1 r 2 z mztx, it 1 x ,mzt 1 x , ty, r 2 z t 1 y , t 1 y , mztz, it 1 z ,mzt 1 z 2565 H462 1 r 2 z mztx, it 1 x ,mzt 1 x , ty, r 2 z t 1 y , t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2566 H460 1 r 2 z tx, r 2 z t 1 x , t 1 x , ty, r 2 z t 1 y , t 1 y , r 2 y tz, it 1 z ,mzt 1 z 2567 H425 1 r 2 z tx, r 2 z t 1 x , t 1 x , ty, r 2 z t 1 y , t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2568 H503 1 r 2 z tx, r 2 z t 1 x , t 1 x , ty, r 2 z t 1 y , t 1 y , r 2 y rztz, r 2 x rztz, r 2 y t 1 z 2569 H519 1 r 2 z tx, r 2 z t 1 x , t 1 x , ty, r 2 z t 1 y , t 1 y , mzr 1 z tz,mzrztz, r 2 y t 1 z [ 10, 35, 76, 131, 203, 291, 395, 515, 651, 803 ] 2570 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, it 1 y , r 2 z rxt 1 y , r 2 x tz,mzr 1 x tz, r 2 z rxt 1 z [ 10, 35, 76, 132, 204, 292, 396, 516, 652, 804 ] 2571 H468 1 r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , mztz, it 1 z ,mzt 1 z 2572 H461 1 r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 x tz,mzt 1 z 2573 H640 hmzr 1 x i r2 x r 2 z rxtx, r 2 y rxt 1 x , r 2 z rxt 1 x ,mzty, r 1 x t 1 y ,mzt 1 y , r 1 x tz,mztz, r 1 x t 1 z 2574 H645 hmzr 1 x i r2 x r 2 z rxtx, r 2 z t 1 x ,mxrxt 1 x ,mzty, r 1 x t 1 y ,mzt 1 y , r 1 x tz,mztz, r 1 x t 1 z [ 10, 36, 75, 131, 203, 291, 395, 515, 651, 803 ] 2575 H631 hmzr 1 x i r2 x r 2 y rxtx, r 2 y rxt 1 x , r 2 z rxt 1 x ,mxty, r 2 y rxt 1 y ,mxt 1 y , itz, r 2 z rxtz, r 2 z rxt 1 z 2576 H631 hmzr 1 x i r2 x r 2 z rxtx, r 2 y rxt 1 x , r 2 z rxt 1 x ,mxty, r 2 y rxt 1 y ,mxt 1 y , itz, r 2 z rxtz, r 2 z rxt 1 z 2577 H471 1 r 2 z mztx, it 1 x ,mzt 1 x ,mzty, it 1 y ,mzt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2578 H433 1 r 2 z mztx, it 1 x ,mzt 1 x , ty, r 2 z t 1 y , t 1 y , itz, it 1 z ,mzt 1 z 2579 H462 1 r 2 z mztx, it 1 x ,mzt 1 x , ty, r 2 z t 1 y , t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 2580 H629 hmzr 1 x i r2 x r 2 y rxtx, r 2 y rxt 1 x , r 2 z rxt 1 x , ty,mzrxt 1 y , t 1 y , r 2 x tz,mzr 1 x tz,mzr 1 x t 1 z 2581 H638 hmzr 1 x i r2 x r 2 y rxtx, r 2 z t 1 x ,mxrxt 1 x , ty,mzrxt 1 y , t 1 y , r 2 x tz,mzr 1 x tz,mzr 1 x t 1 z 2582 H507 1 r 2 z r 2 y rztx, r 2 y rzt 1 x , r 2 x rzt 1 x , r 2 y rzty, r 2 x rzty, r 2 y rzt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2583 H531 1 r 2 z mxrztx,mxrzt 1 x ,mxr 1 z t 1 x ,mxrzty,mxr 1 z ty,mxrzt 1 y , r 2 y tz, it 1 z ,mzt 1 z 144 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 2584 H522 1 r 2 z mxrztx,mxrzt 1 x ,mxr 1 z t 1 x ,mxrzty,mxr 1 z ty,mxrzt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2585 H533 1 r 2 z mxrztx,mxrzt 1 x ,mxr 1 z t 1 x ,mxrzty,mxr 1 z ty,mxrzt 1 y , r 2 y rztz, r 2 x rztz, r 2 y t 1 z 2586 H526 1 r 2 z mxrztx,mxrzt 1 x ,mxr 1 z t 1 x ,mxrzty,mxr 1 z ty,mxrzt 1 y , mzr 1 z tz,mzrztz, r 2 y t 1 z 2587 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, it 1 y , r 2 z rxt 1 y , r 2 x tz,mzr 1 x tz, r 2 z rxt 1 z [ 10, 36, 76, 132, 204, 292, 396, 516, 652, 804 ] 2588 H475 1 r 2 z mztx, it 1 x ,mzt 1 x ,mxty,mxt 1 y ,myt 1 y , mztz, it 1 z ,mzt 1 z 2589 H485 1 r 2 z mztx, it 1 x ,mzt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2590 H466 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , mztz, it 1 z ,mzt 1 z 2591 H427 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 2592 H461 1 r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, it 1 z ,mzt 1 z 2593 H461 1 r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz, it 1 z ,mzt 1 z 2594 H469 1 r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2595 H469 1 r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 2596 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,mxty,mxt 1 y ,myt 1 y , mztz, it 1 z ,mzt 1 z 2597 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,mxty,myty,mxt 1 y , mztz, it 1 z ,mzt 1 z 2598 H486 1 r 2 z r 2 x tx, r 2 y t 1 x , r 2 x t 1 x ,mxty,mxt 1 y ,myt 1 y , mztz, it 1 z ,mzt 1 z 2599 H479 1 r 2 z r 2 x tx, r 2 y t 1 x , r 2 x t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2600 H483 1 r 2 z mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, it 1 z ,mzt 1 z 2601 H482 1 r 2 z mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2602 H529 1 r 2 z mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y rztz, r 2 x rztz, r 2 y t 1 z 2603 H530 1 r 2 z mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y ,myt 1 y , mzr 1 z tz,mzrztz, r 2 y t 1 z [ 10, 36, 77, 132, 204, 292, 396, 516, 652, 804 ] 2604 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , rxty, r 2 z t 1 y ,mxrxt 1 y , r 1 x tz,mztz,mxr 1 x t 1 z [ 10, 37, 76, 132, 204, 292, 396, 516, 652, 804 ] 2605 H473 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , itz, it 1 z ,mzt 1 z 2606 H473 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , mztz, it 1 z ,mzt 1 z 2607 H474 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2608 H474 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 2609 H539 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , mzr 1 z tz,mzrztz, r 2 y rzt 1 z 2610 H539 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 y ty, r 2 x ty, r 2 y t 1 y , mzr 1 z tz,mzrztz, r 2 y rzt 1 z 2611 H475 1 r 2 z mztx, it 1 x ,mzt 1 x ,mxty,mxt 1 y ,myt 1 y , itz, it 1 z ,mzt 1 z 2612 H485 1 r 2 z mztx, it 1 x ,mzt 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 2613 H466 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , itz, it 1 z ,mzt 1 z 2614 H427 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2615 H468 1 r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , itz, it 1 z ,mzt 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 145 Nbr. gr Hi L m X 2616 H461 1 r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 x tz, it 1 z 2617 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2618 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y tz, r 2 x tz, r 2 y t 1 z 2619 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 2620 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 y ty, r 2 x ty, r 2 y t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 2621 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,mxty,mxt 1 y ,myt 1 y , itz, it 1 z ,mzt 1 z 2622 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,mxty,mxt 1 y ,myt 1 y , itz,mztz, it 1 z 2623 H639 hmzr 1 x i r2 x r 2 y rxtx, r 2 y rxt 1 x , r 2 z rxt 1 x , r 2 y ty,mxr 1 x t 1 y , r 2 y t 1 y , mxr 1 x tz, r 2 y tz,mxr 1 x t 1 z 2624 H639 hmzr 1 x i r2 x r 2 z rxtx, r 2 y rxt 1 x , r 2 z rxt 1 x , r 2 y ty,mxr 1 x t 1 y , r 2 y t 1 y , mxr 1 x tz, r 2 y tz,mxr 1 x t 1 z 2625 H429 1 r 2 z r 2 x tx, r 2 y t 1 x , r 2 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2626 H486 1 r 2 z r 2 x tx, r 2 y t 1 x , r 2 x t 1 x ,mxty,mxt 1 y ,myt 1 y , itz, it 1 z ,mzt 1 z 2627 H479 1 r 2 z r 2 x tx, r 2 y t 1 x , r 2 x t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 2628 H640 hmzr 1 x i r2 x r 2 y rxtx, r 2 y rxt 1 x , r 2 z rxt 1 x ,mzty, r 1 x t 1 y ,mzt 1 y , r 1 x tz,mztz, r 1 x t 1 z 2629 H645 hmzr 1 x i r2 x r 2 y rxtx, r 2 z t 1 x ,mxrxt 1 x ,mzty, r 1 x t 1 y ,mzt 1 y , r 1 x tz,mztz, r 1 x t 1 z 2630 H524 1 r 2 z mzrztx,mzr 1 z t 1 x ,mzrzt 1 x ,mzr 1 z ty,mzrzty,mzr 1 z t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2631 H535 1 r 2 z rztx, r 1 z t 1 x , rzt 1 x , r 1 z ty, rzty, r 1 z t 1 y , r 2 y tz, it 1 z ,mzt 1 z 2632 H509 1 r 2 z rztx, r 1 z t 1 x , rzt 1 x , r 1 z ty, rzty, r 1 z t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2633 H512 1 r 2 z rztx, r 1 z t 1 x , rzt 1 x , r 1 z ty, rzty, r 1 z t 1 y , r 2 y rztz, r 2 x rztz, r 2 y t 1 z 2634 H537 1 r 2 z rztx, r 1 z t 1 x , rzt 1 x , r 1 z ty, rzty, r 1 z t 1 y , mzr 1 z tz,mzrztz, r 2 y t 1 z 2635 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , rxty, r 2 z t 1 y ,mxrxt 1 y , r 1 x tz,mztz,mxr 1 x t 1 z 48B [ 9, 35, 79, 134, 204, 292, 396, 516, 652, 804 ] 2636*, [ 9, 36, 79, 133, 204, 292, 396, 516, 652, 804 ] 2637*, 2638*, [ 9, 37, 79, 132, 204, 292, 396, 516, 652, 804 ] 2639*, 2640*, 2641*, 2642*, 2643*, [ 10, 37, 78, 132, 204, 292, 396, 516, 652, 804 ] 2636 H766 hmy, r 2 x i mzr1x mxtx,mxt 1 x , r 2 z rxt 1 x ,myty, r 1 x ty, t 1 y , mzr 1 x t 1 y ,mzr 1 x tz, rxt 1 z [ 10, 38, 77, 132, 204, 292, 396, 516, 652, 804 ] 2637 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty,mzr 1 x ty, it 1 y , r 2 z rxt 1 y ,mzr 1 x tz, r 2 z rxt 1 z 2638 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, rxty, r 2 z t 1 y , mxrxt 1 y , r 1 x tz,mxr 1 x t 1 z [ 10, 39, 76, 132, 204, 292, 396, 516, 652, 804 ] 2639 H768 hmy, r 2 x i mzr1x mxtx,mxt 1 x , r 2 z rxt 1 x , r 2 z ty,mxr 1 x ty,mxt 1 y , r 2 z rxt 1 y , r 2 z rxtz,mxrxt 1 z 2640 H768 hmy, r 2 x i mzr1x r 2 z rxtx,mxt 1 x , r 2 z rxt 1 x , r 2 z ty,mxr 1 x ty,mxt 1 y , r 2 z rxt 1 y , r 2 z rxtz,mxrxt 1 z 2641 H766 hmy, r 2 x i mzr1x r 2 z rxtx,mxt 1 x , r 2 z rxt 1 x ,myty, r 1 x ty, t 1 y , mzr 1 x t 1 y ,mzr 1 x tz, rxt 1 z 2642 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty,mzr 1 x ty, it 1 y , r 2 z rxt 1 y ,mzr 1 x tz, r 2 z rxt 1 z 2643 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, rxty, r 2 z t 1 y , mxrxt 1 y , r 1 x tz,mxr 1 x t 1 z 49 [ 10, 35, 76, 132, 204, 292, 396, 516, 652, 804 ] 2644*, 2645*, 2646*, 2647*, 2648*, 2649*, 2650*, [ 10, 36, 77, 132, 204, 292, 396, 516, 652, 804 ] 146 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 2651*, 2652*, 2653*, 2654*, 2655*, 2656*, 2657*, 2658*, 2659*, 2660*, 2661*, 2662*, 2663*, 2664*, 2665*, 2666*, 2667*, 2668*, 2669*, 2670*, 2671*, 2672*, 2673*, 2674*, 2675*, 2676*, 2677*, 2678*, 2679*, 2680*, 2681*, 2682*, 2683*, 2684*, 2685*, 2686*, 2687*, 2688*, 2689*, 2690*, 2691*, 2692*, 2693*, 2694*, [ 11, 36, 76, 132, 204, 292, 396, 516, 652, 804 ] 2644 H682 hmxi r 2 z itx, r 2 y tx, r 2 x t 1 x , r 2 y t 1 x ,mxty,myt 1 y , mxt 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z 2645 H680 hmxi r 2 z r 2 z tx,mxtx,myt 1 x ,mxt 1 x ,mxty,myt 1 y , mxt 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z 2646 H359 1 mz itx, r 2 z tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , myt 1 y , tz,mzt 1 z , t 1 z 2647 H343 1 mz r 2 z tx, r 2 y rzt 1 x ,mxrzt 1 x , ity, r 2 z ty, r 2 y rzt 1 y , mxrzt 1 y , tz,mzt 1 z , t 1 z 2648 H377 1 mz r 2 y tx,mxtx,mzr 1 z t 1 x , r 1 z t 1 x ,mzrzty, rzty, myt 1 y , tz,mzt 1 z , t 1 z 2649 H690 hmxi r 2 z r 2 z tx,mxtx,myt 1 x ,mxt 1 x ,mxty,myt 1 y , mxt 1 y , r 2 y rztz,mzr 1 z tz, r 2 y t 1 z 2650 H541 1 r 2 z r 2 z rxtx, rxtx, r 2 y rxt 1 x , r 1 x t 1 x ,mxty,mxt 1 y , myt 1 y ,mztz, it 1 z ,mzt 1 z [ 11, 37, 76, 132, 204, 292, 396, 516, 652, 804 ] 2651 H683 hmxi r 2 z itx, r 2 y tx, r 2 x t 1 x , r 2 y t 1 x , r 2 y ty, r 2 x t 1 y , r 2 y t 1 y , r 2 x tz, r 2 y tz, r 2 x t 1 z 2652 H683 hmxi r 2 z itx, r 2 y tx, r 2 x t 1 x , r 2 y t 1 x , r 2 y ty, r 2 x t 1 y , r 2 y t 1 y , r 2 x tz, r 2 y tz, r 2 y t 1 z 2653 H682 hmxi r 2 z itx, r 2 y tx, r 2 x t 1 x , r 2 y t 1 x ,mxty,myt 1 y , mxt 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z 2654 H344 1 mz itx, it 1 x , r 2 z t 1 x , ity, r 2 z ty, it 1 y , r 2 z t 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2655 H344 1 mz itx, r 2 z tx, it 1 x , r 2 z t 1 x , ity, r 2 z ty, it 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2656 H344 1 mz r 2 z tx, it 1 x , r 2 z t 1 x , ity, r 2 z ty, it 1 y , r 2 z t 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2657 H344 1 mz itx, r 2 z tx, it 1 x , r 2 z t 1 x , ity, r 2 z ty, r 2 z t 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2658 H682 hmxi r 2 z itx, r 2 y tx, r 2 x t 1 x , r 2 y t 1 x ,mxty,myt 1 y , mxt 1 y , r 2 x tz, r 2 y tz, r 2 x t 1 z 2659 H683 hmxi r 2 z itx, r 2 y tx, r 2 x t 1 x , r 2 y t 1 x , r 2 y ty, r 2 x t 1 y , r 2 y t 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z 2660 H683 hmxi r 2 z itx, r 2 y tx, r 2 x t 1 x , r 2 y t 1 x , r 2 y ty, r 2 x t 1 y , r 2 y t 1 y , r 2 y tz, r 2 x t 1 z , r 2 y t 1 z 2661 H682 hmyi mx mztx, r 2 y t 1 x ,mzt 1 x , r 2 z ty,myty,mxt 1 y , t 1 y , r 2 y tz, r 2 y t 1 z ,mzt 1 z 2662 H682 hmyi mx mztx, r 2 y t 1 x ,mzt 1 x , r 2 z ty,myty,mxt 1 y , t 1 y ,mztz, r 2 y t 1 z ,mzt 1 z 2663 H680 hmxi r 2 z r 2 z tx,mxtx,myt 1 x ,mxt 1 x ,mxty,myt 1 y , mxt 1 y , r 2 x tz, r 2 x t 1 z , r 2 y t 1 z 2664 H370 1 mz itx, r 2 z tx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y , myt 1 y ,mytz, r 2 x t 1 z ,myt 1 z 2665 H370 1 mz itx, r 2 z tx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , myt 1 y ,mytz, r 2 x t 1 z ,myt 1 z 2666 H359 1 mz itx, r 2 z tx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y , myt 1 y , tz,mzt 1 z , t 1 z 2667 H347 1 mz itx, r 2 y rzt 1 x ,mxrzt 1 x , ity, r 2 z ty, r 2 y rzt 1 y , mxrzt 1 y ,mxr 1 z tz, r 2 x rzt 1 z ,mxr 1 z t 1 z 2668 H347 1 mz r 2 z tx, r 2 y rzt 1 x ,mxrzt 1 x , ity, r 2 z ty, r 2 y rzt 1 y , mxrzt 1 y ,mxr 1 z tz, r 2 x rzt 1 z ,mxr 1 z t 1 z 2669 H343 1 mz itx, r 2 y rzt 1 x ,mxrzt 1 x , ity, r 2 z ty, r 2 y rzt 1 y , mxrzt 1 y , tz,mzt 1 z , t 1 z 2670 H378 1 mz r 2 y tx,mxtx, r 2 y t 1 x ,mxt 1 x , r 2 x ty, r 2 x t 1 y , myt 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2671 H378 1 mz r 2 y tx,mxtx, r 2 y t 1 x ,mxt 1 x ,myty, r 2 x t 1 y , myt 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2672 H382 1 mz r 2 y tx,mxtx,mzr 1 z t 1 x , r 1 z t 1 x ,mzrzty, rzty, r 2 x t 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z 2673 H382 1 mz r 2 y tx,mxtx,mzr 1 z t 1 x , r 1 z t 1 x ,mzrzty, rzty, myt 1 y ,mxrztz, r 2 y rzt 1 z ,mxrzt 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 147 Nbr. gr Hi L m X 2674 H377 1 mz r 2 y tx,mxtx,mzr 1 z t 1 x , r 1 z t 1 x ,mzrzty, rzty, r 2 x t 1 y , tz,mzt 1 z , t 1 z 2675 H453 1 r 2 z mztx, it 1 x ,mzt 1 x , ity,mzty, it 1 y , mzt 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 2676 H539 1 r 2 z itx,mztx, it 1 x ,mzt 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z 2677 H539 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 y ty, r 2 x ty, r 2 y t 1 y , r 2 x t 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z 2678 H690 hmxi r 2 z r 2 z tx,mxtx,myt 1 x ,mxt 1 x ,mxty,myt 1 y , mxt 1 y , r 2 y rztz,mzr 1 z tz, r 2 x t 1 z 2679 H452 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z ty, ty, r 2 z t 1 y , t 1 y , r 2 y rztz, it 1 z ,mzt 1 z 2680 H503 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z ty, ty, r 2 z t 1 y , t 1 y , r 2 y rztz, r 2 y t 1 z , r 2 x t 1 z 2681 H438 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z ty, ty, r 2 z t 1 y , t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 2682 H495 1 r 2 z tx, r 2 z t 1 x , t 1 x , r 2 z ty, ty, r 2 z t 1 y , t 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z 2683 H511 1 r 2 z r 2 z rxtx, rxtx, r 2 y rxt 1 x , r 1 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2684 H511 1 r 2 z r 2 z rxtx, rxtx, r 2 y rxt 1 x , r 1 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 2685 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 y ty, r 2 x ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y tz, r 2 y t 1 z , r 2 x t 1 z 2686 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 y ty, r 2 x ty, r 2 y t 1 y , r 2 x t 1 y , r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 2687 H541 1 r 2 z r 2 z rxtx, rxtx, r 2 y rxt 1 x , r 1 x t 1 x ,mxty,mxt 1 y , myt 1 y , itz, it 1 z ,mzt 1 z 2688 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,mxty,myty,mxt 1 y , myt 1 y , itz, it 1 z ,mzt 1 z 2689 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,mxty,myty,mxt 1 y , myt 1 y ,mztz, it 1 z ,mzt 1 z 2690 H505 1 r 2 z r 2 y tx, r 2 x tx, r 2 y t 1 x , r 2 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 2691 H527 1 r 2 z mxtx,mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y , myt 1 y , r 2 y rztz, it 1 z ,mzt 1 z 2692 H529 1 r 2 z mxtx,mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y , myt 1 y , r 2 y rztz, r 2 y t 1 z , r 2 x t 1 z 2693 H496 1 r 2 z mxtx,mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y , myt 1 y , r 2 y rztz, r 2 y rzt 1 z , r 2 x rzt 1 z 2694 H502 1 r 2 z mxtx,mytx,mxt 1 x ,myt 1 x ,mxty,mxt 1 y , myt 1 y ,mzr 1 z tz,mzrztz, r 2 y rzt 1 z 50 [ 10, 36, 78, 133, 204, 292, 396, 516, 652, 804 ] 2695*, 2696*, 2697*, 2698*, 2699*, 2700*, [ 10, 37, 78, 132, 204, 292, 396, 516, 652, 804 ] 2701*, 2702*, 2703*, 2704*, 2705*, 2706*, 2707*, 2708*, 2709*, 2710*, 2711*, 2712*, 2713*, 2714*, 2715*, 2716*, 2717*, 2718*, 2719*, 2720*, 2721*, 2722*, 2723*, 2724*, 2725*, 2726*, 2727*, 2728*, 2729*, 2730*, 2731*, [ 11, 37, 77, 132, 204, 292, 396, 516, 652, 804 ] 2695 H560 1 mzrx itx, r 2 z rxtx,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz,mzr 1 x tz, r 2 x t 1 z ,mzr 1 x t 1 z 2696 H580 1 mzrx itx, r 2 z rxtx,mxt 1 x ,myty, rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z ,mzt 1 z 2697 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, it 1 y , r 2 z rxt 1 y , r 2 x tz,mzr 1 x tz, it 1 z , r 2 z rxt 1 z 2698 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, rxty, r 2 z t 1 y , mxrxt 1 y , r 1 x tz,mxr 1 x t 1 z , r 2 y t 1 z 2699 H557 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz,mzr 1 x tz, r 2 x t 1 z ,mzr 1 x t 1 z 2700 H578 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z ,mzt 1 z [ 11, 38, 76, 132, 204, 292, 396, 516, 652, 804 ] 2701 H551 1 mzrx itx, it 1 x , r 2 z rxt 1 x , r 2 z rxty, it 1 y , r 2 z rxt 1 y , itz, r 2 z rxtz, it 1 z , r 2 z rxt 1 z 2702 H551 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , r 2 z rxty, it 1 y , r 2 z rxt 1 y , itz, r 2 z rxtz, it 1 z , r 2 z rxt 1 z 148 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 2703 H551 1 mzrx itx, r 2 z rxtx, r 2 z rxt 1 x , ity, r 2 z rxty, it 1 y , r 2 z rxt 1 y , itz, r 2 z rxtz, r 2 z rxt 1 z 2704 H572 1 mzrx itx, it 1 x , r 2 z rxt 1 x , r 2 z ty,mxrxty, r 2 z t 1 y , mxrxt 1 y ,mxr 1 x tz,mxr 1 x t 1 z , r 2 y t 1 z 2705 H572 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , r 2 z ty,mxrxty, r 2 z t 1 y , mxrxt 1 y ,mxr 1 x tz,mxr 1 x t 1 z , r 2 y t 1 z 2706 H555 1 mzrx itx, it 1 x , r 2 z rxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz,mzr 1 x tz, r 2 x t 1 z ,mzr 1 x t 1 z 2707 H555 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz,mzr 1 x tz, r 2 x t 1 z ,mzr 1 x t 1 z 2708 H555 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x , r 2 x ty,mzr 1 x ty, r 2 x t 1 y , mzr 1 x t 1 y , r 2 x tz,mzr 1 x tz,mzr 1 x t 1 z 2709 H574 1 mzrx itx, it 1 x , r 2 z rxt 1 x ,myty, rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z ,mzt 1 z 2710 H574 1 mzrx r 2 z rxtx, it 1 x , r 2 z rxt 1 x ,myty, rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z ,mzt 1 z 2711 H560 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz,mzr 1 x tz, r 2 x t 1 z ,mzr 1 x t 1 z 2712 H560 1 mzrx itx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz,mzr 1 x tz, r 2 x t 1 z ,mzr 1 x t 1 z 2713 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz,mzr 1 x tz, r 2 x t 1 z ,mzr 1 x t 1 z 2714 H560 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty,mzr 1 x ty, r 2 x t 1 y , mzr 1 x t 1 y , r 2 x tz,mzr 1 x tz,mzr 1 x t 1 z 2715 H580 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,myty, rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z ,mzt 1 z 2716 H580 1 mzrx itx, r 2 y rxt 1 x ,mxt 1 x ,myty, rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z ,mzt 1 z 2717 H580 1 mzrx r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z ,mzt 1 z 2718 H556 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z rxty, it 1 y , r 2 z rxt 1 y , itz, r 2 z rxtz, it 1 z , r 2 z rxt 1 z 2719 H556 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z rxty, it 1 y , r 2 z rxt 1 y , itz, r 2 z rxtz, it 1 z , r 2 z rxt 1 z 2720 H556 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , ity, r 2 z rxty, it 1 y , r 2 z rxt 1 y , itz, r 2 z rxtz, r 2 z rxt 1 z 2721 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, it 1 y , r 2 z rxt 1 y , r 2 x tz,mzr 1 x tz, it 1 z , r 2 z rxt 1 z 2722 H561 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty,mzr 1 x ty, it 1 y , r 2 z rxt 1 y , r 2 x tz,mzr 1 x tz, r 2 z rxt 1 z 2723 H561 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 x ty,mzr 1 x ty, it 1 y , r 2 z rxt 1 y , r 2 x tz,mzr 1 x tz, r 2 z rxt 1 z 2724 H576 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z ty,mxrxty, r 2 z t 1 y , mxrxt 1 y ,mxr 1 x tz,mxr 1 x t 1 z , r 2 y t 1 z 2725 H576 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z ty,mxrxty, r 2 z t 1 y , mxrxt 1 y ,mxr 1 x tz,mxr 1 x t 1 z , r 2 y t 1 z 2726 H576 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x , r 2 z ty,mxrxty,mxrxt 1 y , mxr 1 x tz, r 2 y tz,mxr 1 x t 1 z , r 2 y t 1 z 2727 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, rxty, r 2 z t 1 y , mxrxt 1 y , r 1 x tz,mxr 1 x t 1 z , r 2 y t 1 z 2728 H571 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, rxty, r 2 z t 1 y , mxrxt 1 y , r 1 x tz,mztz,mxr 1 x t 1 z 2729 H571 1 mzrx mxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, rxty, r 2 z t 1 y , mxrxt 1 y , r 1 x tz,mztz,mxr 1 x t 1 z 2730 H557 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y ,mzr 1 x t 1 y , r 2 x tz,mzr 1 x tz, r 2 x t 1 z ,mzr 1 x t 1 z 2731 H578 1 mzrx r 2 y rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, rxty,myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z ,mzt 1 z 51A [ 7, 23, 50, 87, 135, 194, 263, 343, 434, 535 ] 2732*, [ 7, 23, 52, 97, 158, 233, 325, 431, 547, 682 ] 2733*, [ 7, 24, 53, 94, 147, 212, 289, 378, 479, 592 ] 2734*, [ 7, 24, 54, 96, 150, 216, 294, 384, 486, 600 ] 2735*, [ 7, 25, 56, 98, 152, 218, 296, 386, 488, 602 ] 2736*, 2737*, [ 7, 27, 62, 109, 172, 249, 338, 444, 564, 695 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 149 Nbr. gr Hi L m X 2738*, 2739*, [ 8, 27, 58, 101, 158, 226, 306, 401, 506, 623 ] 2732 H306 1 i tx, it 1 x , t 1 x , r 2 z rxt 1 y ,mzrxt 1 y ,mzrxtz, r 2 z rxt 1 z [ 8, 27, 60, 109, 172, 246, 338, 446, 564, 701 ] 2733 H321 1 i tx, it 1 x , t 1 x ,mxr 1 x t 1 y , rxt 1 y ,mxrxtz, r 1 x t 1 z [ 8, 28, 60, 107, 166, 239, 325, 425, 537, 664 ] 2734 H307 1 i mzrxtx, r 2 z rxt 1 x ,mzrxt 1 x , r 2 z rxt 1 y ,mzrxt 1 y ,mzrxtz, r 2 z rxt 1 z [ 8, 28, 61, 108, 168, 241, 328, 428, 541, 668 ] 2735 H325 1 i mzrxtx, r 2 z rxt 1 x ,mzrxt 1 x ,mxrxt 1 y , r 1 x t 1 y , rxtz, mxr 1 x t 1 z [ 8, 29, 62, 109, 169, 242, 329, 429, 542, 669 ] 2736 H324 1 i mzrxtx, r 2 z rxt 1 x ,mzrxt 1 x ,mxr 1 x t 1 y , rxt 1 y ,mxrxtz, r 1 x t 1 z 2737 H309 1 i mzrxtx, r 2 z rxt 1 x ,mzrxt 1 x , r 2 y rxt 1 y ,mzr 1 x t 1 y , r 2 y rxtz, mzr 1 x t 1 z [ 8, 31, 66, 115, 181, 258, 349, 459, 578, 711 ] 2738 H310 1 i r 2 x tx,mxt 1 x , r 2 x t 1 x , r 2 z rxt 1 y ,mzrxt 1 y ,mzrxtz, r 2 z rxt 1 z 2739 H322 1 i r 2 x tx,mxt 1 x , r 2 x t 1 x ,mxr 1 x t 1 y , rxt 1 y ,mxrxtz, r 1 x t 1 z 51B [ 7, 24, 62, 118, 190, 278, 382, 502, 638, 790 ] 2740*, [ 7, 24, 62, 120, 192, 280, 384, 504, 640, 792 ] 2741*, [ 7, 25, 59, 109, 175, 257, 355, 469, 599, 745 ] 2742*, [ 7, 25, 61, 116, 188, 276, 380, 500, 636, 788 ] 2743*, [ 7, 25, 65, 124, 196, 284, 388, 508, 644, 796 ] 2744*, [ 7, 25, 66, 127, 200, 288, 392, 512, 648, 800 ] 2745*, [ 7, 26, 63, 117, 187, 273, 375, 493, 627, 777 ] 2746*, [ 7, 26, 64, 121, 194, 282, 386, 506, 642, 794 ] 2747*, [ 7, 26, 68, 128, 200, 288, 392, 512, 648, 800 ] 2748*, 2749*, 2750*, [ 7, 27, 66, 122, 194, 282, 386, 506, 642, 794 ] 2751*, 2752*, [ 7, 27, 68, 126, 198, 286, 390, 510, 646, 798 ] 2753*, [ 7, 27, 69, 128, 200, 288, 392, 512, 648, 800 ] 2754*, [ 7, 28, 69, 126, 198, 286, 390, 510, 646, 798 ] 2755*, [ 7, 28, 70, 128, 200, 288, 392, 512, 648, 800 ] 2756*, 2757*, [ 7, 29, 70, 126, 198, 286, 390, 510, 646, 798 ] 2758*, 2759*, [ 8, 28, 70, 124, 198, 284, 390, 508, 646, 796 ] 2740 H408 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , rxt 1 y , r 2 z tz, r 2 z t 1 z , r 1 x t 1 z [ 8, 28, 70, 126, 198, 286, 390, 510, 646, 798 ] 2741 H403 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mxrxt 1 y ,mytz,mxr 1 x t 1 z , myt 1 z [ 8, 29, 66, 119, 188, 273, 374, 491, 624, 773 ] 2742 H383 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , ty, r 2 z rxt 1 y , t 1 y , r 2 z rxt 1 z [ 8, 29, 68, 124, 196, 284, 388, 508, 644, 796 ] 2743 H385 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mxty,mxt 1 y ,mzr 1 x t 1 y , mzr 1 x t 1 z [ 8, 29, 72, 128, 200, 288, 392, 512, 648, 800 ] 2744 H416 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x ,mxrxt 1 y ,mytz,mxr 1 x t 1 z , myt 1 z [ 8, 29, 73, 130, 202, 290, 394, 514, 650, 802 ] 2745 H411 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , rxt 1 y , r 2 z tz, r 2 z t 1 z , r 1 x t 1 z [ 8, 30, 69, 124, 195, 282, 385, 504, 639, 790 ] 2746 H387 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , ty, r 2 z rxt 1 y , t 1 y , r 2 z rxt 1 z [ 8, 30, 70, 127, 199, 287, 391, 511, 647, 799 ] 2747 H399 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x ,mxty,mxt 1 y ,mzr 1 x t 1 y , mzr 1 x t 1 z [ 8, 30, 74, 130, 202, 290, 394, 514, 650, 802 ] 2748 H417 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , rxt 1 y , r 2 z tz, r 2 z t 1 z , r 1 x t 1 z 150 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 2749 H423 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,mxrxt 1 y ,mytz,mxr 1 x t 1 z , myt 1 z 2750 H405 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x ,mxrxt 1 y ,mytz,mxr 1 x t 1 z , myt 1 z [ 8, 31, 71, 127, 199, 287, 391, 511, 647, 799 ] 2751 H397 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x ,mxty,mxt 1 y ,mzr 1 x t 1 y , mzr 1 x t 1 z 2752 H390 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , ty, r 2 z rxt 1 y , t 1 y , r 2 z rxt 1 z [ 8, 31, 73, 129, 201, 289, 393, 513, 649, 801 ] 2753 H406 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 z rxt 1 y , tz, r 2 z rxt 1 z , t 1 z [ 8, 31, 74, 130, 202, 290, 394, 514, 650, 802 ] 2754 H423 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,mzty,mxrxt 1 y ,mzt 1 y , mxr 1 x t 1 z [ 8, 32, 73, 129, 201, 289, 393, 513, 649, 801 ] 2755 H406 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , ty, r 2 z rxt 1 y , t 1 y , r 2 z rxt 1 z [ 8, 32, 74, 130, 202, 290, 394, 514, 650, 802 ] 2756 H420 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mxrxt 1 y ,mytz,mxr 1 x t 1 z , myt 1 z 2757 H421 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , rxt 1 y , r 2 z tz, r 2 z t 1 z , r 1 x t 1 z [ 8, 33, 73, 129, 201, 289, 393, 513, 649, 801 ] 2758 H384 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , ty, r 2 z rxt 1 y , t 1 y , r 2 z rxt 1 z 2759 H386 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mxty,mxt 1 y ,mzr 1 x t 1 y , mzr 1 x t 1 z 52A [ 8, 26, 56, 98, 152, 218, 296, 386, 488, 602 ] 2760*, [ 8, 27, 59, 105, 166, 240, 328, 430, 545, 675 ] 2761*, [ 8, 28, 62, 109, 172, 249, 338, 444, 564, 695 ] 2762*, [ 9, 29, 62, 109, 169, 242, 329, 429, 542, 669 ] 2760 H752 hmy, r 2 z i i r 2 z rxtx,mxr 1 x tx,mzr 1 x t 1 x , r 1 x t 1 x , r 2 y rxty,mxrxt 1 y , mxrxt 1 z , r 2 y rxt 1 z [ 9, 30, 64, 113, 176, 251, 342, 446, 563, 696 ] 2761 H697 hmxi r 2 z mxr 1 z tx, rzt 1 x , rzt 1 y ,mxr 1 z t 1 y , r 2 y rztz,mzr 1 z tz, r 2 x t 1 z , r 2 y t 1 z [ 9, 31, 66, 115, 181, 258, 349, 459, 578, 711 ] 2762 H751 hmy, r 2 z i i r 2 y tx, r 2 z tx,mzt 1 x , r 2 z t 1 x , r 2 y rxty,mxrxt 1 y , mxrxt 1 z , r 2 y rxt 1 z 52B [ 8, 27, 68, 128, 200, 288, 392, 512, 648, 800 ] 2763*, [ 8, 28, 66, 122, 194, 282, 386, 506, 642, 794 ] 2764*, [ 8, 28, 69, 128, 200, 288, 392, 512, 648, 800 ] 2765*, 2766*, [ 8, 29, 68, 124, 196, 284, 388, 508, 644, 796 ] 2767*, [ 8, 29, 69, 126, 198, 286, 390, 510, 646, 798 ] 2768*, [ 8, 29, 70, 128, 200, 288, 392, 512, 648, 800 ] 2769*, [ 8, 30, 70, 126, 198, 286, 390, 510, 646, 798 ] 2770*, [ 9, 30, 74, 130, 202, 290, 394, 514, 650, 802 ] 2763 H653 hmxi r 2 y rx itx, r 2 z rxtx, r 2 x t 1 x , r 2 z rxt 1 x ,mxr 1 x t 1 y ,mytz, mxrxtz,myt 1 z [ 9, 31, 71, 127, 199, 287, 391, 511, 647, 799 ] 2764 H647 hmxi r 2 y rx itx, r 2 z rxtx, r 2 x t 1 x , r 2 z rxt 1 x ,mxty,mzrxt 1 y , mxt 1 y ,mzrxtz [ 9, 31, 74, 130, 202, 290, 394, 514, 650, 802 ] 2765 H423 1 r 2 z rx r 2 z tx, r 1 x tx, r 2 y t 1 x , rxt 1 x ,mxrxt 1 y ,mytz, mxr 1 x t 1 z ,myt 1 z 2766 H581 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x , r 1 x t 1 y ,mytz, rxtz,myt 1 z [ 9, 32, 72, 128, 200, 288, 392, 512, 648, 800 ] 2767 H552 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x , ty,mzrxt 1 y , t 1 y ,mzrxtz [ 9, 32, 73, 129, 201, 289, 393, 513, 649, 801 ] 2768 H406 1 r 2 z rx r 2 z tx, r 1 x tx, r 2 y t 1 x , rxt 1 x , ty, r 2 z rxt 1 y , t 1 y , r 2 z rxt 1 z [ 9, 32, 74, 130, 202, 290, 394, 514, 650, 802 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 151 Nbr. gr Hi L m X 2769 H651 hmxi r 2 y rx r 2 y rxtx,mxtx,mzrxt 1 x ,mxt 1 x ,mxr 1 x t 1 y ,mytz, mxrxtz,myt 1 z [ 9, 33, 73, 129, 201, 289, 393, 513, 649, 801 ] 2770 H646 hmxi r 2 y rx r 2 y rxtx,mxtx,mzrxt 1 x ,mxt 1 x ,mxty,mzrxt 1 y , mxt 1 y ,mzrxtz 53A [ 7, 27, 64, 113, 179, 260, 354, 467, 593, 734 ] 2771*, [ 8, 31, 68, 119, 188, 269, 368, 481, 609, 752 ] 2771 H393 1 r 2 z rx r 2 y rztx, r 1 y r 1 z t 1 x , r 2 y rzty, ryrxty, r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz 53B [ 7, 24, 55, 98, 153, 222, 303, 396, 503, 622 ] 2772*, [ 7, 27, 60, 103, 160, 230, 312, 407, 515, 635 ] 2773*, [ 8, 28, 63, 110, 172, 249, 338, 441, 560, 691 ] 2772 H393 1 r 2 z rx r 2 y rztx, ryrxtx, r 2 y rzty, r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz, r 1 y r 1 z t 1 z [ 8, 31, 65, 113, 176, 253, 343, 447, 566, 698 ] 2773 H393 1 r 2 z rx r 2 x ryt 1 x , r 1 y r 1 z t 1 x , ryrxty, r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz, r 2 x ryt 1 z 53C [ 7, 29, 77, 134, 203, 294, 395, 515, 654, 803 ] 2774*, [ 8, 33, 79, 131, 205, 292, 395, 517, 652, 803 ] 2774 H393 1 r 2 z rx ryrxtx, r 2 x ryt 1 x , r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz, r 2 x ryt 1 z , r 1 y r 1 z t 1 z 54A [ 8, 32, 76, 131, 203, 294, 395, 515, 654, 803 ] 2775*, [ 9, 35, 76, 131, 205, 292, 395, 517, 652, 803 ] 2775 H393 1 r 2 z rx r 2 y rztx, r 1 y r 1 z t 1 x , r 2 y rzty, ryrxty, r 2 x ryt 1 y , r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz 54B [ 8, 27, 59, 103, 160, 230, 312, 407, 515, 635 ] 2776*, [ 9, 30, 65, 113, 176, 253, 343, 447, 566, 698 ] 2776 H393 1 r 2 z rx r 2 y rztx, ryrxtx, r 2 y rzty, r 2 x ryt 1 y , r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz, r 1 y r 1 z t 1 z 55A [ 9, 31, 69, 123, 193, 279, 381, 499, 633, 783 ] 2777*, [ 9, 32, 72, 128, 200, 288, 392, 512, 648, 800 ] 2778*, 2779*, [ 9, 32, 73, 130, 202, 290, 394, 514, 650, 802 ] 2780*, 2781*, [ 9, 33, 74, 130, 202, 290, 394, 514, 650, 802 ] 2782*, 2783*, 2784*, 2785*, 2786*, 2787*, 2788*, 2789*, [ 9, 33, 75, 132, 204, 292, 396, 516, 652, 804 ] 2790*, 2791*, [ 9, 34, 76, 132, 204, 292, 396, 516, 652, 804 ] 2792*, 2793*, 2794*, 2795*, 2796*, 2797*, 2798*, 2799*, 2800*, 2801*, [ 10, 33, 72, 127, 198, 285, 388, 507, 642, 793 ] 2777 H777 hmzrx, ryrxi r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mxr 1 z ty,mxr 1 y r 1 x t 1 y ,mxr 1 z t 1 y , r 1 y r 1 z tz, r 2 x ryt 1 z , r 1 y r 1 z t 1 z [ 10, 34, 74, 130, 202, 290, 394, 514, 650, 802 ] 2778 H301 1 i tx, it 1 x , t 1 x , ty, it 1 y , t 1 y , mytz, r 2 y t 1 z ,myt 1 z 2779 H658 hmzrxi r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 1 x ty, r 2 z t 1 y ,mzt 1 y , mytz,mxr 1 x t 1 z , rxt 1 z [ 10, 34, 75, 131, 203, 291, 395, 515, 651, 803 ] 2780 H406 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , ty, r 2 z rxt 1 y , t 1 y , tz, r 2 z rxt 1 z , t 1 z 2781 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 y rxty, it 1 y ,mzrxt 1 y , mzrxtz, r 2 y rxt 1 z , r 2 x t 1 z [ 10, 35, 75, 131, 203, 291, 395, 515, 651, 803 ] 2782 H655 hmzrxi r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mzrxty, it 1 y , t 1 y , tz, r 2 z rxt 1 z ,mzrxt 1 z 2783 H302 1 i tx, it 1 x , t 1 x ,mzty, r 2 z t 1 y ,mzt 1 y , r 2 z tz, r 2 z t 1 z ,mzt 1 z 2784 H312 1 i tx, it 1 x , t 1 x ,mzty, r 2 z t 1 y ,mzt 1 y , mxtz,mxt 1 z , r 2 x t 1 z 2785 H656 hmzrxi r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 y rxty,mxt 1 y , r 2 x t 1 y , mxtz, r 2 y rxt 1 z ,mzr 1 x t 1 z 152 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 2786 H657 hmzrxi r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 y rxty,mxt 1 y , r 2 x t 1 y , mxtz, r 2 y rxt 1 z ,mzr 1 x t 1 z 2787 H315 1 i mztx, r 2 z t 1 x ,mzt 1 x ,mzty, r 2 z t 1 y ,mzt 1 y , mytz, r 2 y t 1 z ,myt 1 z 2788 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 y rxt 1 z , r 2 x t 1 z 2789 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 y rxty, it 1 y ,mzrxt 1 y , mzrxtz, r 2 y rxt 1 z , r 2 x t 1 z [ 10, 35, 76, 132, 204, 292, 396, 516, 652, 804 ] 2790 H423 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,mzty,mxrxt 1 y ,mzt 1 y , mytz,mxr 1 x t 1 z ,myt 1 z 2791 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mxr 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz,mxrxt 1 z ,mzt 1 z [ 10, 36, 76, 132, 204, 292, 396, 516, 652, 804 ] 2792 H659 hmzrxi r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mxr 1 x ty, r 2 y t 1 y ,myt 1 y , r 2 z tz,mxrxt 1 z , r 1 x t 1 z 2793 H314 1 i tx, it 1 x , t 1 x , r 2 y ty, r 2 y t 1 y ,myt 1 y , mxtz,mxt 1 z , r 2 x t 1 z 2794 H316 1 i mztx, r 2 z t 1 x ,mzt 1 x , r 2 y ty, r 2 y t 1 y ,myt 1 y , r 2 z tz, r 2 z t 1 z ,mzt 1 z 2795 H317 1 i mztx, r 2 z t 1 x ,mzt 1 x , r 2 y ty, r 2 y t 1 y ,myt 1 y , mytz, r 2 y t 1 z ,myt 1 z 2796 H660 hmzrxi r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 1 x ty, r 2 z t 1 y ,mzt 1 y , mytz,mxr 1 x t 1 z , rxt 1 z 2797 H734 hr 1 y r 1 z i i mztx, r 2 z t 1 x ,mzt 1 x ,mxr 1 y rxty,mxr 1 y rxt 1 y , ryr 1 x t 1 y , mxryr 1 z tz, ryrzt 1 z ,mxryr 1 z t 1 z 2798 H778 hmzrx, ryrxi r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 1 y r 1 x ty, r 1 y r 1 x t 1 y , r 1 z t 1 y , r 1 y mytz, r 1 y myt 1 z ,mxr 1 y r 1 z t 1 z 2799 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,mxr 1 x ty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mxrxt 1 z ,mzt 1 z 2800 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mxr 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz,mxrxt 1 z ,mzt 1 z 2801 H393 1 r 2 z rx r 2 y rztx, r 2 x ryt 1 x , r 1 y r 1 z t 1 x , r 2 y rzty, ryrxty, r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz, r 2 x ryt 1 z 55B [ 9, 34, 76, 132, 204, 292, 396, 516, 652, 804 ] 2802*, [ 9, 35, 77, 132, 204, 292, 396, 516, 652, 804 ] 2803*, 2804*, [ 9, 35, 79, 134, 204, 292, 396, 516, 652, 804 ] 2805*, [ 9, 36, 78, 132, 204, 292, 396, 516, 652, 804 ] 2806*, 2807*, 2808*, 2809*, 2810*, [ 9, 36, 79, 133, 204, 292, 396, 516, 652, 804 ] 2811*, [ 10, 36, 76, 132, 204, 292, 396, 516, 652, 804 ] 2802 H759 hmz, r 2 x i r2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , r 2 z ty, r 1 x ty,mxt 1 y , mzr 1 x t 1 y ,mzr 1 x tz, rxt 1 z [ 10, 37, 76, 132, 204, 292, 396, 516, 652, 804 ] 2803 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 y rxty, r 2 x ty, it 1 y , mzrxt 1 y ,mzrxtz, r 2 y rxt 1 z 2804 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , r 1 x t 1 y , rxtz,mxrxt 1 z [ 10, 37, 78, 132, 204, 292, 396, 516, 652, 804 ] 2805 H393 1 r 2 z rx r 2 y rztx, ryrxtx, r 2 x ryt 1 x , r 2 y rzty, r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz, r 2 x ryt 1 z , r 1 y r 1 z t 1 z [ 10, 38, 76, 132, 204, 292, 396, 516, 652, 804 ] 2806 H760 hmz, r 2 x i r2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 z ty, r 1 x ty,mxt 1 y , mzr 1 x t 1 y ,mzr 1 x tz, rxt 1 z 2807 H758 hmz, r 2 x i r2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mxrxty, r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y ,mxr 1 x tz, r 2 y rxt 1 z 2808 H757 hmz, r 2 x i r2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x ,mxrxty, r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y ,mxr 1 x tz, r 2 y rxt 1 z 2809 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 y rxty, r 2 x ty, it 1 y , mzrxt 1 y ,mzrxtz, r 2 y rxt 1 z 2810 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , r 1 x t 1 y , rxtz,mxrxt 1 z [ 10, 38, 77, 132, 204, 292, 396, 516, 652, 804 ] 2811 H393 1 r 2 z rx ryrxtx, r 2 x ryt 1 x , r 1 y r 1 z t 1 x , ryrxty, r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz, r 2 x ryt 1 z , r 1 y r 1 z t 1 z 56A [ 10, 34, 74, 130, 202, 290, 394, 514, 650, 802 ] K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 153 Nbr. gr Hi L m X 2812*, 2813*, 2814*, [ 10, 35, 76, 132, 204, 292, 396, 516, 652, 804 ] 2815*, 2816*, 2817*, 2818*, 2819*, 2820*, 2821*, 2822*, 2823*, 2824*, 2825*, 2826*, 2827*, 2828*, 2829*, [ 11, 35, 75, 131, 203, 291, 395, 515, 651, 803 ] 2812 H745 hmzr 1 x , r 2 z rxi i r 2 y rxtx, r 2 z rxtx,mzrxt 1 x , r 2 z rxt 1 x ,mxty,mzrxt 1 y , mxt 1 y , r 2 x tz, r 2 z rxtz, r 2 z rxt 1 z 2813 H746 hmzr 1 x , r 2 z rxi i r 2 z tx,mxrxtx,myt 1 x , rxt 1 x ,mxty,mzrxt 1 y , mxt 1 y , r 2 x tz, r 2 z rxtz, r 2 z rxt 1 z 2814 H638 hmzr 1 x i r2 x r 2 y rxtx, r 2 z rxtx, r 2 z t 1 x ,mxrxt 1 x , ty,mzrxt 1 y , t 1 y , r 2 x tz,mzr 1 x tz,mzr 1 x t 1 z [ 11, 36, 76, 132, 204, 292, 396, 516, 652, 804 ] 2815 H615 hmzi r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , itz,mztz, r 2 z t 1 z , t 1 z 2816 H305 1 i tx, it 1 x , t 1 x , ity, ty, it 1 y , t 1 y ,mxr 1 z tz, r 2 y rzt 1 z ,mxr 1 z t 1 z 2817 H747 hmzr 1 x , r 2 z rxi i r 2 y rxtx, r 2 z rxtx,mzrxt 1 x , r 2 z rxt 1 x , r 2 y ty,mxr 1 x t 1 y , r 2 y t 1 y ,mxr 1 x tz, r 2 y tz,mxr 1 x t 1 z 2818 H616 hmzi r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 x tz,mxt 1 z ,myt 1 z 2819 H748 hmzr 1 x , r 2 z rxi i r 2 z tx,mxrxtx,myt 1 x , rxt 1 x , r 2 y ty,mxr 1 x t 1 y , r 2 y t 1 y ,mxr 1 x tz, r 2 y tz,mxr 1 x t 1 z 2820 H308 1 i mztx, r 2 z t 1 x ,mzt 1 x , r 2 z ty,mzty, r 2 z t 1 y , mzt 1 y ,mxr 1 z tz, r 2 y rzt 1 z ,mxr 1 z t 1 z 2821 H323 1 i r 2 z tx,mztx, r 2 z t 1 x ,mzt 1 x ,mxryty, r 2 x ryt 1 y , mxryt 1 y ,mxtz,mxt 1 z , r 2 x t 1 z 2822 H326 1 i mzrxtx, r 2 z rxt 1 x ,mzrxt 1 x , r 2 y ty,myty, r 2 y t 1 y , myt 1 y , r 2 z tz, r 2 z t 1 z ,mzt 1 z 2823 H539 1 r 2 z mztx, it 1 x ,mzt 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , mzr 1 z tz,mzrztz, r 2 y rzt 1 z , r 2 x rzt 1 z 2824 H540 1 r 2 z mztx, it 1 x ,mzt 1 x , ryty, r 2 x ryt 1 y , r 1 y t 1 y , mxtz,mytz,mxt 1 z ,myt 1 z 2825 H461 1 r 2 z tx, r 2 z t 1 x , t 1 x ,mxty,mxt 1 y ,myt 1 y , r 2 y tz, r 2 x tz, it 1 z ,mzt 1 z 2826 H504 1 r 2 z tx, r 2 z t 1 x , t 1 x , ryty, r 2 x ryt 1 y , r 1 y t 1 y , r 2 z tz, tz, r 2 z t 1 z , t 1 z 2827 H511 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x , r 2 y ty, r 2 y t 1 y , r 2 x t 1 y , r 2 y tz, r 2 x tz, r 2 y t 1 z , r 2 x t 1 z 2828 H541 1 r 2 z r 2 z rxtx, rxtx, r 1 x t 1 x ,mxty,mxt 1 y ,myt 1 y , itz,mztz, it 1 z ,mzt 1 z 2829 H645 hmzr 1 x i r2 x r 2 y rxtx, r 2 z rxtx, r 2 z t 1 x ,mxrxt 1 x ,mzty, r 1 x t 1 y , mzt 1 y , r 1 x tz,mztz, r 1 x t 1 z 56B [ 10, 37, 78, 132, 204, 292, 396, 516, 652, 804 ] 2830*, [ 11, 38, 76, 132, 204, 292, 396, 516, 652, 804 ] 2830 H779 hi, r 2 x , r 2 z i r2 z rx mxtx,mzr 1 x tx, t 1 x , r 2 z rxt 1 x ,myty,mxr 1 x ty, t 1 y , r 2 z rxt 1 y , r 2 z rxtz,mxrxt 1 z 57 [ 10, 35, 76, 132, 204, 292, 396, 516, 652, 804 ] 2831*, 2832*, 2833*, 2834*, [ 10, 36, 77, 132, 204, 292, 396, 516, 652, 804 ] 2835*, 2836*, 2837*, 2838*, 2839*, 2840*, 2841*, 2842*, 2843*, 2844*, 2845*, 2846*, 2847*, 2848*, 2849*, 2850*, 2851*, 2852*, 2853*, 2854*, 2855*, 2856*, 2857*, 2858*, [ 10, 36, 78, 133, 204, 292, 396, 516, 652, 804 ] 2859*, 2860*, [ 11, 36, 76, 132, 204, 292, 396, 516, 652, 804 ] 2831 H388 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mzrxty, it 1 y ,mzrxt 1 y , itz,mzrxtz, it 1 z ,mzrxt 1 z 2832 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 y rxty, r 2 x ty, it 1 y , mzrxt 1 y ,mzrxtz, r 2 y rxt 1 z , r 2 x t 1 z 2833 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , r 1 x t 1 y , rxtz,mxrxt 1 z ,mzt 1 z 2834 H407 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 z ty, r 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z , rxt 1 z [ 11, 37, 76, 132, 204, 292, 396, 516, 652, 804 ] 2835 H389 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x ,mzrxty, it 1 y ,mzrxt 1 y , itz,mzrxtz, it 1 z ,mzrxt 1 z 2836 H414 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , r 2 z ty, r 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z , rxt 1 z 154 Art Discrete Appl. Math. 4 (2021) #P2.04 Nbr. gr Hi L m X 2837 H415 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x ,mxr 1 x ty,myty,mxr 1 x t 1 y , myt 1 y ,mxrxtz,mxrxt 1 z ,mzt 1 z 2838 H396 1 r 2 z rx mzrxtx, it 1 x ,mzrxt 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 y rxt 1 z , r 2 x t 1 z 2839 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,mxr 1 x ty,myty,mxr 1 x t 1 y , myt 1 y ,mxrxtz,mxrxt 1 z ,mzt 1 z 2840 H422 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x ,mxr 1 x ty,mxr 1 x t 1 y ,myt 1 y , mxrxtz,mztz,mxrxt 1 z ,mzt 1 z 2841 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 y rxt 1 z , r 2 x t 1 z 2842 H409 1 r 2 z rx r 2 z tx, r 1 x tx, rxt 1 x , r 2 y rxty, r 2 x ty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 y rxt 1 z , r 2 x t 1 z 2843 H394 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 y rxty, it 1 y ,mzrxt 1 y , itz,mzrxtz, r 2 y rxt 1 z , r 2 x t 1 z 2844 H412 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , r 1 x t 1 y , r 2 y tz, rxtz,mxrxt 1 z 2845 H402 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x ,mxr 1 x ty,myty,mxr 1 x t 1 y , myt 1 y ,mxrxtz,mxrxt 1 z ,mzt 1 z 2846 H391 1 r 2 z rx tx, r 2 z rxt 1 x , t 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 y rxt 1 z , r 2 x t 1 z 2847 H398 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x ,mzrxty, it 1 y ,mzrxt 1 y , itz,mzrxtz, it 1 z ,mzrxt 1 z 2848 H410 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , r 2 z ty, r 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z , rxt 1 z 2849 H404 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x ,mxr 1 x ty,myty,mxr 1 x t 1 y , myt 1 y ,mxrxtz,mxrxt 1 z ,mzt 1 z 2850 H392 1 r 2 z rx r 2 x tx, r 2 y rxt 1 x , r 2 x t 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 y rxt 1 z , r 2 x t 1 z 2851 H400 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mzrxty, it 1 y ,mzrxt 1 y , itz,mzrxtz, it 1 z ,mzrxt 1 z 2852 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 y rxty, it 1 y ,mzrxt 1 y , itz,mzrxtz, r 2 y rxt 1 z , r 2 x t 1 z 2853 H395 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 y rxty, r 2 x ty, it 1 y , mzrxt 1 y ,mzrxtz, r 2 y rxt 1 z , r 2 x t 1 z 2854 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , r 1 x t 1 y , r 2 y tz, rxtz,mxrxt 1 z 2855 H413 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mxr 1 x ty,myty, r 2 z t 1 y , r 1 x t 1 y , rxtz,mxrxt 1 z ,mzt 1 z 2856 H418 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 z ty, r 1 x ty, r 2 z t 1 y , r 1 x t 1 y , rxtz, r 2 y t 1 z , rxt 1 z 2857 H419 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x ,mxr 1 x ty,myty,mxr 1 x t 1 y , myt 1 y ,mxrxtz,mxrxt 1 z ,mzt 1 z 2858 H401 1 r 2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 y rxt 1 z , r 2 x t 1 z [ 11, 37, 77, 132, 204, 292, 396, 516, 652, 804 ] 2859 H393 1 r 2 z rx r 2 y rztx, r 2 x ryt 1 x , r 1 y r 1 z t 1 x , r 2 y rzty, ryrxty, r 2 x ryt 1 y , r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz, r 2 x ryt 1 z 2860 H393 1 r 2 z rx r 2 y rztx, ryrxtx, r 1 y r 1 z t 1 x , r 2 y rzty, ryrxty, r 2 x ryt 1 y , r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz, r 1 y r 1 z t 1 z 58 [ 11, 36, 76, 132, 204, 292, 396, 516, 652, 804 ] 2861*, 2862*, 2863*, [ 12, 36, 76, 132, 204, 292, 396, 516, 652, 804 ] 2861 H779 hmz,mzrx, r 2 x i r2 z rx mzr 1 x tx,mxt 1 x ,mzr 1 x t 1 x , r 2 z ty, r 2 x ty,mxt 1 y , mzr 1 x t 1 y ,mxr 1 x tz,mzr 1 x tz, r 2 y rxt 1 z , r 2 x t 1 z 2862 H750 hmy,mzi r 2 z mytx,mxt 1 x ,myt 1 x , r 2 z ty,myty,mxt 1 y , myt 1 y , r 2 y rztz,mzrztz,mxrzt 1 z , rzt 1 z 2863 H690 hmxi r 2 z r 2 z tx,mxtx,myt 1 x ,mxt 1 x ,mxty,myt 1 y , mxt 1 y , r 2 y rztz,mzr 1 z tz, r 2 x t 1 z , r 2 y t 1 z 59 [ 11, 37, 77, 132, 204, 292, 396, 516, 652, 804 ] 2864*, 2865*, 2866*, 2867*, 2868*, 2869*, 2870*, 2871*, 2872*, [ 12, 37, 76, 132, 204, 292, 396, 516, 652, 804 ] 2864 H649 hmxi r 2 y rx itx, r 2 z rxtx, r 2 x t 1 x , r 2 z rxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 z rxtz, r 2 x t 1 z , r 2 z rxt 1 z 2865 H652 hmxi r 2 y rx itx, r 2 z rxtx, r 2 x t 1 x , r 2 z rxt 1 x ,myty,mxrxty, myt 1 y ,mxrxt 1 y ,mxr 1 x tz,mxr 1 x t 1 z , r 2 y t 1 z K. V. Kostousov: Symmetrical 2-extensions of the 3-dimensional grid 155 Nbr. gr Hi L m X 2866 H422 1 r 2 z rx r 2 z tx, r 1 x tx, r 2 y t 1 x , rxt 1 x ,mxr 1 x ty,myty, mxr 1 x t 1 y ,myt 1 y ,mxrxtz,mxrxt 1 z ,mzt 1 z 2867 H409 1 r 2 z rx r 2 z tx, r 1 x tx, r 2 y t 1 x , rxt 1 x , r 2 y rxty, r 2 y rxt 1 y , r 2 x t 1 y , r 2 y rxtz, r 2 x tz, r 2 y rxt 1 z , r 2 x t 1 z 2868 H648 hmxi r 2 y rx r 2 y rxtx,mxtx,mzrxt 1 x ,mxt 1 x , r 2 z rxty, r 2 x t 1 y , r 2 z rxt 1 y , r 2 x tz, r 2 z rxtz, r 2 x t 1 z , r 2 z rxt 1 z 2869 H650 hmxi r 2 y rx r 2 y rxtx,mxtx,mzrxt 1 x ,mxt 1 x ,myty,mxrxty, myt 1 y ,mxrxt 1 y ,mxr 1 x tz,mxr 1 x t 1 z , r 2 y t 1 z 2870 H393 1 r 2 z rx r 2 y rztx, ryrxtx, r 2 x ryt 1 x , r 1 y r 1 z t 1 x , r 2 y rzty, ryrxty, r 1 y r 1 z t 1 y , r 2 y rztz, ryrxtz, r 2 x ryt 1 z , r 1 y r 1 z t 1 z 2871 H560 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,mzr 1 x ty, r 2 x t 1 y , mzr 1 x t 1 y , r 2 x tz,mzr 1 x tz, r 2 x t 1 z ,mzr 1 x t 1 z 2872 H580 1 mzrx itx, r 2 z rxtx, r 2 y rxt 1 x ,mxt 1 x ,myty, rxty, myt 1 y , rxt 1 y , r 1 x tz, r 1 x t 1 z ,mzt 1 z ORCID iDs Kirill V. Kostousov https://orcid.org/0000-0002-8955-5912 References [1] B. Eick, F. Gähler and W. Nickel, Cryst manual. a gap4 package, version 4.1.10: [e-resource], https://www.gap-system.org/Manuals/pkg/cryst/htm/chapters.htm. [2] gap-system.org, Gap - groups, algorithms, programming - a system for computational discrete algebra, ver. 4.5.7: [e-resource], http://www.gap-system.org. [3] E. A. Konovalchik and K. V. Kostousov, Symmetrical 2-extensions of the 2-dimensional grid. i, Trudy Inst. Mat. Mekh. UrO RAN 22 (2016), 159–179, In Russian, http://journal.imm. uran.ru/sites/default/files/archive/trudy_imm-2016-1.pdf. [4] E. A. Konovalchik and K. V. Kostousov, Symmetrical 2-extensions of the 2-dimensional grid. ii, Trudy Inst. Mat. Mekh. UrO RAN 23 (2017), 192–211, doi:10.21538/ 0134-4889-2017-23-4-192-211, In Russian. [5] E. A. Neganova and V. I. Trofimov, Symmetrical extensions of graphs, Izv. Math. 78 (2014), 809–835, doi:10.1070/IM2014v078n04ABEH002707. [6] V. I. Trofimov, Symmetrical extensions of graphs and some other topics in graph theory related with group theory, Proc. Steklov Inst. Math. 279 (2012), 107–112, doi:10.1134/ S0081543812090088. [7] V. I. Trofimov, The finiteness of the number of symmetrical 2-extensions of the d-dimensional grid and similar graphs, Proc. Steklov Inst. Math. 285 (2014), 169–182, doi:10.1134/ s0081543814050198. ISSN 2590-9770 The Art of Discrete and Applied Mathematics 4 (2021) #P2.05 https://doi.org/10.26493/2590-9770.1365.884 (Also available at http://adam-journal.eu) Infinite Paley graphs Gareth A. Jones⇤ Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, UK Received 15 March 2020, accepted 8 June 2020, published online 10 March 2021 Abstract Infinite analogues of the Paley graphs are constructed, based on uncountably many locally finite fields. By using character sum estimates due to Weil, they are shown to be isomorphic to the countable random graph of Erdős, Rényi and Rado. Keywords: Paley graph, random graph, universal graph, quadratic residue, character sum. Math. Subj. Class.: 05C63, 03C13, 03C15, 05C80, 05E18, 11L40, 12E20, 20B25, 20B27. 1 Introduction In 1963 Erdős and Rényi [14] described two constructions of graphs which have subse- quently become well-known and well-understood parts of the landscape of graph theory. One construction gave a countably infinite family of finite graphs, defined deterministically, which later became known as the Paley graphs P (q). The other gave a single countably in- finite graph R (or more precisely an uncountable family of mutually isomorphic countably infinite graphs), defined randomly and later variously named after Erdős, Rényi and Rado, who gave an alternative construction in [24] the following year. It is perhaps surprising that in the following half-century and more, a strong connection between these very different graphs seems to have received little notice, except in the world of model theory (see [20, Examples 1.3.6 and 1.8.3]), though there are hints to be found in papers such as [2, 3]. Perhaps this lacuna is less surprising when one realises that an essential ingredient in this connection comes from algebraic geometry, namely Weil’s estimate for character-sums, used in his proof of the Riemann hypothesis for curves over finite fields. The first aim of this paper is give a more combinatorial explanation of this connection by constructing, for each odd prime p, infinite analogues of the Paley graphs, defined over uncountably many locally finite fields of characteristic p, and its second aim is to show that these graphs are all isomorphic to R. The finite and infinite Paley graphs are described in ⇤The author is grateful to Peter Cameron, Dugald Macpherson, Yaokun Wu and the referees for some very helpful comments. E-mail address: G.A.Jones@maths.soton.ac.uk (Gareth A. Jones) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 2 Art Discrete Appl. Math. 4 (2021) #P2.05 Sections 2 and 3, and R is described in Section 4. The isomorphism is proved in Section 5, with remarks on the proof in Section 6. The automorphism groups of these finite and infinite graphs are compared in Section 7, and the construction and the isomorphism with R are extended in Section 8 to the generalised Paley graphs introduced by Lim and Praeger in [19]. The paper [14] is revisited in Section 9. 2 The Paley graphs and their inclusions For each prime power q = pe ⌘ 1 mod (4) the Paley graph P (q) has as its vertex set the field Fq of q elements, with vertices x and y adjacent if and only if x y is a quadratic residue (non-zero square) in Fq . It is an undirected strongly regular graph with parameters v = q (the number of vertices), k = (q 1)/2 (their common valency), = (q 5)/4 and µ = (q 1)/4 (the number of common neighbours of two adjacent or non-adjacent vertices). See [4] for further basic properties of the Paley graphs. These graphs were introduced, not (as is often asserted) by Paley [23] in 1933, but in the case e = 1 in 1962 by Sachs [25], as examples of self-complementary graphs, and in the general case e 1 in 1963 by Erdős and Rényi [14, §1], as part of their study of asymmetric graphs. Neither paper attached a name to these graphs; they appear to have been named around 1970, no doubt by analogy with Paley designs (see [12]) which, like the orthogonal matrices constructed by Paley in [23], are based on properties of quadratic residues in finite fields. See [18] for a discussion of the history of these graphs. A finite field Fq is a subfield of Fq0 if and only if q0 is a power qf of q, in which case the subfield is unique, consisting of the solutions of the equation xq = x. In this case, if q ⌘ 1 mod (4) then q0 ⌘ 1 mod (4), so we have Paley graphs P (q) and P (q0). Clearly each quadratic residue in Fq is also a quadratic residue in Fq0 , so P (q) is a subgraph of P (q0). If f is even then every element of Fq has a square root in Fq0 (in fact, in the quadratic subfield Fq2 ✓ Fq0 ) so the subgraph of P (q0) induced by P (q) is a complete graph Kq . However, if f is odd then an element of Fq has a square root in Fq0 if and only if it has one in Fq , so the induced subgraph is simply P (q), that is, P (q) is a full subgraph of P (q0). 3 Infinite Paley graphs Let E be any set of odd e 2 N which is closed under taking divisors and least common multiples. For any prime p ⌘ 1 mod (4) let FpE := [ e2E Fpe , the direct limit of the direct system of fields Fpe for e 2 E and inclusions between them. This is a subfield of the algebraic closure Fp of Fp, infinite if and only if E is, and locally finite in the sense that each finite subset is contained in a finite subfield. The finite subfields of FpE are just the fields Fpe for e 2 E, so distinct sets E determine distinct (and non- isomorphic) fields FpE . There are uncountably many sets E satisfying the above conditions (consider, for example, the set of integers e whose prime factors all belong to a given set of odd primes), so for each p we obtain uncountably many non-isomorphic fields FpE . Now let us define P (pE) := [ e2E P (pe), G. A. Jones: Infinite Paley graphs 3 the direct limit of the Paley graphs P (pe) for e 2 E, with respect to the embeddings P (pe) ✓ P (pe0) where e divides e0 in E. By our remarks in Section 2, each P (pe) is a full subgraph of P (pE). If E is finite then E is just the set of all divisors of l := lcm(E), so that P (pE) is just another Paley graph P (pl). We will therefore assume from now on that E is infinite, in which case we will call P (pE) an infinite Paley graph (see [20, Example 1.8.3] for a similar construction by Macpherson and Steinhorn, though the exponents e = 2i used there should be replaced with odd integers). In Section 5 we will use the fact that if, as assumed, E is infinite then each e 2 E divides infinitely many elements e0 2 E: only finitely many elements of E can have a given least common multiple e0 with e, so e0 takes infinitely many distinct values, all divisible by e. Thus each finite subfield Fpe of FpE or Paley subgraph P (pe) of P (pE) is contained in infinitely many others. In the same way as this one can construct infinite Paley graphs P (p2E) for primes p ⌘ 1 mod (4) as unions of Paley subgraphs P (p2e) where e is odd. Although they are constructed from uncountably many mutually non-isomorphic fields, these infinite Paley graphs P (pE) and P (p2E) are all isomorphic to each other. In fact, we shall prove: Theorem 3.1. Each infinite Paley graph P (prE) for r = 1, 2 is isomorphic to the random graph R. 4 The countable random graph The countable random graph, or universal graph R was introduced by Erdős and Rényi [14, §3] in 1963 and Rado [24] in 1964. For details of its properties see [6, 7, 8] or [13, Sec- tion 9.6], and for some recent generalisations see [1, 15]. Theorem 3.1 should not be as surprising as it might at first appear, since in a sense we shall now explain ‘almost all’ countably infinite graphs are isomorphic to R. (However, the isomorphism class contain- ing R is very far from being random, in the colloquial sense of being typical, since it is just one among uncountably many isomorphism classes of countably infinite graphs.) Erdős and Rényi showed that if a graph has a countably infinite vertex set, and its edges are chosen randomly, then with probability 1 it has the following property U : given any two disjoint finite sets A and B of vertices of , there is a vertex which is a neighbour of each vertex in A and a non-neighbour of each vertex in B. They used this to show that is symmetric (has a non-identity automorphism) with probability 1 (by contrast with the finite case, where they showed that a random graph of order n is symmetric with probabil- ity approaching 0 as n ! 1). In fact, a similar argument shows that any two countably infinite graphs with property U are isomorphic: one can construct an isomorphism between them by using U to extend, by a back-and-forth argument, one vertex at a time, any iso- morphism between finite induced subgraphs, such as a single vertex in each of them. (See, for example, [21, Theorem 2.4.2], which in the language of model theory shows that the theory of graphs with property U is satisfiable and @0-categorical, and hence complete and decidable.) Thus any two graphs constructed randomly as above are isomorphic with probability 1. As a model of R one can therefore take any countably infinite graph with property U . For instance, Rado [24] constructed a ‘universal graph’, in which every countable graph is embedded as an induced subgraph, by using the vertex set V = N (including 0), with vertices x < y adjacent if and only if 2x appears in the binary representation of y as a sum of distinct powers of 2; this easily implies property U . 4 Art Discrete Appl. Math. 4 (2021) #P2.05 For an alternative model of R, let the vertex set V be the (countably infinite) set of all primes p ⌘ 1 mod (4), and define distinct vertices p and q to be adjacent if and only if q is a quadratic residue mod (p), that is, the Legendre symbol ( qp ) = 1. By quadratic reciprocity, which states that (pq )( q p ) = 1 for primes p, q ⌘ 1 mod (4), this is a symmetric relation, so it defines an undirected graph. To show that this graph has property U , given disjoint finite subsets A and B of V , for each prime a 2 A choose an integer na which is a quadratic residue mod (a), and for each prime b 2 B choose an integer nb which is a non-residue mod (b). By the Chinese Remainder Theorem, the simultaneous congruences n ⌘ 1 mod (4) and n ⌘ nc mod (c) for all c 2 C := A [ B have a unique solution n mod (d) where d = 4 Q c2C c, and by a theorem of Dirichlet this congruence class contains a prime (infinitely many, in fact). This gives a vertex in V adjacent to all the vertices a 2 A and to none of the vertices b 2 B, as required. 5 Proof of Theorem 3.1 In order to prove Theorem 3.1 it is sufficient to prove that the infinite Paley graphs P (prE) all have property U . Any pair of disjoint finite sets of vertices A and B are contained in some finite subfield Fq of FprE . As noted earlier, since E is infinite Fq is contained in finite subfields Fq0 (q0 = qf ) of FprE for infinitely many odd f . It is therefore sufficient for us to show that for all sufficiently large powers q0 of q there is an element x 2 Fq0 such that x a is a quadratic residue in Fq0 for all a 2 A and x b is a non-residue in Fq0 for all b 2 B. We will adapt an argument used by Blass, Exoo and Harary [2] to obtain a similar result concerning the family of Paley graphs P (p) for primes p ⌘ 1 mod (4). Given such subsets A and B of Fq , let S be the set of all x 2 Fq0 satisfying the above condition. Let C := A [ B, let n = |C|, and let : Fq0 ! C be the quadratic residue character of Fq0 , defined by (x) = 1, 1 or 0 as x is a quadratic residue, a non-residue or 0. Note that (xy) = (x)(y) for all x, y 2 Fq0 . For each x 2 Fq0 \ C we have Y a2A (1 + (x a)). Y b2B (1 (x b)) = ( 2n if x 2 S, 0 otherwise. (5.1) It follows that S is non-empty if and only if s := X x 62C Y a2A (1 + (x a)). Y b2B (1 (x b)) ! > 0. Summing over all x 2 Fq0 instead, let us define t := X x2Fq0 Y a2A (1 + (x a)). Y b2B (1 (x b)) ! . Expanding the product on the right-hand side, we have t = X x 1 + X x X a (x a) X x X b (x b) + · · · , G. A. Jones: Infinite Paley graphs 5 where the first term is q0 and the second and third are 0. To aid our consideration of the remaining terms, let us write C = {c1, . . . , cn}. Then it follows from the above that |tq0|  X i1 0 for all sufficiently large q0, as required. 6 Remarks on the proof 1. Weil proved in [28] that if is a multiplicative character of order d of a finite field Fq (one whose values are the dth roots of 1 in C), and f(x) is a polynomial of degree k over Fq not of the form cg(x)d for any c 2 Fq and g(x) 2 Fq[x], then X x2Fq (f(x))  (k 1)pq. (6.1) (See [26, p. 53], for example.) Replacing q with q0, taking to be the quadratic residue character, which has degree d = 2, and taking f(x) = (x ci1) · · · (x cik) we obtain the estimate (5.2) used above. In fact, when k = 2 we have an exact value in (5.2): this is Jacobsthal’s Lemma [17], used by Paley in [23], which states that X x2Fq (x u)(x v) = 1 for all u 6= v in Fq . This can be proved in a few simple lines by using the substitution w = (x v)/(x u). The parameters = (q 5)/4 and µ = (q 1)/4 for the strongly regular graph P (q) then follow by a simple version of the calculation used in Section 5. 2. The argument used to prove Theorem 3.1 in fact shows that |S| = s 2n ⇠ q 0 2n as q0 ! 1, n fixed, 6 Art Discrete Appl. Math. 4 (2021) #P2.05 which is what one would expect for Paley graphs on heuristic grounds, regarding adjacency or non-adjacency of vertices as independent events with equiprobable outcomes. Bollobás and Thomason [3] have given a more precise estimate, equivalent in our notation to |S| q 0 2n  1 2 (n 2 + 21n) p q0 + n 2 . 3. In [2] Blass, Exoo and Harary, working with the Paley graphs P (p) for primes p ⌘ 1 mod (4), needed to show that given any integer n 1, if p is sufficiently large then for any disjoint n-element sets A and B of vertices of P (p) there is a vertex x adjacent to every a 2 A and to no b 2 B. Their argument (based on one for tournaments by Graham and Spencer [16]) was similar to that used in Section 5, except that in place of Weil’s character sum estimate for fields Fq they used one by Burgess [5], that if p is prime and c1, . . . , ck are distinct elements of Fp, then X x2Fp (x c1) . . .(x ck)  (k 1)pp where is the quadratic residue character (Legendre symbol) mod (p). 4. Chung [11] has given some generalisations of the character sum estimates by Weil and Burgess, with applications to the discrepancy of finite graphs, including the Paley graphs; for any graph this is the maximum, over all s, of the difference between the maximum number of edges of an s-vertex subgraph and the average for that s. Estimating character sums is a major activity; Paley himself was an early contributor in [22], but this was in connection with number theory (specifically Dirichlet series), not graph theory. 5. For prime powers q ⌘ 1 mod (4) the construction in Section 2 yields the Paley tour- nament T (q), a complete graph Kq with directed edges, and the construction in Section 3 yields, for each prime p ⌘ 1 mod (4) and infinite set E satisfying the conditions given there, an infinite Paley tournament T (pE). Again, there are uncountably many of these objects, but a slight adaptation of the preceding arguments shows that they are all isomor- phic to the countable random tournament; a model of this can be obtained by applying the construction in Section 4 to primes p, q ⌘ 1 mod (4), where quadratic reciprocity now gives (pq )( q p ) = 1. 6. Peter Cameron [9] has suggested a more general construction using ultraproducts of finite fields, rather than direct limits, together with Łoś’s Theorem, to approximate the ran- dom graph (see also [20, Example 1.3.6], based on asymptotic classes and ultraproducts); this has the advantage of allowing finite fields of different characteristics to be used, thus yielding fields of characteristic 0. 7 Automorphism groups It follows from a theorem of Carlitz [10] that the automorphism group AutP (q) of P (q) is the subgroup AL1(q) of index 2 in AL1(q) consisting of the transformations t 7! at + b (a, b 2 Fq, (a) = 1, 2 GalFq) of the vertex set Fq , where GalFq is the Galois group or automorphism group of Fq , a cyclic group of order logp q generated by the Frobenius automorphism t 7! tp. The affine G. A. Jones: Infinite Paley graphs 7 transformations (those elements with = 1) and the translations (those with = 1 and a = 1) form normal subgroups AHL1(q) (‘H’ for ‘half’) and T1(q) of AL1(q) with AL1(q) AHL1(q) > T1(q) > 1, and the abelian quotients in this series show that AL1(q) is solvable, of derived length at most 3. One might hope that the automorphism group of P (prE) for r = 1 or 2 would have a similar structure. Clearly it contains the subgroup AL1(prE) of index 2 in AL1(prE) consisting of the transformations t 7! at + b (a, b 2 FprE , (a) = 1, 2 GalFprE ). Here GalFprE is not the direct limit of the groups GalFpre for e 2 E, but their in- verse limit: this can be identified with the (uncountable) subgroup of the cartesian productQ e2E GalFpre consisting of those elements whose coordinates re 2 GalFpre are consis- tent with the restriction mappings GalFprf ! GalFpre induced by inclusions Fpre ✓ Fprf for e dividing f 2 E. As in the finite case, this group AL1(prE) is solvable, of derived length 3. However, the facts that P (prE) ⇠= R and that AutR acts transitively on isomorphism classes of finite induced subgraphs of R (by the back-and-forth argument used in Section 4) destroy any hope that this subgroup might be the whole of AutP (prE). Indeed, far from being solvable, AutR has been shown by Truss [27] to be a simple group, and to contain a subgroup isomorphic to the symmetric group on a countably infinite set. 8 Generalised Paley graphs In 2009 Lim and Praeger [19] introduced generalised Paley graphs Pd(q), where q is a prime power pe and d divides q 1 (for convenience, we have changed their notation). Again the vertex set is Fq , but now vertices x and y are adjacent if and only if x y is contained in the unique subgroup D of index d in the multiplicative group F⇤q , consisting of the non-zero dth powers. To give an undirected graph we assume that if q is odd then the order (q 1)/d of D is even. For example, taking d = 2 gives the Paley graphs P (q) = P2(q). The construction in Section 3 carries through in the obvious way to give infinite gener- alised Paley graphs Pd(prE) where r is the multiplicative order of the prime p mod (2d) (or mod (d) if p = 2), except that we now need E to consist of integers e coprime to d. The proof of Theorem 3.1 also carries through, provided we take to be a multiplicative char- acter of Fqe of degree d (equivalently with kernel D), and replace the factor 1 + (x a) in equation (5.1) with 1 + (x a) + (x a)2 + · · ·+ (x a)d1 = d1Y j=1 ((x a) !j) where ! is a primitive dth root of 1 in C; again we can apply Weil’s estimate, now in the more general form (6.1) given in Remark 1, to show that Pd(prE) ⇠= R. The remarks in Section 7 about automorphism groups also apply here, though it should be noted that, as shown in [19], there are examples where d does not divide p 1 and AutPd(q) is significantly larger than the obvious analogue of AL1(q). 8 Art Discrete Appl. Math. 4 (2021) #P2.05 9 Symmetry versus asymmetry The main aim of Erdős and Rényi in [14] was to consider, in the contexts of finite and countably infinite graphs, the balance between symmetric and asymmetric graphs, those with and without a non-identity automorphism. Most of the paper concerns finite graphs, and here they proved, in a very precise sense, that not only are most graphs asymmetric, but in fact they are on average a long way from being symmetric. For a finite graph G = (V,E) they defined A(G) to be the least number of edge-changes (insertions or deletions) required to convert G into a symmetric graph on V . We may identify G with its edge set E, regarded as an element of the power set P(V (2)) = (F2)V (2) of the set V (2) of 2-element subsets of V ; the Hamming distance between two graphs (V,E) and (V,E0), with respect to the basis consisting of the graphs with one edge, is |EE0| where denotes symmetric difference, so A(G) is the distance from G to the nearest symmetric graph on V . For distinct vertices u and v in G Erdős and Rényi defined uv to be the number of vertices w 6= u, v adjacent to just one of u and v. By making uv edge-changes one can give u and v the same neighbours, allowing an automorphism transposing them and fixing all other vertices, so A(G)  min u 6=v uv. By a simple counting argument they showed that if G has order n then min u 6=v uv  b n 1 2 c, (9.1) so that A(G)  bn 1 2 c. They then showed that ‘most’ graphs G of order n have A(G) close to b(n 1)/2c, so that they are very far from being symmetric. As an aside they defined a -graph to be one achieving equality in (9.1), and noted that the graphs P (q) have this property: indeed, uv = (q 1)/2 for all pairs u 6= v in P (q), another simple consequence of Jacobsthal’s Lemma. Of course, these graphs are exceptional from this point of view, in that they satisfy A(P (q)) = 0. By contrast, Erdős and Rényi showed in the last part of their paper that ‘most’ count- ably infinite graphs are symmetric. Indeed, it follows from their construction of R and the alternative one due to Rado [24] that most such graphs are isomorphic to R and are there- fore highly symmetric: for example, AutR acts transitively on isomorphic finite induced subgraphs, and hence has rank 3 on the vertices. In fact, one can show that this group is un- countable, for instance by choosing a prime p ⌘ 1 mod (4) and taking E = {qn | n 1} in Section 3 for some odd prime q, so that by our remarks in Section 7 AutR contains a copy of GalP (pE) = lim GalP (pq n ) ⇠= lim Z/qnZ ⇠= Zq, the uncountable group of q-adic integers. ORCID iDs Gareth A. Jones https://orcid.org/0000-0002-7082-7025 G. A. Jones: Infinite Paley graphs 9 References [1] O. Angel and Y. Spinka, Geometric random graphs on circles, 2019, https://arxiv.org/ abs/1912.06770. [2] A. Blass, G. Exoo and F. Harary, Paley graphs satisfy all first-order adjacency axioms, J. Graph Theory 5 (1981), 435–439, doi:10.1002/jgt.3190050414. [3] B. Bollobás and A. Thomason, Graphs which contain all small graphs, European J. Combin. 2 (1981), 13–15, doi:10.1016/s0195-6698(81)80015-7. [4] A. Brouwer, Paley graphs, https://www.win.tue.nl/˜aeb/graphs/Paley. html. [5] D. A. Burgess, On character sums and primitive roots, Proc. London Math. Soc. 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ISSN 2590-9770 The Art of Discrete and Applied Mathematics 4 (2021) #P2.06 https://doi.org/10.26493/2590-9770.1367.cb8 (Also available at http://adam-journal.eu) Minimal generating orbit sets of torsion elements in GLn(Z)⇤ Jianchun Wu† , Yonghu Zheng Department of Mathematics, Soochow University, Suzhou 215006, CHINA Received 11 May 2020, accepted 11 August 2020, published online 15 March 2021 Abstract We construct minimal generating orbit sets for torsion elements in GLn(Z) for n  4. Keywords: generating orbit sets, rank, torsion, generalized linear group. Math. Subj. Class.: 15B36, 11C20 1 Introduction Let Zn = Z ⇥ · · · ⇥ Z be a direct product of n copies of the ring of integers Z. We consider Zn as an additive group and it is known as the free abelian group of rank n. An element v 2 Zn is called an integer vector of size n and can be written as a column vector v = [v1, ..., vn]T . The standard basis of Zn is {e1, ..., en} where ei = [0, ..., 1, ..., 0]T (i = 1, ..., n) having 1 in the i-th position and 0 otherwise. Denote the set of n⇥ n matrices over Z by Mn(Z). An unimodular matrix of size n is an element A 2 Mn(Z) having determinant ±1. All unimodular matrices of size n form a group with the operation of matrix multiplication. It is known as the general linear group GLn(Z). That is to say GLn(Z) = {A 2 Mn(Z)| detA = ±1}. An element A 2 GLn(Z) induces an automorphism of Zn by A : Zn ! Zn v 7! Av and in fact GLn(Z) is the automorphism group of Zn. The orbit of v 2 Zn by A is the set {Akv | k 2 Z}. It generates a subgroup of Zn whose elements are integral linear combinations of finitely many elements in {Akv | k 2 Z}. ⇤The authors thank the referees for their work and suggestions that have helped improve this paper. †The author is partially supported by National Natural Science Foundation of China (No. 11571246). E-mail addresses: wujianchun@suda.edu.cn (Jianchun Wu), 20174207001@stu.suda.edu.cn (Yonghu Zheng) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 2 Art Discrete Appl. Math. 4 (2021) #P2.06 Definition 1.1. Let S be a subset of Zn and A 2 GLn(Z). The orbit subgroup OGA(S) on S by A is the subgroup of Zn generated by {Akv | k 2 Z, v 2 S}. If OGA(S) is the full group Zn, we call S a generating orbit set of A. Remark 1.2. If S generates Zn, then it is a generating orbit set for any A 2 GLn(Z) while any generating orbit set of the identity matrix In must generate Zn. We consider the question that if S is a generating orbit set of a fixed element A 2 GLn(Z), how many elements must S contain. Among all generating orbit sets of A those having minimal cardinalities are called minimal generating orbit sets. So the question is: Question 1.3. Let A 2 GLn(Z), what is mA = min{#S | OGA(S) = Zn} the cardinality of a minimal generating orbit set of A? In this paper, we determine mA and construct an explicit minimal generating orbit set for each torsion element (defined below) A in GLn(Z) (n  4). Definition 1.4. A 2 GLn(Z) is called a torsion element if Am is the identity matrix In for some positive integer m. A torsion element A has order d if d is the minimal positive integer such that Ad = In. There are two other interpretations of mA. The first one is from ring and module theory. Fixing A 2 GLn(Z), Zn can be viewed as a module over the ring of integral polynomials Z[X] by defining p(X) · v = p(A)v (p(x) 2 Z[X], v 2 Zn). Denote the rank (i.e., the minimal number of generators) of the Z[X]-module Zn by rkZ[X](Zn), then mA = rkZ[X](Zn). Since A is invertible, we can also define Zn as a module over the ring of integral Laurent polynomials Z[X,X1] by the same rule and mA is equal to the rank of the Z[X,X1]-module Zn. The second one is from combinatorial group theory. Let G = Zn oA Z be the semidirect product of Zn and Z determined by A, the rank (i.e., the minimal number of generators) of G is denoted by rk(G), then mA = rk(G) 1 (see [6, Corollary 2.4]). It is obvious that mA  n since we can choose S to be a basis of Zn. mA is computable for each A 2 GL2(Z) ([6, Corollary 3.3]) because in this case mA = 1 or 2 and by Lemma 2.4 we know when mA=1. To the authors’ knowledge, it is not known whether mA is computable for each non-torsion element A 2 GLn(Z) even when n = 3. Another fact is that Proposition 1.5. mA is a conjugacy invariant. That is to say mXAX1 = mA for any X 2 GLn(Z). Proof. If S is a generating orbit set of A, then {Xv | v 2 S} is a generating orbit set for XAX 1 and has the same cardinality as S. So mXAX1  mA. For the same reason mA = mX1XAX1X  mXAX1 . Remark 1.6. Throughout this paper, conjugation always means integral conjugation. This is to say, B is conjugate to A by X if and only if B = XAX1 for some X 2 GLn(Z). If the basis of Zn is changed, an automorphism of Zn may correspond to different matrices, but they are (integrally) conjugate to each other. J. Wu and Y. Zheng: Minimal generating orbit sets of torsion elements in GLn(Z) 3 The problem of the classification of conjugacy classes in GLn(Z) has long history and is not completely solved, see [1, 4, 5, 8]. For more results about classifying torsion elements up to conjugacy, see [7, 9, 10]. After introducing some facts in Section 2, we use classification results (Theorems 2.6, 2.8 and 2.10) from [7, 9, 10] and construct a minimal generating orbit set for each represen- tative of conjugacy class of torsion elements in Section 3. The results are listed in Table 1, Table 2 and Table 3 respectively. 2 Preliminary Definition 2.1. The companion matrix of a monic polynomial f(x) = a0 + a1x + · · · + an1x n1 + xn is the square matrix C(f) = 2 666664 0 0 . . . 0 a0 1 0 . . . 0 a1 0 1 . . . 0 a2 ... ... . . . ... ... 0 0 . . . 1 an1 3 777775 . Remark 2.2 ([3, p147]). The characteristic and minimal polynomials of a companion ma- trix do coincide. Definition 2.3. The companion matrix of the characteristic polynomial of A 2 GLn(Z) is called the companion of A. The following observation is standard (see [6]), we prefer to write down a proof here by using our notations. Lemma 2.4. Suppose A 2 GLn(Z), then mA = 1 if and only if A is conjugate to its companion. Proof. Denote the characteristic polynomial of A by p(x) and the companion of A by C. Note that C = C(p). If mA = 1, suppose S = {v} is a minimal generating orbit set of A, then {Akv|k 2 Z} generates Zn. By Cayley-Hamilton theorem, for any k 2 Z, Akv is an integral linear combination of {v,Av, ..., An1v}. That is to say {v,Av, ..., An1v} is a basis of Zn. The matrix of the automorphism A : Zn ! Zn under this basis is the companion matrix C(p) and so A is conjugate to C(p). Conversely, if A is conjugate to its companion C = C(p), it is easy to check that {e1, Ce1, ..., Cn1e1} generates Zn where e1 = [1, 0, ..., 0]T . So mC = 1 and mA = mC by Proposition 1.5. For convenience, suppose A 2 GLm(Z), B 2 GLn(Z), denote by A B the matrix A 0 0 B 2 GLm+n(Z). Similarly, for u 2 Zm, v 2 Zn, the column vector  u v 2 Zm Zn = Zm+n is denoted by u v. We have (AB)(u v) = AuBv. 4 Art Discrete Appl. Math. 4 (2021) #P2.06 Lemma 2.5. (1) mA = mA; (2) max{mA,mB}  mAB  mA +mB . Proof. (1) A and A have same generating orbit sets. (2) For A 2 GLm(Z), B 2 GLn(Z), suppose S is a minimal generating orbit set of A B and each element s 2 S is written as  sA sB where sA 2 Zm, sB 2 Zn. Then SA = {sA|s 2 S} is a generating orbit set for A and so mA  |SA|  |S| = mAB . Similarly, mB  mAB . Suppose SA and SB are minimal generating orbit sets of A 2 GLm(Z) and B 2 GLn(Z) respectively. Let S = {u 0|u 2 SA} [ {0 v|v 2 SB} ⇢ Zm+n. Then S is a generating orbit set for AB and mAB  |S| = |SA|+ |SB | = mA +mB . Theorem 2.6 ([7], [8, Chapter IX], [10, Lemma 1.6]). Each torsion element in GL2(Z) is conjugate to one of the matrices listed in the second row of the table below where K =  1 0 0 1 , U =  1 1 0 1 , W =  0 1 1 1 , J =  0 1 1 0 . Order d = 1 d = 2 d = 3 d = 4 d = 6 Representative of conjugacy class I2 I2, K, U W J W Remark 2.7. In some literatures, the representative U is replaced by V =  0 1 1 0 . Theorem 2.8 ([9, page 173, 174, 184]). Each torsion element in GL3(Z) is conjugate to one of the matrices listed in the second column of the table below where K =  1 0 0 1 , V =  0 1 1 0 , W =  0 1 1 1 , J =  0 1 1 0 , E1 = ⇥ 0 1 ⇤ . Order Representative of conjugacy class d = 1 I3 d = 2 I3, I1 (I2), (I1) V , (I1) I2, I1 (V ) d = 3 I1 W ,  0 I2 I1 0 d = 4 I1 J ,  I1 E1 0 J , (I1 J),  I1 E1 0 J d = 6 I1 (W ), (I1)W , (I1 W ),  0 I2 I1 0 Remark 2.9. Since  0 1 1 1 is conjugate to W =  0 1 1 1 by  1 1 1 0 , the order 6 element W1 = 2 4 1 0 0 0 0 1 0 1 1 3 5 in [9, page 184] is conjugate to I1 (W ) and we choose the latter as a representative for simplicity and replace W2 (in the same page) by (I1)W for the same reason. J. Wu and Y. Zheng: Minimal generating orbit sets of torsion elements in GLn(Z) 5 Theorem 2.10 ([10, page 492]). Each torsion element in GL4(Z) is conjugate to one of the matrices listed in the second column of the table below where K =  1 0 0 1 , U =  1 1 0 1 , W =  0 1 1 1 , J =  0 1 1 0 , E =  0 1 0 0 and C5, C8, C10, C12 are the companion matrices of the cyclotomic polynomials 5(x) = x4 + x3 + x2 + x+1, 8(x) = x4+1, 10(x) = x4x3+x2x+1, 12(x) = x4x2+1 respectively. Order Representative of conjugacy class d = 1 I4 d = 2 I4, K (I2), U (I2), I2 (I2), K U , U U , I2 K, I2 U d = 3 W W , I2 W ,  I2 E 0 W d = 4 J J , I2 J ,  I2 E 0 J , (I2) J ,  I2 E 0 J , K J ,  K E 0 J ,  K I2 0 J ,  K I2 E 0 J , U J ,  U E 0 J ,  U I 0 J d = 5 C5 d = 6 (W W ), I2 (W ), (I2)W , (I2 W ),  I2 E 0 W , K W ,  K E 0 W , U W ,  U E 0 W , (K W ),  K E 0 W , (U W ),  U E 0 W , W (W ),  W E 0 W d = 8 C8 d = 10 C10 d = 12 C12, J W , J (W ) 3 Constructing minimal generating orbit sets In this section, we determine mA and construct a minimal generating orbit set SA for A being a representative of conjugacy class in GLn(Z) (n  4). Results for other torsion elements can be obtained as follows: Given a torsion element B in GLn(Z) (n  4), then B is conjugate to some represen- tative A listed in Theorem 2.6, 2.8, 2.10 by some X 2 GLn(Z). There is an algorithm for deciding whether two elements in GLn(Z) are conjugate. Theorem 3.1 ([2, Theorem A]). Given two matrices A,B 2 GLn(Z), there is an algo- rithm to deciding whether there exists a matrix X 2 GLn(Z) such that B = XAX1. If 6 Art Discrete Appl. Math. 4 (2021) #P2.06 the answer is “yes” the algorithm constructs a conjugating matrix X . So we can determine A and construct X at the same time by the algorithm from The- orem 3.1 through enumerating A in the lists. Now mB = mA and a minimal generating orbit set for B can be obtained as {Xv | v 2 SA}. 3.1 Conjugacy classes of companion matrices In Theorem 2.6, 2.8 and 2.10, the representatives of conjugacy classes of torsion elements in the tables are not always companion matrices. The algorithm in Theorem 3.1 for deciding whether A 2 GLn(Z) is conjugate to its companion is hard to be conducted by hand. But for torsion elements, especially when n is small, there is a simpler method to handle most cases. We describe it as the following three steps. Step 1: Compute the characteristic polynomial and minimal polynomial of A, if they are not the same, then A is not conjugate to its companion, moreover mA 2, otherwise go to Step 2. Step 2: If there is only one representative with the same characteristic and minimal polynomials as those of A in the table, then A is conjugate to its companion because the companion of A is also a torsion element and has the same characteristic and minimal polynomials, otherwise go to Step 3. Step 3: If we can find a minimal generating orbit set with only one element for A, then by Lemma 2.4 A is conjugate to its companion, otherwise we apply the algorithm in Theorem 3.1. 3.2 Direct sum We already know mAB  mA +mB by Lemma 2.5, the equality is not always true, for instance: Example 3.2. Suppose W =  0 1 1 1 , the companion matrix of f(x) = x2 + x + 1, so mW = 1 and by Lemma 2.5, mW = 1. Now for I1 (W ) = 2 4 1 0 0 0 0 1 0 1 1 3 5, one can check {[1, 1, 0]T } is a minimal generating orbit set. Another example is shown in Example 3.9. But for some special cases the equality can be obtained. We have: Proposition 3.3. If mA = 1 then mAA = 2. Proof. By Lemma 2.5, mAA  2. The characteristic polynomial and minimal polyno- mial of AA are not the same, so AA is not conjugate to its companion and mAA 2 by Lemma 2.4. 3.3 Construction Let e1 = [1, 0, 0, 0] T , e2 = [0, 1, 0, 0] T , e3 = [0, 0, 1, 0] T , e4 = [0, 0, 0, 1] T be the standard basis elements of Z4. J. Wu and Y. Zheng: Minimal generating orbit sets of torsion elements in GLn(Z) 7 Proposition 3.4. Suppose A = I2 W = 2 664 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 3 775, then mA = 3. Proof. Suppose some generating orbit set of A contains only u = [u1, u2, u3, u4]T and v = [v1, v2, v3, v4]T , that is to say, {Aku,Akv|k 2 Z} generates Z4. By Cayley-Hamilton the- orem, for any k 2 Z, w 2 Z4, Akw is an integral linear combination of w,Aw,A2w,A3w. Since A3 = I4, we have hu,Au,A2u, v, Av,A2vi = Z4. Note that A2 = 2 664 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 3 775 so h 2 664 u1 u2 u3 u4 3 775 , 2 664 u1 u2 u4 u3 u4 3 775 , 2 664 u1 u2 u3 + u4 u3 3 775 , 2 664 v1 v2 v3 v4 3 775 , 2 664 v1 v2 v4 v3 v4 3 775 , 2 664 v1 v2 v3 + v4 v3 3 775i = Z 4 , if and only if h 2 664 u1 u2 u3 u4 3 775 , 2 664 0 0 u3 + u4 2u4 u3 3 775 , 2 664 0 0 2u3 u4 u4 + u3 3 775 , 2 664 v1 v2 v3 v4 3 775 , 2 664 0 0 v3 + v4 2v4 v3 3 775 , 2 664 0 0 2v3 v4 v4 + v3 3 775i = Z 4 , if and only if h 2 664 u1 u2 u3 u4 3 775 , 2 664 0 0 u3 + u4 2u4 u3 3 775 , 2 664 0 0 3u3 3u4 3 775 , 2 664 v1 v2 v3 v4 3 775 , 2 664 0 0 v3 + v4 2v4 v3 3 775 , 2 664 0 0 3v3 3v4 3 775i = Z 4 , if and only if  u1 v1 u2 v2 2 GL2(Z) and {  u3 + u4 2u4 u3 ,  3u3 3u4 ,  v3 + v4 2v4 v3 ,  3v3 3v4 } generates Z2. We take mod 3, then {  u3 + u4 2u4 u3 ,  v3 + v4 2v4 v3 } generates Z23. It is impossible because the determinant of  u3 + u4 v3 + v4 2u4 u3 2v4 v3 is 0 mod 3. We have proved that mA > 2. One can check S = {e1, e2, e4} is a generating orbit set for A. So S is minimal and mA = 3. Remark 3.5. Suppose A = (Im) Inm 2 GLn(Z), then mA = n. More or less this fact is trivial, it can be proved by using similar method for proving Proposition 3.4. 8 Art Discrete Appl. Math. 4 (2021) #P2.06 3.4 Summary Now we can determine mA for every torsion element A 2 GLn(Z) (n  4) by using methods discussed in Section 3.1, 3.2, 3.3. We summarize them as the following reasons: Trivial: For some trivial cases, we have mA = n, for instance A = In, A = (Im) Inm and so on. R1: A is conjugate to its companion or we can find a generating orbit set for A with only one element, so mA = 1. It can be done through the three steps in Section 3.1. R2: A is not conjugate to its companion and we can find a generating orbit set for A with two elements, so mA = 2. It can be done through steps in Section 3.1 and for some special cases by Proposition 3.3. R3: We can prove mA > 2 and find a generating orbit set for A with three elements, so mA = 3. We have shown how to do it through an example by Proposition 3.4 in Section 3.3. The argument there can be applied to similar cases. We will show part of the procedure by Example 3.9, 3.10 after the statements of main results in Section 3.5. 3.5 Determination of mA for torsion elements in GLn(Z) (n  4) 3.5.1 Torsion elements in GL2(Z) The representatives of conjugacy classes of torsion elements in GL2(Z) are already listed in Theorem 2.6. It is easy to find minimal generating orbit sets for these elements and we have Theorem 3.6. For a given torsion element A 2 GL2(Z), mA is determined and a mini- mal generating orbit set is constructed explicitly. The results are listed in Table 1 where K =  1 0 0 1 , U =  1 1 0 1 , W =  0 1 1 1 , J =  0 1 1 0 , e1 = [1, 0]T , e2 = [0, 1]T . Table 1: Minimal generating orbit sets for torsion elements in GL2(Z) Order Representative ofConjugacy Class mA Minimal Generating Orbit Set Characteristic polynomial; Minimal Polynomial Reason d=1 I2 2 {e1, e2} (x 1)2; x 1 Trivial d=2 I2 2 {e1, e2} (x+ 1)2; x+ 1 Trivial K = I1 (I1) 2 {e1, e2} x2 1; x2 1 Trivial U 1 {e2} x2 1; x2 1 R1 d=3 W 1 {e1} x2 + x+ 1; x2 + x+ 1 R1 d=4 J 1 {e1} x2 + 1; x2 + 1 R1 d=6 W 1 {e1} x2 x+ 1; x2 x+ 1 R1 In Table 1 (also Table 2 and 3 below), the representatives of conjugacy classes of torsion J. Wu and Y. Zheng: Minimal generating orbit sets of torsion elements in GLn(Z) 9 elements are listed in the second column and their orders in the first column, mA and explicit minimal generating orbit sets are listed in the third and fourth columns respectively. For the convenience of checking the results through the procedure in Section 3.4, we record the characteristic and minimal polynomials in the fifth column and the reasons for determining mA in the last column. 3.5.2 Torsion elements in GL3(Z) The representatives of conjugacy classes of torsion elements in GL3(Z) are already listed in Theorem 2.8 and we have Theorem 3.7. For a given torsion element A 2 GL3(Z), mA is determined and a mini- mal generating orbit set is constructed explicitly. The results are listed in Table 2 where K =  1 0 0 1 , V =  0 1 1 0 , W =  0 1 1 1 , J =  0 1 1 0 , E1 = ⇥ 0 1 ⇤ , e1 = [1, 0, 0]T , e2 = [0, 1, 0]T , e3 = [0, 0, 1]T . Table 2: Minimal generating orbit sets for torsion elements in GL3(Z) Order Representative ofConjugacy Class mA Minimal Generating Orbit Set Characteristic polynomial; Minimal Polynomial Reason d=1 I3 3 {e1, e2, e3} (x 1)3; x 1 Trivial d=2 I3 3 {e1, e2, e3} (x+ 1)3; x+ 1 Trivial I1 (I2) 3 {e1, e2, e3} x3 + x2 x 1; x2 1 Trivial (I1) V 2 {e1, e2} x3 + x2 x 1; x2 1 R2 (I1) I2 3 {e1, e2, e3} x3 x2 x+ 1; x2 1 Trivial I1 (V ) 2 {e1, e2} x3 x2 x+ 1; x2 1 R2 d=3 I1 W 2 {e1, e2} x3 1; x3 1 R2 0 I2 I1 0 1 {e1} x3 1; x3 1 R1 d=4 I1 J 2 {e1, e3} x3 x2 + x 1; x3 x2 + x 1 R2  I1 E1 0 J 1 {e2} x3 x2 + x 1; x3 x2 + x 1 R1 (I1) J 2 {e1, e3} x3 + x2 + x+ 1; x3 + x2 + x+ 1 R2  I1 E1 0 J 1 {e2} x3 + x2 + x+ 1; x3 + x2 + x+ 1 R1 d=6 I1 (W ) 1 {e1 + e2} x3 2x2 + 2x 1; x3 2x2 + 2x 1 R1 (I1)W 1 {e1 + e2} x3 + 2x2 + 2x+ 1; x3 + 2x2 + 2x+ 1 R1 10 Art Discrete Appl. Math. 4 (2021) #P2.06 Order Representative ofConjugacy Class mA Minimal Generating Orbit Set Characteristic polynomial Minimal Polynomial Reason (I1 W ) 2 {e1, e2} x3 + 1; x3 + 1 R2  0 I2 I1 0 1 {e1} x3 + 1; x3 + 1 R1 3.5.3 Torsion elements in GL4(Z) The representatives of conjugacy classes of torsion elements in GL4(Z) are already listed in Theorem 2.10 and we have Theorem 3.8. For a given torsion element A 2 GL4(Z), mA is determined and a mini- mal generating orbit set is constructed explicitly. The results are listed in Table 3 where K =  1 0 0 1 , U =  1 1 0 1 , W =  0 1 1 1 , J =  0 1 1 0 , C5, C8, C10, C12 are the companion matrices of the cyclotomic polynomials 5(x) = x4 + x3 + x2 + x+1, 8(x) = x4+1, 10(x) = x4x3+x2x+1, 12(x) = x4x2+1 respectively and e1 = [1, 0, 0, 0] T ,e2 = [0, 1, 0, 0] T , e3 = [0, 0, 1, 0] T , e4 = [0, 0, 0, 1] T . Table 3: Minimal generating orbit sets for torsion elements in GL4(Z) Order Representative ofConjugacy Class mA Minimal Generating Orbit Set Characteristic polynomial Minimal Polynomial Reason d=1 I4 4 {e1, e2, e3, e4} (x 1)4; x 1 Trivial d=2 I4 4 {e1, e2, e3, e4} (x+ 1)4; x+ 1 Trivial K (I2) 4 {e1, e2, e3, e4} x4 + 2x3 2x 1; x2 1 Trivial U (I2) 3 {e2, e3, e4} x4 + 2x3 2x 1; x2 1 R3 I2 (I2) 4 {e1, e2, e3, e4} x4 2x2 + 1; x2 1 Trivial K U 3 {e1, e2, e4} x4 2x2 + 1; x2 1 R3 U U 2 {e2, e4} x4 2x2 + 1; x2 1 R2 I2 K 4 {e1, e2, e3, e4} x4 2x3 + 2x 1; x2 1 Trivial I2 U 3 {e1, e2, e4} x4 2x3 + 2x 1; x2 1 R3 d=3 W W 2 {e1, e3} x4 + 2x3 + 3x2 + 2x+ 1; x2 + x+ 1 R2 I2 W 3 {e1, e2, e4} x4 x3 x+ 1; x3 1 R3 J. Wu and Y. Zheng: Minimal generating orbit sets of torsion elements in GLn(Z) 11 Order Representative ofConjugacy Class mA Minimal Generating Orbit Set Characteristic polynomial Minimal Polynomial Reason  I2 E 0 W 2 {e2, e4} x4 x3 x+ 1; x3 1 R2 d=4 J J 2 {e1, e3} x4 + 2x2 + 1; x2 + 1 R2 I2 J 3 {e1, e2, e4} x4 2x3 + 2x2 2x+ 1; x3 x2 + x 1 R3  I2 E 0 J 2 {e2, e3} x4 2x3 + 2x2 2x+ 1; x3 x2 + x 1 R2 (I2) J 3 {e1, e2, e4} x4 + 2x3 + 2x2 + 2x+ 1; x3 + x2 + x+ 1 R3  I2 E 0 J 2 {e2, e3} x4 + 2x3 + 2x2 + 2x+ 1; x3 + x2 + x+ 1 R2 K J 3 {e1, e2, e4} x4 1; x4 1 R3 K E 0 J 2 {e2, e3} x4 1; x4 1 R2  K I2 0 J 2 {e1, e3} x4 1; x4 1 R2  K I2 E 0 J 2 {e1, e3} x4 1; x4 1 R2 U J 2 {e2, e3} x4 1; x4 1 R2 U E 0 J 2 {e2, e3} x4 1; x4 1 R2  U I 0 J 1 {e4} x4 1; x4 1 R1 d=5 C5 1 {e1} x4 + x3 + x2 + x+ 1; x4 + x3 + x2 + x+ 1 R1 d=6 (W W ) 2 {e1, e3} x4 2x3 + 3x2 2x+ 1; x2 x+ 1 R2 I2 (W ) 2 {e1, e2 + e3} x4 3x3 + 4x2 3x+ 1; x3 2x2 + 2x 1 R2 (I2)W 2 {e1, e2 + e3} x4 + 3x3 + 4x2 + 3x+ 1; x3 + 2x2 + 2x+ 1 R2 (I2 W ) 3 {e1, e2, e4} x4 + x3 + x+ 1; x3 + 1 R3  I2 E 0 W 2 {e2, e4} x4 + x3 + x+ 1; x3 + 1 R2 K W 2 {e1, e2 + e3} x4 + x3 x 1; x4 + x3 x 1 R2  K E 0 W 2 {e1, e2 + e4} x4 + x3 x 1; x4 + x3 x 1 R2 12 Art Discrete Appl. Math. 4 (2021) #P2.06 Order Representative ofConjugacy Class mA Minimal Generating Orbit Set Characteristic polynomial Minimal Polynomial Reason U W 2 {e2, e3} x4 + x3 x 1; x4 + x3 x 1 R2  U E 0 W 1 {e2 e3} x4 + x3 x 1; x4 + x3 x 1 R1 (K W ) 2 {e1, e2 + e3} x4 x3 + x 1; x4 x3 + x 1 R2  K E 0 W 2 {e2, e4} x4 x3 + x 1; x4 x3 + x 1 R2 (U W ) 2 {e2, e3} x4 x3 + x 1; x4 x3 + x 1 R2  U E 0 W 1 {e2 + e3} x4 x3 + x 1; x4 x3 + x 1 R1 W (W ) 2 {e1, e3} x4 + x2 + 1; x4 + x2 + 1 R2  W E 0 W 1 {e4} x4 + x2 + 1; x4 + x2 + 1 R1 d=8 C8 1 {e1} x4 + 1; x4 + 1 R1 d=10 C10 1 {e1} x4 x3 + x2 x+ 1; x4 x3 + x2 x+ 1 R1 d=12 C12 1 {e1} x4 x2 + 1; x4 x2 + 1 R1 J W 1 {e1 + e3} x4 + x3 + 2x2 + x+ 1; x4 + x3 + 2x2 + x+ 1 R1 J (W ) 1 {e1 + e3} x4 x3 + 2x2 x+ 1; x4 x3 + 2x2 x+ 1 R1 Example 3.9. The last row in Table 3 is an order 12 representative J (W ) = 2 664 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 1 3 775 , its characteristic polynomial and minimal polynomial are both x4 x3 + 2x2 x + 1, so it may be conjugate to its companion. But there are no other elements in Table 3 with the same characteristic and minimal polynomials as those of J (W ), so it must be conjugate to its companion. By Lemma 2.4, J (W ) has a minimal generating orbit set with only one element and we can easily find it. J. Wu and Y. Zheng: Minimal generating orbit sets of torsion elements in GLn(Z) 13 Example 3.10. In Table 3, A =  U I 0 J = 2 664 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 0 3 775 is of order 4, the characteristic polynomial and minimal polynomial are both x41, but other representatives (including KJ , U J and so on) have the same characteristic and minimal polynomials x 4 1, so we do Step 3 in Section 3.1 and find the orbit of {e4 = [0, 0, 0, 1]T } by A generates Z4: he4, Ae4, A2e4, A3e4i = * 2 664 0 0 0 1 3 775 , 2 664 0 1 1 0 3 775 , 2 664 0 1 0 1 3 775 , 2 664 1 0 1 0 3 775 + = Z4. This is to say, A is conjugate to its companion, but we don’t need this fact since we already found a minimal generating orbit set. ORCID iDs Jianchun Wu https://orcid.org/0000-0002-0802-5291 Yonghu Zheng https://orcid.org/0000-0002-8152-8100 References [1] E. C. Dade, The maximal finite groups of 4 ⇥ 4 integral matrices, Illinois J. Math. 9 (1965), 99–122, doi:10.1215/ijm/1256067584. [2] F. J. 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ISSN 2590-9770 The Art of Discrete and Applied Mathematics 4 (2021) #P2.07 https://doi.org/10.26493/2590-9770.1375.12a (Also available at http://adam-journal.eu) On deformation of polygonal dendrites preserving the intersection graph Dmitry Drozdov Novosibirsk State University, Novosibirsk, Russia Mary Samuel Indian Institute of Information Technology Lucknow, Lucknow, India Andrei Tetenov⇤ Novosibirsk State University, Novosibirsk, Russia, and Gorno-Altaisk State University, Gorno-Altaisk, Russia Received 6 August 2020, accepted 29 September 2020, published online 15 March 2021 Abstract Let S = S1, ..., Sm be a system of contracting similarities of R2. The attractor K(S) of the system S is a non-empty compact set satisfying K = S1(K) [ ... [ Sm(K). We consider contractible polygonal systems S which are defined by a finite family of polygons whose intersection graph is a tree and therefore the attractor K(S) is a dendrite. We find conditions under which a deformation S 0 of a contractible polygonal system S has the same intersection graph and therefore the attractor K(S 0) is a self-similar dendrite which is isomorphic to the attractor K of the system S . Keywords: Self-similar dendrite, generalized polygonal system, attractor, intersection graph, index diagram. Math. Subj. Class.: 28A80 Introduction Our work is part of a series of works aimed at studying self-similar dendrites in Rd. A dendrite K is called self-similar, if it can be represented as an union K = S1(K) [ ... [ ⇤Supported by Russian Foundation of Basic Research project 18-01-00420 E-mail addresses: dimalek97@yandex.ru (Dmitry Drozdov), marysamuel@iiitl.ac.in (Mary Samuel), atet@mail.ru (Andrei Tetenov) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 2 Art Discrete Appl. Math. 4 (2021) #P2.07 Sm(K) of its images under a finite system S = {S1, ..., Sm} of contracting similarities of Rd. This representation defines a self-similar structure on K. The history of fractal geometry contains remarkable examples of dendrites such as Hata tree [8], Vicsek set, Pentadendrite and others, as well as certain important theorems obtained by various authors. In 1985, M. Hata [8] obtained a criterion for the connectedness of self-similar sets and proved that if a dendrite K is the attractor of a system of weak contractions in a complete metric space, then the set of its endpoints is infinite. In 1990, C. Bandt showed in [3] that the Jordan arcs connecting pairs of points of a postcritically finite self-similar dendrite are self-similar, and the set of possible dimensions of such arcs is finite. J. Kigami in his work [11] considered finite topological subgraphs of a dendrite generated by a given set of points and studied shortest path metrics on self-similar dendrites. Many examples of dendrites are defined as fractal squares [14]. A special case of such squares, fractal labyrinths, were studied in [5, 6, 7]. A systematic approach to the study of self-similar dendrites required finding answers to the following questions: What topological constraints characterize the class of dendrites generated by systems of similarities in Rd? What are the explicit algorithms for construct- ing self-similar dendrites? What are the metric and analytical properties of morphisms of self-similar structures on dendrites? Starting from the simplest and most obvious settings that have been used implicitly by many authors [3, 19], we considered systems S = {S1, .., Sm} of contracting similarities in R2, defined by some polygon P 2 R2 and a family of polygons Pi = Si(P )⇢P , which (i) intersect each other only by their vertices; (ii) whose union eP = nS i=1 Pi is a contractible set; and (iii) whose set of vertices contains all the vertices of P . We called such systems S contractible P -polygonal systems [15, 16, 17, 18]. The attractor K of each such system S is a dendrite. The properties of such dendrites directly follow from the geometry of the system of polygons Pi and from the properties of the intersection graph (S) of this system. For example, the upper bound for the order of ramification points of K depends only on the values of angles and on the number of vertices of P . The addresses of ramification points of K and their order can be derived from the intersection graph (S) of the system S. Moreover, by [18, Theorem 27], if two contractible polygonal systems S, S0 have iso- morphic intersection graphs (S),(S0), then there is a homeomorphism ' : K ! K 0 of their attractors, which defines the isomorphism of self-similar structures (K, S) and (K 0, S0). As we define in Section 2, the system S is called a generalised P -polygonal system if the system of polygons Pi = Si(P ) satisfies the conditions (ii)–(iii), but the requirement Pi⇢P is omitted. If S is a contractible P -polygonal system and S0 is a generalized P 0-polygonal system, the intersection graphs (S),(S0) are isomorphic and for any i, j, P 0 i \ P 0 j = S0 i (K 0) \ S 0 j (K 0), then there is a homeomorphism ' : K ! K 0 which defines the isomorphism of (K, S) and (K 0, S0). In the current paper we find the conditions under which a small deformation S0 of a contractible polygonal system S preserves the intersection graph of the system S. In Section 1 we expound basic definitions of the theory of self-similar sets and the D. Drozdov, M. Samuel and A. Tetenov: On deformation of polygonal dendrites 3 definition of contractible polygonal systems. In Section 2 we make a reference to some of our results from [20] on self-similar sets possessing one-point intersection property. Then we proceed to generalized polygonal systems and prove the Theorem 2.6 providing the condition D0 under which the attractor of a generalized polygonal system is a dendrite. Then we give the Definition 2.7 of -deformations of a contractible P -polygonal system S and prove the Theorem 2.9 which shows that if a -deformation S0 of the polygonal system S satisfies the condition D0 then the self-similar structures (K, S) and (K 0, S0) are isomorphic. So the question arises, how to ensure that the system S0 satisfies Condition D0 which guarantees that the deformation preserves the intersection graph of the system S. The answer is given by two following statements. First is the Parameter Matching Theorem 3.12 which gives necessary condition for one- point intersections at common vertices P 0 i \ P 0 j . The second is the Small Deformations Theorem 4.6 which states that there is a number ⌫ > 0, depending on the system S such that if is not greater than ⌫ and if the Parameter Matching condition holds, then the system S0 has the same intersection graph as S. In Section 3 we introduce the Index Diagram G(S) of a P -polygonal system S. This allows us to understand the role of cyclic vertices of the polygon P and to show that each vertex A of P is subordinate to some cyclic vertex B. In subsection 3.2 we define the standard neighbourhood UA of a non-cyclic vertex A and therefore of any of its images Sj(A). Proceeding to generalized polygonal systems we prove the existence of invariant arcs at cyclic points of these systems and prove the Parameter Matching Theorem. Section 4 is rather technical. We find the estimates for -deformations, and in the end we prove the Small Deformations Theorem 4.6. 1 Preliminaries 1.1 Self-similar sets Definition 1.1. Let S = {S1, S2, . . . , Sm} be a system of (injective) contraction maps on the complete metric space (X, d). A nonempty compact set K⇢X is called the attractor of the system S, if K = mS i=1 Si(K). Throughout the whole paper, the maps Si 2 S are supposed to be similarities and the set X to be R2. We will use complex notation for the point on the plane, so each similarity will be written as Sj(z) = qjei↵j (z zj) + zj , where qj = LipSj and zj = fix(Sj). For a system S, we denote qmin = min{qj , j = 1, ...,m} and qmax = max{qj , j = 1, ...,m}. The system S defines its Hutchinson operator T by the equation T (A) = mS i=1 Si(A). By Hutchinson’s Theorem [9], the attractor K is unique for S and for any compact set A⇢X the sequence Tn(A) converges to K. We also call the set K self-similar with respect to S. The set I = {1, 2, ...,m} is called the set of indices, while I⇤ = 1S n=1 I n is the set of all finite I-tuples, or multi-indices j = j1j2...jn. The length n of the multi-index j = j1...jn is denoted by |j| and ij denote the concatenation of respective multi-indices. We say i @ j, if j = il for some l 2 I⇤; if i 6@ j and j 6@ i, i and j are incomparable. For any j 2 I⇤ we write Sj = Sj1j2...jn = Sj1Sj2 ...Sjn ; given a set A, we denote Sj(A) by Aj. The set GS = {Sj, j 2 I⇤} is the semigroup, generated by S. 4 Art Discrete Appl. Math. 4 (2021) #P2.07 The set of all infinite sequences I1 = {↵ = ↵1↵2 . . . ,↵i 2 I} is called the index space of the system S; the map ⇡ : I1 ! K called the index map, sends each ↵ to the point ⇡(↵) = 1T n=1 K↵1...↵n . If ⇡(↵) = x, then ↵ is called an address of the point x. For each address ↵ of a point x 2 K, the point xn = S1↵1...↵n(x) is called the n-th predecessor of the point x, and the sequence x1, x2, ... is called the sequence of predecessors of x. Along with the system S we will consider its n-th refinement S(n) = {Sj, j 2 In}. The Hutchinson operator of the system S(n) is equal to Tn. The pair (K, S) is called a self-similar structure. A map f : K ! K 0 agrees with the structures (K, S), (K 0, S0) if for any x 2 K and any i 2 I , f Si(x) = S0i f(x). If the map f is a homeomorphism, then it defines the isomorphism of self-similar structures (K, S) and (K 0, S0). Definition 1.2. The system S satisfies the open set condition (OSC) if there exists a non- empty open set O⇢X such that the sets Si(O), {1  i  m} are pairwise disjoint and are contained in O. For any i, j 2 I⇤, i @ j iff Si(O)Sj(O) and i and j are incomparable, iff Si(O) \ Sj(O) = ?. The union C of all intersections Si(K) \ Sj(K), i, j 2 I, i 6= j is called the critical set of the system S. The set of all predecessors of the points in C, @K = {x 2 K : for some j 2 I⇤, Sj 2 C is called the self-similar boundary of the set K. The post-critical set P of the system S is the set of all ↵ 2 I1 such that ⇡(↵) 2 @K. A system S is called post-critically finite (PCF) if its post-critical set P is finite [12]. Thus, if the system S is post-critically finite then the self-similar boundary @K is a finite set V = ⇡(P) such that for any non-comparable i, j 2 I⇤, Ki \Kj = Si(V) \ Sj(V). 1.2 Dendrites A dendrite is a locally connected continuum containing no simple closed curve [4, 13]. The order Ord(p,X) of the point p with respect to a dendrite X is the number of components of the set X \ {p}. Points of order 1 in a dendrite X are called end points of X; a point p 2 X is called a cut point of X if X \ {p} is disconnected; points of order at least 3 are called ramification points of X . A continuum X is a dendrite iff X is locally connected and uniquely arcwise connected. 1.3 Contractible polygonal systems Let P ⇢ R2 be a finite polygon homeomorphic to a disk, VP = {A1, . . . , AnP } be the set of its vertices. Let ⌦(P,A) denote the angle with vertex A in the polygon P . We consider a system of similarities S = {S1, . . . , Sm} in R2 such that: (D1) for any i 2 I set Pi = Si(P ) ⇢ P ; (D2) for any i 6= j, i, j 2 I, Pi T Pj = VPi T VPj and #(VPi T VPj ) < 2; (D3) VP ⇢ S i2I Si(VP ); (D4) the set eP = mS i=1 Pi is contractible. D. Drozdov, M. Samuel and A. Tetenov: On deformation of polygonal dendrites 5 K1 K2 K3 K4 K5 K6 K7 K8 K9 K1 K2 K3 K4 K5 K6 K7 K8 K9 Figure 1: A polygonal system and its intersection graph. Definition 1.3. The system S satisfying the conditions (D1)–(D4), is called a contractible P -polygonal system of similarities. Theorem 1.4 ([18, Theorem 12]). The attractor K of a contractible P -polygonal system of similarities S is a dendrite. It was proved in [16, Theorem 4], that: 1. Every contractible polygonal system satisfies (OSC), where we can take Ṗ as an open set. 2. Pj⇢Pi iff j A i, and if i @ j, then Si(VP ) \ Pj⇢Sj(VP ). If i, j 2 I⇤ are incom- parable, then Pi \ Pj is either empty or is a common vertex of the polygons Pi and Pj. 3. All the vertices of P lie in K, therefore the set GS(VP ) of vertices of the polygons Pj is contained in K and dense in K. 4. Every point x 2 K\GS(VP ) has a unique address. 2 Generalized polygonal systems 2.1 One-point intersection systems and intersection graph When considering generalized polygonal systems, we will rely on a number of definitions and statements from our paper [20]: Definition 2.1. Let A = {Ai, i 2 I} be a finite system of compact sets such that for any i 6= j 2 I , #Ai \ Aj  1. Then A is a system of sets with one-point intersection (or a FIP1-system of sets). Let B be the set of points of pairwise intersection of the sets Ai, and Bi = Ai \B. Definition 2.2. The intersection graph (A) of a FIP1-system of sets A is a bipartite graph (A,B;E) for which e = {Ai, B} 2 E if and only if B 2 Ai. We call Ai 2 A white vertices and B 2 B - black vertices. The set N(Ai) of the neighbors of a white vertex Ai is Bi, whereas for a black vertex B, N(B) = {Ai : B 2 Ai}. Since B is the intersection point of at least two sets Ai, we have deg(B) 2. 6 Art Discrete Appl. Math. 4 (2021) #P2.07 P2 P P1 K1 K2 Figure 2: A generalized polygonal system and its attractor. Definitions 2.1 and 2.2 can be applied to the systems of contractions and their attractors. Let S = {S1, ..., Sm} be a system of injective contractions in a complete metric space X and K be its attractor. Let A(S) = {K1, ...,Km} and An(S) = {Ki : i 2 In}. Definition 2.3. A system of injective contractions S is called a system with one-point in- tersection property (or FIP1-system of injective contractions), if the system of sets A(S) = {S1(K), ..., Sm(K)} is a FIP1-system of sets. Theorem 2.4 ([20, Th.1.7]). Let S be a system of injective contraction maps in a complete metric space X such that the intersection graph (S) is a tree. Then the attractor K of the system S is a dendrite. 2.2 Generalized polygonal systems If we omit the condition (D1) in the definition of contractible P -polygonal system S, we get the definition of a generalized P -polygonal system: Definition 2.5. A system S = {S1, ..., Sm} satisfying the conditions D2-D4, is called a generalized P -polygonal system of similarities. Theorem 2.6. Let S be a generalized P -polygonal system. If for any i, j 2 I Si(K) \ Sj(K) = Pi \ Pj , (D0) then (i) the attractor K of the system S is a dendrite; (ii) the system S satisfies OSC; (iii) the set of addresses ⇡1(x) of any point x 2 K is finite. Remark 1. If the generalized polygonal system S satisfies Condition D0, then S is a system with a connected attractor K which has one-point intersection property and whose self- similar boundary @K = VP is finite. Such systems were studied in our paper [20]. It was proved there that for any two incomparable multi-indices i, j 2 I⇤, the intersection Ki \Kj = Si(VP ) \ Sj(VP ) is either empty set or a singleton. D. Drozdov, M. Samuel and A. Tetenov: On deformation of polygonal dendrites 7 P1 P2 P3 P4 P5P6 P7 P8 P9 K1 K2 K3 K4 K5K6 K7 K8 K9 Figure 3: Example for the Remark 2 P3 P5 P2 P4 P1 P 0 3 P 0 5 P 0 2 P 0 4 P 0 1 Figure 4: Polygonal system S and its -deformation S0 Proof. (i) Indeed, it follows from formula (D0) that the intersections graphs ({Ki}) and ({Pi}) are isomorphic. Also, it follows from the properties (D2)-(D4) that the intersec- tion graph of generalized polygonal system ({Pi}) is a tree. Then, by Theorem 2.4, K is a dendrite. (ii) Since S is a system of contracting similarities in R2 which has finite intersection property and has connected attractor K then, by [2, Theorem 2], it satisfies the Open Set Condition. (iii) Finiteness of the set ⇡1(x) follows from (ii) and [20, Proposi- tion 2.3]. Remark 2. It is possible for a generalized P -polygonal system S not to satisfy the Condi- tion D0 and to have the attractor K which is a dendrite. The attractor K of a generalized polygonal system S in the picture below is a dendrite, but P7 \ P9 = ?, whereas K7 \K9 is a line segment. 2.3 -deformations of contractible polygonal systems Definition 2.7. Let > 0. A generalized P 0-polygonal system S0 = {S01, ..., S0m} is called a -deformation of the P -polygonal system S = {S1, ..., Sm}, if there is a bijection f : mS k=1 VPk ! mS k=1 VP 0k such that a) f |VP extends to a homeomorphism f̃ : P ! P 0; b) |f(x) x| < for any x 2 mS k=1 VPk ; c) f(Sk(x)) = S0k(f(x)) for any k 2 I and x 2 VP . 8 Art Discrete Appl. Math. 4 (2021) #P2.07 Since f̂ is a homeomorphism of polygons mapping vertices to vertices, we can assume that f̂ is a simplicial isomorphism of some triangulation of the polygon P whose vertex set is VP to equivalent triangulation of P 0 whose vertex set is VP 0 . By condition c), if i, j 2 I , A1, A2 2 VP and Si(A1) = Sj(A2), then S0i(f(A1)) = S 0 j (f(A2)). The same relation is fulfilled in the case when i, j 2 I⇤ are multi-indices: Lemma 2.8. If A1, A2 2 VP , i, j 2 I⇤ and Si(A1) = Sj(A2), then S0i(f(A1)) = S 0 j(f(A2)). Proof. Suppose that Si(A) = B 2 V eP for some A 2 VP , and let i = i1i2...in. We denote Sik+1...in(A) by Ak. Then we have a finite sequence of relations between B 2 V eP and the vertices Ak 2 VP : B = Si1(A1), A1 = Si2(A2), . . . An1 = Sin(A). (2.1) By c), the map f transforms these relations into B 0 = S0 i1 (A01), A 0 1 = S 0 i2 (A02), . . . A 0 n1 = S 0 in (A0). (2.2) This implies S0i(A 0) = B0. Moreover, if Si(A1) = Sj(A2) 2 V eP , then S 0 i(f(A1)) = S 0 j(f(A2)). Now suppose that Si(A1) = Sj(A2) and i = li0, j = lj0 and Si(A1) = Sj(A2) = Sl(B) for some B 2 V eP . Then Si0(A1) = Sj0(A2) = B, therefore S 0 i0(f(A1)) = S 0 j0(f(A2)) = f(B) and S 0 i(f(A1)) = S 0 j(f(A2)) = S 0 l(f(B)). Theorem 2.9. Let S0 be a -deformation of a contractible P -polygonal system S defined by the map f and let ⇡ : I1 ! K,⇡0 : I1 ! K 0 be respective address maps. (i) f has unique continuous extension f̂ : K ! K 0 such that f̂ ⇡ = ⇡0; (ii) if S0 satisfies Condition D0, then f̂ is a homeomorphism. Remark 3. The equality f̂ ⇡ = ⇡0 holds if and only if for any z 2 K and any i 2 I⇤, f̂(Si(z)) = S 0 i(f̂(z)). (2.3) Proof. The proof is similar to (cf. [1, Lemma 1]). First, we define the function f̂ which is a surjection of the dense subset GS(VP )⇢K to the dense subset GS0(VP 0)⇢K 0. Second, we show that f̂ is Hölder continuous on GS(VP ) and therefore has unique continuous extension to a surjection from K to K 0, which we denote by the same symbol f̂ . Thirdly, we show that the Condition D0 implies that f̂ is injective and therefore is a homeomorphism. 1. Define a map f̂(z) : GS(VP ) ! GS0(VP 0) by f̂(z) = S0i(f(S 1 i (z)), where z 2 Si(VP ). (2.4) As it follows from Lemma 2.8, if Si(A1) = Sj(A2) = z, then S0i(f(S 1 i (z))) = S 0 j(f(S 1 j (z))), so the map f̂ is well defined. Obviously f̂(GS(VP )) = GS0(VP 0), because if A0 2 VP 0 and z0 = S0i(A0), then there is a vertex A = f1(A0) 2 VP , therefore z0 = f̂(Si(A)). Moreover, for any z 2 GS(VP ) and i 2 I⇤, f̂(Si(z)) = S0i(f̂(z)) and if z1, z2 2 GS(VP ), i, j 2 I⇤ and Si(z1) = Sj(z2), then S0i(f̂(z1)) = S0j(f̂(z2)). D. Drozdov, M. Samuel and A. Tetenov: On deformation of polygonal dendrites 9 1 2 3 P1 P2 1 1 2 2 2 3 Figure 5: An example of a generalized polygonal system and its index diagram. It contains 3 equivalent cyclic vertices. 2. Let qk = LipSk, q0k = LipS 0 k , = min k2I log q0 k log qk . Then, following the proof [18, Theorem 27, step 4], in which we use K 0 instead of P 0, one can see that for any z1, z2 2 GS(VP ), |z01 z02|  2|K 0| (⇢0 · sin (↵0/2)) |z1 z2| . Therefore, the map f̂ can be extended to a Hölder continuous surjective mapping of the set K in K 0. Since for any z 2 K and any k 2 I , f̂(Sk(z)) = S0k(f(z)), then f̂ ⇡ = ⇡0. 3. Now suppose the system S0 satisfies Condition D0. Suppose that for some = i1i2... 2 I1and ⌧ = j1j2... 2 I1 f̂ ⇡() = f̂ ⇡(⌧ ). Then, if i1 6= j1, then, by Condition D0 , P 0 i1 \ P 0 j1 6= ?, resulting in Pi1 \ Pj1 = {B} for some B 2 V eP and ⇡() = ⇡(⌧ ) = B. Let now = l0 and ⌧ = l⌧ 0, and f̂ ⇡() = f̂ ⇡(⌧ ). Then, by the formula 2.3, f̂ ⇡(0) = f̂ ⇡(⌧ 0), so if the first indices in 0 and ⌧ 0 are different then ⇡() = ⇡(⌧ ) = Sl(B) for some B 2 V eP . This implies that the mapping f̂ is injective. Thus, f̂ is a homeomorphism of the compact sets K and K 0. 3 Parameter matching theorem 3.1 Cyclic vertices and the index diagram Definition 3.1. Let S = {Si, i 2 I} be a generalized P -polygonal system. The index diagram of the system S is an edge-labeled directed multigraph G = (VP , E, µ), where the vertex set of G is the set VP . Given A,B 2 VP , there is an edge e 2 E which is directed from A to B and is labeled by an index i 2 I , iff there is Si such that Si(B) = A. The labeling map µ : E ! I is defined by the equation µ(e) = i. We use the following notation for edges in directed graphs: if e is an edge in a graph G, directed from A to B, then ↵(e) = A and !(e) = B. A walk in G is a sequence of edges e1e2 . . . en . . . such that ↵(ek) = !(ek1) for all k > 1. The walk starts at ↵() = ↵(e1) and if the walk ends at an edge en, then !() = !(en). For a walk = e1e2 . . . en . . . we define µ() = µ(e1)µ(e2) . . . = i1i2 . . . By condition D3 for each vertex A 2 VP there is at least one edge starting from A, so the outdegree of each vertex 1, and for any vertex A 2 VP the set of infinite walks = e1e2..., starting from A is nonempty. 10 Art Discrete Appl. Math. 4 (2021) #P2.07 1 2 34 P1 P2 P3 P4 P5 1 2 34 4 12 3 2 3 4 1 2 1 3 4 1 2 3 4 1 2 34 324 1 Figure 6: Each walk in the index diagram arrives to some cyclic vertex 1 2 34 1 2 3 4 P1 P2 P3 P7 P8 P9 31 2 8 7 9 Figure 7: An example of disconnected index diagram. The vertex 3 is non-cyclic and is subordinate to the vertex 1 by the maps S7, S8 and S9. Consider the sequence A,A1, . . . , An, . . . of vertices of the walk . By finiteness of VP , there are k and l such that Al = Al+k. Then Al = Sil+1...il+k(Al). In this case Al is a cyclic vertex. It follows from D3 that all the predecessors of a vertex A 2 VP are also vertices of P . That is, if A 2 Si1...in(P ), then there is a vertex An such that A = Si1...in(An). Therefore for any two vertices A,B 2 VP the equality A = Si(B) holds iff there is a walk AB = e1e2e3 . . . en from A to B such that i = µ(AB). Consider some infinite walk = e1e2e3 . . ., and let An = !(en) be its vertices. As A = Si1...in(An), the sequence {An} is a sequence of predecessors of the vertex A. Due to equality A = lim n!1 Si1...in(P ), the infinite sequence µ() = i1 . . . in . . . 2 I1 is an address of the point A defined by the walk . Conversely, each address i1 . . . in, . . . of the point A is equal to µ() for some infinite walk starting from A. Definition 3.2. A vertex B 2 VP is called a cyclic vertex if there is a cycle B = e1 . . . ek such that ↵(B) = !(B) = B. The length k of the cycle B is called the order of the vertex B. We call B a basepoint of the cycle B . Remark 4. As we show in Proposition 3.5 (i), each cyclic vertex has outdegree 1 and therefore belongs to only one cycle. The multi-index j = µ(B) is the shortest of the multi-indices k satisfying Sk(B) = B. Conversely, if B is a vertex of P and j = j1 . . . jk is the shortest multi-index such that Sj(B) = B then j = µ(B) for some cycle B in G. D. Drozdov, M. Samuel and A. Tetenov: On deformation of polygonal dendrites 11 3 1 5 2 4 1 2 34 5 P1 P2 P3 P4 P5 P6 16 4 3 6 4 5 3 2 Figure 8: A more complicated example of index diagram. The vertex 5 is subordinate to the vertex 2 by the maps S3 and S56 and to the vertex 1 by the maps S2 and S54. Definition 3.3. A finite path ⌘ in G is called a pre-cyclic path if !(⌘) is a unique cyclic vertex in ⌘. Let i = µ(⌘). Then we say the point A = ↵(⌘) is subordinate to the point B = !(⌘) by means of the map Si. In other words, each pre-cyclic path ⌘ starting from a vertex A defines a pair (B,Si) = (!(⌘), Sµ(⌘)) such that B is a cyclic vertex and A = Si(B). Proposition 3.4. Let S be a generalised P -polygonal system of similarities. For each vertex A 2 VP there is a finite number of pairs (Bk, Sik), such that for each k the vertex A is subordinate to Bk by means of the map Sik . Proof. Let A 2 VP be a non-cyclic vertex. Consider the set {⌘1, . . . ⌘n} of all pre-cyclic paths starting from A. Let (Bk, Sik) be the pair, defined by ⌘k. Notice that if for some k, l, !(⌘k) = !(⌘l), then µ(⌘k) 6= µ(⌘l). Therefore all the pairs (Bk, Sik) are different. In the case when S is a generalized polygonal system, which satisfies condition (1) and particularly in the case when S is a contractible polygonal system, the vertex set VP and the index diagram G(S) have the following properties: Proposition 3.5. Let S be a generalized polygonal system satisfying the condition (1), then: (i) All the cyclic vertices of the index diagram G(S) have the outdegree equal to 1. (ii) Each cyclic vertex B is a basepoint of a unique cycle B . (iii) There is n such that all cyclic vertices of the system S(n) = {Si, i 2 In} have the order 1. Proof. Let the outdegree of some cyclic vertex B be greater than 1. This implies that there is a cycle B in G with a basepoint B and an edge e1 : ↵(e1) = B such that e1 /2 B . Consider an infinite walk ⌧ = e1e2e3..., starting from B. Then all the walks nB⌧ start from B and are pairwise different, therefore B has infinite set of addresses. Since a contractible polygonal system has finite intersection property and satisfies OSC, it follows from [20, Theorem 1.7], that the set of addresses of each vertex B is finite. The contradiction obtained proves (i). Let B be a cyclic vertex and let ⌧ be the periodic walk generated by the cycle B . By (i) it is the only infinite walk originating from B, which proves (ii). Moreover, µ(⌧) is the unique address of the point B. 12 Art Discrete Appl. Math. 4 (2021) #P2.07 W S1(W ) S 2 1(W ) S 3 1(W ) B A AC Figure 9: The decomposition (3.1) of the attractor K at the cyclic vertex B (left) and the decomposition (3.2) of the standard neighborhood UA for a non-cyclic vertex A (right). Notice that the order of the point B is 3, and the order of the point A is 9. Let G be the index diagram of the system S. Since S(n) is a generalized polygonal system satisfying condition (1), the set of vertices of its index diagram Gn is also VP , and the edges e0 going from A to B correspond to the walks ⌘0 = e1...en of length n for which ↵(⌘0) = A, !(⌘0) = B. Let n be the least common multiple of the orders of all cyclic vertices. Then for every walk ⌘ of length n outgoing from a cyclic vertex B, !(⌘) = B. Therefore, the order of any cyclic vertex in the system S(n) is equal to 1. 3.2 The structure of neighborhoods of points of the attractor of a contractible polyg- onal system 1. Cyclic vertices. Let B 2 VP be a cyclic vertex of the contractible polygonal system S and let B be the cycle with the basepoint B. Let j = µ(B). Then the similarity Sj is a homothety, and the angle ⌦B formed by the sides of P adjacent to B contains P . Moreover, assuming W = K \Sj(K) we obtain a decomposition of the set K to a disjoint union K = {B} [ 1G n=0 S n j (W ). (3.1) 2. Non-cyclic vertices. Let A 2 VP be a non-cyclic vertex. Let {⌘1, . . . ⌘n} be the set of all pre-cyclic paths starting from A and {(Bk, Sik), k = 1, ..., n} be the corresponding set of pairs (Bk, Sik). Notice that if k 6= l, then Kik \Kil = {A}. Suppose contrary. Then by Remark 1, one of the multi-indices, say ik, satisfies ik @ il. Then ⌘k is a subpath of ⌘l, which is impossible because ⌘l is pre-cyclic. The sets Sik(K \ Bk) are pairwise disjoint and lie inside pairwise disjoint angles Sik(⌦Bk) for which A is the common vertex. The set UA = nS k=1 Sik(K) is a neighborhood of A in K. It is called the standard neighbor- hood of the vertex A. Let k be the cycle whose basepoint is Bk and let jk = µ(k). Assuming Wk = D. Drozdov, M. Samuel and A. Tetenov: On deformation of polygonal dendrites 13 C Figure 10: The decomposition (3.2) of the standard neighborhood UC of a point C 2 GS(VP ). The order of the point C is 24. K\Sjk(K), we get the canonical decomposition of the standard neighborhood UA: UA = {A} [ nG k=1 Sik 1G l=0 S l jk(Wk) (3.2) 3. The points of the set GS(VP ). Consider the point A 2 GS(VP ). Definition 3.6. We say that A 2 GS(VP ) is subordinate to a cyclic vertex B by means of a map Si = Si1...ik if A = Si1...ik(B) and for any l < k the point Sil+1...ik(B) is not a cyclic vertex. Proposition 3.7. For any point A 2 GS(VP ) there is a unique finite set of pairs UA = {(Bk, Sik), k = 1, . . . , NA} such that the point A is subordinate to a cyclic vertex B by means of a map Si if and only if the pair (B,Si) lies in UA. For any non-equal k, l  NA, Sik(K) \ Sil(K) = {A}. The set UA = NAS k=1 Sik(K) is a neighborhood of A in K; it admits the decomposi- tion (3.2). 3.3 Parameter matching theorem Let A be a cyclic vertex of a generalized P-polygonal system S. In this case the map Si for which Si(A) = A, need not be a homothety and we have to define the rotation parameter for such map. Though the rotation angle ↵i of the map Si is formally defined up to 2n⇡, in the case of polygonal systems the integer n is uniquely defined by the set eP and depends on its geometric configuration. Proposition 3.8. Let S be a generalized P -polygonal system satisfying Condition D0 and let A be a cyclic vertex of the polygon P . Then there is a vertex B 2 VP and a multi- index i 2 I⇤ such that Si(A) = A and the Jordan arc AB⇢K satisfies the inclusion Si(AB)⇢AB . 14 Art Discrete Appl. Math. 4 (2021) #P2.07 Proof. Let j be the shortest multi-index for which A = Sj(A). Let W = K\Sj(K), then K = {A} [ 1G n=0 S n j (W ). Let Q = S1j (W̄ \ Sj(K)). From the Remark 1 we see that Q⇢VP \{A}. The vertex A cannot belong to Q. Otherwise, there would be a piece Ki such that Ki \ Kj = {A}, so for any k, l, Skj (Ki) \ Slj(Ki) = {A}. In this case A would be a infinite order ramification point in K, which is impossible by Theorem 2.6. Since K is a dendrite, for any B 2 Q there is a unique arc AB⇢K. Let B0B be the smallest of those subarcs of AB for which B0 2 Kj. Then B0 2 Sj(Q). Define a map : Q ! Q by the formula (B) = S1j (B0). Then for any B 2 Q, Sj(A (B))⇢AB . Further, for any n, Snj (A n(B))⇢AB . Thus is a mapping of a finite set Q to itself. There is n and B 2 Q, such that n(B) = B. Then Snj (AB)⇢AB . So we put Si = Snj . Definition 3.9. The arc AB is called an invariant arc of the cyclic vertex A. Let A be a cyclic vertex and AB be its invariant arc and Si(A) = A. Let B0 = Si(B). We denote by ↵ the total increase of the argument of z A as z travels along AB from B to B0. This gives a unique representation Si(z) = qiei↵(z A) +A. Remark 5. The following picture shows how the angle ↵ depends on the geometric con- figuration of the system S, though the similarity which fixes A and sends B to B0 is the same. ↵ = ⇡ A A B B0 B0 B ↵ = ⇡ Definition 3.10. The number A = ↵ ln qi is called the parameter of the cyclic vertex A. Definition 3.11. Generalized P -polygonal system S of similarities satisfies the Parameter Matching Condition, if for any B 2 [m i=1VPi and for any cyclic vertices A,A0 such that for some i, j 2 I⇤, Si(A) = Sj(A0) = B, the equality A = A0 holds. From Propositions 3.4 and 3.8 and V.V.Aseev’s Lemma on disjoint periodic arcs [1, Lemma 3.1] we come to the following Parameter Matching Theorem: Theorem 3.12. Let S be a generalized P -polygonal system whose attractor K is a dendrite. Then the system S satisfies the Parameter Matching Condition. Proof. Let S be a generalized polygonal system whose attractor K is a dendrite. Let C 2 [m i=1VPi and A,A0 2 VP be such cyclic vertices that for some i, j 2 I , Si(A) = Sj(A0) = C. Denote the images Si(K) and Sj(K) by Ki, Kj respectively. Without loss of generality we can suppose that the point C has coordinate 0 in C. Since for some D. Drozdov, M. Samuel and A. Tetenov: On deformation of polygonal dendrites 15 ⇢2 ⇢1 ↵0 ⇢0 Figure 11: The choice of parameters ↵0, ⇢0, ⇢1 and ⇢2 for the polygonal system. i, j 2 I⇤, Si(A) = A and Sj(A0) = A0, the maps Sb1 = SiSiS1i and Sb2 = SjSjS 1 j have C as their fixed point and Sb1(Ki)⇢Ki and Sb2(Kj)⇢Kj . Let Sb1(z) = qiei↵iz and Sb2(z) = qjei↵jz. So the parameters of the vertices A and A0 will be 1 = ↵i log qi and 2 = ↵j log qj . Let AB⇢K and A0B0⇢K be invariant arcs for the vertices A and A0. Let also 1 = Si(AB) and 2 = Sj(A0B0). Then Sb1(1)⇢1 and Sb2(2)⇢2. From [1, Lemma 3.1] it follows that if 1 \ 2 = {C}, then 1 = 2. 4 Small deformation theorem 4.1 Main parameters of a contractible polygonal system For any set X⇢R2 or point A by V"(X) (resp.V"(A)) we denote "-neighborhood of the set X (resp. of the point A) in the plane. Parameter ⇢0: By ⇢0 > 0, we denote the smallest of all distances between the points of non-intersecting polygons Pi, Pj and distances between the vertices A 2 VP and the points of polygons Pi 63 A: (i) for any vertex A 2 VP , V⇢(A) T Pk 6= ? ) A 2 Pk; (ii) for any x, y 2 P , such that there are Pk, Pl : x 2 Pk, y 2 Pl and Pk T Pl = ?, d(x, y) ⇢0. Parameters ⇢1,⇢2: Let C 2 VP̃ be a non-cyclic vertex. Let UC be its standard neighborhood and ŨC = kS l=1 Sil(Wl). Then ⇢1 and ⇢2 are chosen so that for any C 2 VP̃ , the set ŨC is contained in the ring ⇢1 < |z C|  ⇢2. Parameter ↵0: ↵0 is the smallest possible angle between those sides of the polygons Pi, Pj , i, j 2 I , which have a common vertex. Notation for maps of cyclic vertices. In the case when S is a contractible P -polygonal system all of whose cyclic vertices are of order 1, we order the indices in I and enumerate the vertices in VP in such a way that each cyclic vertex Al corresponds to a homothety Sl(z) = ql(z Al) + Al. In this case, K is within the angle ⌦(P,Al) and K\{Al} = 1F n=0 S n l (Wl). 16 Art Discrete Appl. Math. 4 (2021) #P2.07 4.2 Estimate of and Main Theorem Initial assumptions. Let S = {S1, ..., Sm} be a contractible P -polygonal system, and the map f defines a -deformation of the system S to a generalized P 0-polygonal system S0 = {S01, ..., S0m}, where Sk(z) = qkei↵k(z zk)+ zk and S0k(z) = q0kei↵ 0 k(z z0 k )+ z0 k . We assume that diamP = 1, and > 0 is such that < qmin/8 and < (1 qmax)/8. (4.1) The estimates for qk = |q0k qk| and ↵k = |↵0k ↵k| under the deformation f are given by the following Lemma. Lemma 4.1. For any k 2 I , qk < 3 and ↵k < C↵, where C↵ = 2.1(1 + 1/qmin). (4.2) Proof. Choose vertices A,B of the polygon P such that |B A| = 1. For the images of A we use the notation Ak = Sk(A), A0 = f(A) and A0k = f(Ak) = S 0 k (A0), and similar notation for the vertex B. We estimate the increments qk = |q0k qk| and ↵k = |↵0k ↵k| for Bk Ak B A = qke i↵k and B 0 k A0 k B0 A0 = q 0 k e i↵ 0 k . Since f is a -deformation, |(Bk Ak)| 2  |(B0k A0k)|  |(Bk Ak)| + 2. This implies 3qmin/5 < qmin 2 1 + 2 < qk 2 1 + 2  q0 k  qk + 2 1 2 < qmax + 2 1 2 < 1 + 3qmax 3 + qmax . (4.3) Since ↵ 0 i ↵i = arg B 0 k A0 k Bk Ak arg B 0 A0 B A , (4.4) we get |↵0 k ↵k|  arcsin 2 + arcsin 2 qk . (4.5) Substituting the inequalities (4.1) to (4.3) and (4.5) and taking into account the inequal- ity 0 < arcsinx < 1.05x, which holds for any 0 < x < 0.5, we obtain the estimates qk = |q0k qk| < 2(1 + qk) 1 2 < 3 and ↵k = |↵ 0 k ↵k| < C↵, (4.6) where C↵ = 2.1(1 + 1/qmin). Let V(P ) denote the -neighborhood of the polygon P . Lemma 4.2. Let 1 = 4 1 qmax , and U = V1(P ). Then (1) For any k 2 I , Sk(U)⇢U and S0k(U)⇢U . (2) For any z 2 U , |S0 k (z) Sk(z)| < C, where C = 6.5 + 1.5C↵. D. Drozdov, M. Samuel and A. Tetenov: On deformation of polygonal dendrites 17 Proof. (1) By Definition 2.7, V(Pk)P 0k, V(P 0k)Pk, V(P )P 0 and V(P 0)P . Thus, S0 k (P 0)⇢V(Pk)⇢V(P ). Therefore S0k(P )⇢V2(Pk)⇢V2(P ). Since S0 k is a similarity, S0 k (V⇢(P ))⇢V2+q0k⇢(P ) for any positive ⇢. If ⇢ 2 1 q0 k , then 2 + q0 k ⇢  ⇢. Therefore S0 k (V⇢(P ))⇢V⇢(P ). From the inequality q0 k < qmax + 2 1 2 it follows that 2 1 q0 k  2(1 2) 1 qmax 4 < 4 1 qmax . The latter implies (1). Moreover, due to (4.1), 1 < 1/2. (2) Take z 2 U and consider the difference S0 k (z) Sk(z). It can be represented in the form S0 k (A) Sk(A) + (q0kei↵ 0 k qkei↵k)(z A). Therefore |S0 k (z) Sk(z)| < |S0k(A) Sk(A)|+ (|q0k qk|+ qk|ei↵ 0 k ei↵k |)|z A|. (4.7) Since |zA| < 1+ 1 < 1.5, |S0k(A) Sk(A)| < 2, and |ei↵ 0 k ei↵k |  |↵0 k ↵k|, the right hand side of (4.7) is no greater than 2 + 1.5(3 + C↵). Applying the Displacement Theorem [10, Theorem 17] to S, S0 and U we obtain the following statement. Proposition 4.3. Let ⇡,⇡0 be the address maps for the systems S and S0 respectively. 1. For any 2 I1, |⇡0() ⇡()| < CK where CK = 2C 1 qmax . (4.8) 2. If the system S 0(n) satisfies D2–D4, then it is a (CK)-deformation of the system S(n). Remark 6. Let S0 = {S01, ..., S0m} be a -deformation of the contractible P -polygonal system S. Let A 2 Sj(VP ) for some j 2 I . Let g(z) = zA+A0 and S00k = g S0k g1. Then S00 = {S001 , ..., S00m} is a 2-deformation of the system S, for which A00 = A, K 00 = g(K 0), P 00j = g(Pj). Since g is a translation, the estimates (4.1) and (4.2) for S 00 remain the same with the same , while |⇡00() ⇡()| < (CK + 1). Thus we will denote 2 = (CK + 1). Taking into account the Propositions 3.4 and 4.3, it is sufficient to prove the Theorem for the case when all cyclic vertices of the system S have order 1. Proposition 4.4. Let the initial assumptions be fulfilled, and let Sk(z) = qk(zAk)+Ak be the homothety fixing Ak 2 VP . Then the parameter k of the similarity S0k satisfies the inequality |k| < C, where C = 2.1(1 + 1/qmax) log(3 + qmax) log(3qmax + 1) . (4.9) Proof. From Lemma 4.1 we have |k|  arcsin 2 + arcsin 2 qk | log(qk + 2) log(1 2)| . (4.10) Given the inequalities (4.3) and (4.5), we get (4.9). 18 Art Discrete Appl. Math. 4 (2021) #P2.07 Lemma 4.5. Let S be a contractible P -polygonal system whose cyclic vertices have order 1 and S0 be its -deformation. Then if 2.1 2 ⇢1 + log ⇢2 + 2 ⇢1 2 < ↵0 and 22 < ⇢0, (4.11) then the system S0 satisfies condition (D0). Remark 7. On the assumption that 2 < ⇢1/4, and 2 < (1 ⇢2)/4, the inequality (4.11) holds if 2.1 2 ⇢1 + log 1 + 3⇢2 3⇢1 < ↵0. Proof. Take a vertex B 2 V eP . Taking into account Remark 6, we can assume that B = 0 and f(0) = 0, so B0 = B = 0. The decomposition of a standard neighborhood UB of the point B has the form UB = {B} [ kG l=1 Sjl 1G n=0 S n l (Wl). (4.12) The maps S̄l = SjlSilS 1 jl are homotheties with a fixed point B such that Kjl\{B} = 1G n=0 S̄ n l (Wl). (4.13) Similarly, let W 0 l = f̂(Wl) and S̄0l = S 0 jl S 0 il S 01 jl . Then K 0 jl\{B} = 1G n=0 S̄ 0 n l (W 0 l ). (4.14) Notice that for any l, S̄l(z) = qilz and S̄0l(z) = q 0 il e i↵il z, and due to Parameter Matching Condition, there is such , that for any l, ↵il = log q0il . Consider the map z = exp(w) of the plane (w = % + i') as a universal cover of the punctured plane C\{0}. Consider the polygons Pjl and choose their liftings in the plane (w = % + i'). We may suppose that these liftings lie in respective horizontal strips ✓ l  '  ✓+ l , where 0 < ✓ l < ✓ + l < 2⇡ and ✓+ l + ↵0 < ✓ l+1 for any l < k and ✓ + k + ↵0 < ✓ 1 + 2⇡. We also consider liftings of Kjl , Wl, K 0jl and W 0 l . We denote these liftings by Kjl , Wl, K0jl and W0 l . It follows from the equations (4.13) and (4.14) that Kjl = 1G n=0 T̄ n l (Wl) and K0jl = 1G n=0 T̄ 0 n l (W0 l ), (4.15) where Tl(w) = w + log ql and T 0l (w) = w + (1 + i) log q 0 l are parallel translations for which Tl(Kl)⇢Kl and T 0l (K0l)⇢K0l. The sets Kl lie in the half-strips %  log ⇢2, ✓l  '  ✓ + l , while the sets Wl are contained in the rectangles Rl = {log ⇢1  %  log ⇢2, ✓l  '  ✓ + l }. D. Drozdov, M. Samuel and A. Tetenov: On deformation of polygonal dendrites 19 B O Figure 12: The images of the set K 0 under the map w = log(z O) and the map w = log(z B) Then the sets W0 l lie in the rectangle R 0 l = ⇢ log(⇢1 2)  %  log(⇢2 + 2), ✓l 1.05 2 ⇢1  '  ✓+ l + 1.05 2 ⇢1 . (4.16) Each union 1S n=0 T 0 n l (R0 l ) lies in a half-strip 8 < : %  log(⇢2 + 2) ✓ l 1.05 2 ⇢1 log(⇢2 + 2)  ' %  ✓+l + 1.05 2 ⇢1 log(⇢1 2). (4.17) Therefore, the set K0jl also lies in this half-strip. So if ✓ + l1 + 1.05 2 ⇢1 log(⇢1 2) < ✓l 1.05 2 ⇢1 log(⇢2 + 2), (4.18) then K0jl1 \K 0 jl = ?. The inequality (4.18) holds for any l if 2.1 2 ⇢1 + log ⇢2 + 2 ⇢1 2 < ↵0. If, moreover, 22 < ⇢0, then for any i1, i2 2 I such that Pi1 \Pi2 = ?, P 0i1 \P 0 i2 = ? and K 0 i1 \K 0 i2 = ?. This implies that condition (D0) is fulfilled. Theorem 4.6. Let S be a contractible P -polygonal system. There is > 0 such that for any -deformation S0 of the system S satisfying parameter matching condition the attractor K(S0) is a dendrite, homeomorphic to K(S). Proof. Let all the cyclic vertices of the P -polygonal system S have order 1. If we combine inequalities 4.2, 4.8, 4.9, 4.11 and take into account Remark 7, we see that if the following inequality holds: < min min(qmin, 1 qmax) 8 , min(⇢0, ⇢1, 1 ⇢2) 2(CK + 1) , ↵0 2.1(CK+1) ⇢1 + C log 1+3⇢2 3⇢1 ! , (4.19) 20 Art Discrete Appl. Math. 4 (2021) #P2.07 then the attractor K 0 of a -deformation S0 of the system S satisfies condition (D0). There- fore K 0 is a dendrite. By Theorem 2.9, the map f̂ : K ! K 0 is a bijection and therefore it is a homeomorphism. Suppose now that S has cyclic vertices of order greater than 1. There is such n, that all the cyclic vertices of the system S(n) have order 1. Suppose any -deformation of the system S(n) generates a dendrite. Then for any /CK-deformation S0 of the system S, the system S0(n) is a -deformation of the system S(n). ORCID iDs Dmitry Drozdov https://orcid.org/0000-0002-8099-808X Mary Samuel https://orcid.org/0000-0001-9645-1006 Andrei Tetenov https://orcid.org/0000-0001-5419-5710 References [1] V. V. Aseev, A. V. Tetenov and A. S. Kravchenko, Self-similar Jordan curves on the plane, Sibirsk. Mat. Zh. 44 (2003), 481–492, doi:10.1023/a:1023848327898. [2] C. Bandt and H. Rao, Topology and separation of self-similar fractals in the plane, Nonlinearity 20 (2007), 1463–1474, doi:10.1088/0951-7715/20/6/008. [3] C. Bandt and J. Stahnke, Self-similar sets 6. interior distance in deterministic fractals, 1990, preprint, Greifswald. [4] J. J. Charatonik and W. J. Charatonik, Dendrites, in: XXX National Congress of the Mexican Mathematical Society (Spanish) (Aguascalientes, 1997), Soc. Mat. Mexicana, México, volume 22 of Aportaciones Mat. Comun., pp. 227–253, 1998, https://www. researchgate.net/publication/285354319_Dendrites. [5] L. L. Cristea and B. Steinsky, Curves of infinite length in 4 ⇥ 4-labyrinth fractals, Geom. Dedicata 141 (2009), 1–17, doi:10.1007/s10711-008-9340-3. [6] L. L. Cristea and B. Steinsky, Curves of infinite length in labyrinth fractals, Proc. Edinb. Math. Soc. (2) 54 (2011), 329–344, doi:10.1017/s0013091509000169. [7] L. L. Cristea and B. Steinsky, Mixed labyrinth fractals, Topology Appl. 229 (2017), 112–125, doi:10.1016/j.topol.2017.06.022. [8] M. Hata, On the structure of self-similar sets, Japan J. Appl. Math. 2 (1985), 381–414, doi: 10.1007/bf03167083. [9] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713–747, doi: 10.1512/iumj.1981.30.30055. [10] K. G. Kamalutdinov and A. V. Tetenov, Twofold Cantor sets in R, Sib. Èlektron. Mat. Izv. 15 (2018), 801–814, doi:10.17377/semi.2018.15.066. [11] J. Kigami, Harmonic calculus on limits of networks and its application to dendrites, J. Funct. Anal. 128 (1995), 48–86, doi:10.1006/jfan.1995.1023. [12] J. Kigami, Analysis on fractals, volume 143 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2001, doi:10.1017/cbo9780511470943. [13] K. Kuratowski, Topology. Vol. I & 2, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966, doi:10.1016/C2013-0-11023-9. [14] K.-S. Lau, J. J. Luo and H. Rao, Topological structure of fractal squares, Math. Proc. Cam- bridge Philos. Soc. 155 (2013), 73–86, doi:10.1017/s0305004112000692. D. Drozdov, M. Samuel and A. Tetenov: On deformation of polygonal dendrites 21 [15] M. Samuel, A. Tetenov and D. Mekhontsev, On dendrites generated by symmetric polygo- nal systems: The case of regular polygons, in: Advances in Algebra and Analysis. Trends in Mathematics, Springer, volume 1, pp. 17–25, 2019, doi:10.1007/978-3-030-01120-8 4. [16] M. Samuel, A. Tetenov and D. Vaulin, On dendrites generated by polyhedral systems and their ramification points, Proc. Krasovskii Inst. Math. Mech. UB RAS 23 (2017), 281–291, doi:10.21538/0134-4889-2017-23-4-281-291. [17] M. Samuel, A. Tetenov and D. Vaulin, On dendrites generated by polyhedral systems and their ramification points, 2017, https://arxiv.org/abs/1707.02875v1. [18] M. Samuel, A. Tetenov and D. Vaulin, Self-similar dendrites generated by polygonal systems in the plane, Sib. Èlektron. Mat. Izv. 14 (2017), 737–751, doi:10.17377/semi.2017.14.063. [19] R. S. Strichartz, Isoperimetric estimates on Sierpinski gasket type fractals, Trans. Amer. Math. Soc. 351 (1999), 1705–1752, doi:10.1090/s0002-9947-99-01999-6. [20] A. Tetenov, Finiteness properties for self-similar sets, 2020, https://arxiv.org/abs/ 2003.04202v1. ISSN 2590-9770 The Art of Discrete and Applied Mathematics 4 (2021) #P2.08 https://doi.org/10.26493/2590-9770.1420.b57 (Also available at http://adam-journal.eu) Three conjectures of Ostrander on digraph Laplacian eigenvectors* Yaokun Wu† , Da Zhao School of Mathematical Sciences and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China. Received 28 May 2021, accepted 15 June 2021, published online 18 September 2021 Abstract Ostrander proposed three conjectures on the connections between topological proper- ties of a weighted digraph and combinatorial properties of its Laplacian eigenvectors. We verify one of his conjectures, give counterexamples to the other two, and then seek for re- lated valid connections and generalizations to Schrödinger operators on countable digraphs. We suggest the open question of deciding if the countability assumption can be dropped from our main results. Keywords: Alexandrov topology, harmonic function, nonnegative eigenvector. Math. Subj. Class.: 05C50, 15A18. 1 Background An eigenfunction of a linear operator can be viewed as a fixed point, namely a time- invariant point, of the operator in a corresponding projective space. Many dynamical pro- cesses on a geometric domain, including diffusion processes and consensus processes [18, 24, 29, 40, 42, 44, 57, 58], are driven by a Laplacian operator that reflects the local con- nectivity scenarios of the space. It is natural to expect that the shape of an eigenfunction of a Laplacian operator should somehow follow the shape of the underlying space; That is, you may be able to tell/predict the shape of space from some time-invariant data. Classical Fourier analysis provides deep understanding of signals over regular domains, to process graph-supported data we should accordingly develop spectral graph theory or the theory of graph Fourier transforms [3, 9, 12, 21, 27, 39, 41, 46, 50]. There are already quite some *Supported by NSFC 11971305. †Corresponding author. E-mail addresses: ykwu@sjtu.edu.cn (Yaokun Wu), jasonzd@sjtu.edu.cn (Da Zhao) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 2 Art Discrete Appl. Math. 4 (2021) #P2.08 results and questions on the relationship between the oscillations of Laplacian eigenvec- tor and the landscape of the underlying digraph, say those related to Courant’s nodal line theorem [55], those related to Fiedler’s monotonicity theorem [32, 48, 50], those related to Rauch’s hot spots conjecture [16, 22] and others [52, 56]. To describe the influence of eigenstructures on connectivity patterns, various concepts on the digraphs, say diffusion distance [16] or Alexandrov topology [45], have been considered. Especially, Ostrander listed three conjectures on digraph Laplacians and Alexandrov topology [45], which is the starting point of this work. 2 Alexandrov topologies and Laplacian eigenvectors We write R+ for the set of positive reals, R0 for the set of nonnegative reals, N0 for the set of nonnegative integers, N for the set of positive integers, and K for a field being either R or C. For each n 2 N, [n] refers to the set of first n positive integers. Let S be a set and let M be a nonnegative function on S⇥S, namely M : S⇥S ! R0. We naturally view this function M as a weighted digraph on S and say that S is the vertex set of M , denoted by V(M), and let the arc set of M be A(M) := {(v, w) : M(v, w) > 0} ✓ V(M)⇥ V(M). For each v 2 V(M), its out-degree in M is deg+ M (v) := P w2S M(v, w) and its in- degree in M is deg M (v) := P w2S M(w, v). If the summation does not converge, it is regarded as +1. We say that a digraph M is a finite-out-degree (FOD) digraph if every vertex of M has a finite out-degree. Moreover, if the out-degrees of vertices of M is bounded, we say that M is a bounded-out-degree (BOD) digraph. The out-neighbors of a vertex v in M is N+ M (v) := {w : M(v, w) > 0} and the in-neighbors of v in M is N M (v) := {w : M(w, v) > 0}. We call a digraph M a finite-out-neighbor (FON) digraph if every vertex of M has a finite set of out-neighbors. An FON digraph is surely a BOD digraph, and a BOD digraph is trivially an FOD digraph. We name a digraph a finite/countable digraph if its vertex set is a finite/countable set. Surely, every finite digraph is FON, and hence FOD. A path of length ` from a vertex v to a vertex w in M is a sequence of vertices v = v0, v1, . . . , v` = w such that (vi1, vi) 2 A(M) for i 2 [`]. We denote by v ⇢ w that (v, w) 2 A(M) and by v ⇣ w that there is a path from v to w in M . The future of v in M is defined to be v"M := {w : v ⇣ w} while the history of v in M is defined as v#M := {w : w ⇣ v}. In general, for any subset T of V(M), let T"M := S v2T v"M and T#M := S v2T v#M . If M takes value in {0, 1}, we call M transitive provided M(u, v) = M(v, w) = 1 im- plies M(u,w) = 1. In general, we refer to a digraph M as a transitive digraph provided M(v, w) > 0 implies that M(u, v)  M(u,w) for all u, v, w 2 V(M). The Alexandrov topology on S = V(M) induced by M has a set T ✓ V(M) as an open set if and only if T#M = T . We will simply speak of an open set or a closed set of M to mean such a set in the Alexandrov topology induced by M . To see the naturalness of the concept of Alexandrov topology, you can recall that basically the step from a digraph to its lattice of open sets is a Birkhoff transform and this operation produces all distributive lattices [47]. A strongly connected component of M is a minimal nonempty subset of V(M) which is the intersection of a closed set and an open set of M . A strongly connected component of Y. Wu and D. Zhao: Three conjectures of Ostrander on digraph Laplacian eigenvectors 3 M is called a sink component if it is a closed set in M . We write the union of the set of sink components of a weighted digraph M by ⌅M . The Laplacian LM of a digraph M is a linear operator from a suitable linear subspace U of KV(M) to some other subspace of KV(M) such that (LM f)(v) = X w2V(M) M(v, w) f(v) f(w) (2.1) for every f 2 U . In general, the summation in Equation (2.1) may not converge, and so, in order to make Equation (2.1) well-defined, we will specify the domain of the Laplacian for various digraph classes later. In particular, if the digraph M is viewed as an FOD digraph, the domain will be chosen as `1(V(M)); If the digraph M is viewed as a BOD digraph, the domain will be chosen as `1(V(M)); If the digraph M is viewed as an FON digraph, the domain will be chosen as all K-valued functions on V(M). Additionally, when the digraph is finite, the Laplacian can be represented by the matrix LM = DM M , where DM is the diagonal matrix on V(M) whose v-th diagonal element is deg+M (v). For any linear map L from U to U 0, KerL stands for the right null (kernel) space {x 2 U : L(x) = 0 2 U 0}. For each weighted digraph M , we refer to each f 2 KerLM as a harmonic function with respect to M . We say that f is harmonic at v 2 V(M) with respect to M if (LM f)(v) = 0. u v w u v w Figure 1: A transitive digraph and its reverse. Example 2.1. Let M be the transitive digraph as shown on the left of Figure 1 and let M> be its transpose, also called its reverse digraph, which is the transitive digraph as shown on the right of Figure 1. We display a basis for the left eigenspaces of LM and LM> as follows: LM : 2 4 0 1 1 2 1 1 0 1 1 3 5 2 4 u v w u 2 1 1 v 0 1 1 w 0 1 1 3 5 = 2 4 0 2 2 3 5 2 4 0 1 1 2 1 1 0 1 1 3 5 ; LM> : 2 4 2 0 0 2 1 1 0 1 1 3 5 2 4 u v w u 0 0 0 v 1 2 1 w 1 1 2 3 5 = 2 4 0 1 3 3 5 2 4 2 0 0 2 1 1 0 1 1 3 5 . We display a basis for the right eigenspaces of LM and LM> as below: LM : 2 4 u v w u 2 1 1 v 0 1 1 w 0 1 1 3 5 2 4 1 1 0 1 0 1 1 0 1 3 5 = 2 4 1 1 0 1 0 1 1 0 1 3 5 2 4 0 2 2 3 5 ; 4 Art Discrete Appl. Math. 4 (2021) #P2.08 LM> : 2 4 u v w u 0 0 0 v 1 2 1 w 1 1 2 3 5 2 4 1 0 0 1 1 1 1 1 1 3 5 = 2 4 1 0 0 1 1 1 1 1 1 3 5 2 4 0 1 3 3 5 . In general, for a matrix A, its Jordan canonical form J and an invertible matrix P , it surely holds PA = JP if and only if AP1 = P1J. Therefore, if we can say something on the relationship between the combinatorial patterns of a matrix and its inverse matrix, we may be able to somehow link the patterns of the left eigenspace and the right eigenspace of that matrix. Ostrander proposed three conjectures in [45]. He stated each conjecture in two parts but one can easily check that the two parts are equivalent to each other. We summarize his conjectures as below and invite the readers verify the conjecture for Example 2.1. Conjecture 2.2. Let M be a finite digraph. (1) ([45, Conjecture 1]) If M is transitive and f is a nonnegative right eigenvector of LM , then f(v) f(w) for every v, w 2 V(M) such that v ⇢ w. (2) ([45, Conjecture 2]) If f is a nonnegative right eigenvector of LM , then supp(f) is open in M . (3) ([45, Conjecture 3]) A set P ✓ V(M) is the support of a nonnegative harmonic function with respect to M if and only if P = S#M for some sink component S of M . As with Conjecture 2.2 (2), it has a very simple proof. Theorem 2.3. [45, Conjecture 2] is correct. That is, for any nonnegative square matrix M and any nonnegative right eigenvector f of LM , it holds supp(f) = supp(f)#M . Proof. By the definition of open set, our task is to deduce f(v) > 0 from the assumption that f(w) > 0 and M(v, w) > 0. We assume that LM f = f . Evaluating both sides at the vertex v yields (deg+ M (v) )f(v) = X u2V(M) M(v, u)f(u) M(v, w)f(w) > 0. This implies f(v) 6= 0 and so, since f is nonnegative, f(v) > 0 follows. Unfortunately, Conjecture 2.2 (1) does not hold true. Example 2.4. Let m 3 and let M be the matrix on {v1, . . . , vm} with M(vi, vj) = ( 1, if i = 1 and j > 1; 0, otherwise. Y. Wu and D. Zhao: Three conjectures of Ostrander on digraph Laplacian eigenvectors 5 Then the function f with f(v1) = 1, f(v2) = m 1 and f(vi) = 0 for i = 3, . . . ,m, is a nonnegative harmonic function with respect to M ; See Figure 2. Note that f(v1) < f(v2) while (v1, v2) 2 A(M). v1 : 1 v2 : m 1 v3 : 0 · · · vm : 0 Figure 2: A digraph and a harmonic function with respect to it. Let us further demonstrate that one can even see arbitrary oscillation pattern in a Lapla- cian eigenvector of a transitive digraph along a path of the digraph. tm tm1 tj+1 tj t1 t0 `m `m1 `j+1 `j `1 Figure 3: The Hasse diagram of a poset. Example 2.5. Note that a poset is a transitive acyclic digraph. Consider a digraph M which induces a poset whose Hasse diagram is as depicted in Figure 3. In other words, M(v, w) = 1 if and only if v 6= w and there is a path from v to w in the Hasse digram. We show that, for every element 2 {0,±1}m, there is a nonnegative harmonic function f on M such that = (sgn f(tm) f(tm1) , sgn f(tm1) f(tm2) , . . . , sgn f(t1) f(t0) ), where sgn is the sign function. For each j 2 [m], we choose fj 2 RV(M) by setting fj(v) = 8 >>>< >>>: 2j, if v = `j ; 1, if v = `k, k > j; 1, if v = tk, k j; 0, otherwise. It is obvious that fj(`j) = 2j > 1 = fj(tj) and that fj is harmonic with respect to M . It follows that a convex combination of such fj and the constant function can give us a required harmonic function. Example 2.6. Consider a digraph M which has more than one sink component. The all- ones function on V(M) is surely harmonic with respect to M . But its support, namely V(M), is not the history of any single sink component of M . In order to exclude Example 2.6 as a counterexample, we have to modify the statement of Conjecture 2.2 (3). Let us propose the following variant of Conjecture 2.2 (3), which will be proved in Section 3. Theorem 2.7. Let M be a finite digraph and let P ✓ V(M). Then the following are equivalent. 6 Art Discrete Appl. Math. 4 (2021) #P2.08 (1) The set P is the support of a nonnegative harmonic function with respect to M . (2) The set P is the history of the union of several sink components of M . (3) The set P is the support of a nonnegative right eigenvector of LM and P T ⌅M 6= ;. Here is a left eigenvector counterpart of Conjecture 2.2, which we will also prove in Section 3. Theorem 2.8. Let M be a finite digraph and let f be a nonnegative left eigenvector of the Laplacian matrix LM . Then the following hold. (1) supp(f) is closed in M . (2) If M is transitive, then for every v, w 2 V(M) with v ⇢ w it holds f(v)  f(w). (3) A set P ✓ V(M) is the support of a nonnegative left eigenvector of M if and only if P is the union of several sink components of M . 3 Eigenfunctions of finite digraph Laplacians To prove Theorems 2.7 and 2.8, we need to recall some facts on the right and left null spaces of digraph Laplacians. These facts are in the folklore in various guises for many years. For the purpose of our subsequent proof and for the convenience of readers, we will briefly outline a proof of these facts; Whenever possible, we will try to avoid the most well-known treatments. But let us also point out here several references where such facts and their proofs can be found explicitly. Agaev and Chebotarev [1, Theorem 4] obtained the rank of a Laplacian matrix. The right null space of a Laplacian matrix was determined by Caughman and Veerman [13, Theorem 3.3], by Gunawardena [25, p. 5], and by Mirzaev and Gunawardena [42, Proposition 5]. Ostrander [45, Proposition 16] described the structure of the right null space of the Laplacian of a transitive digraph. Veerman and Kummel [57, Theorem 4.5] presented the structure of the left null space of a Laplacian matrix. The spectral radius of a matrix A is denoted by ⇢(A). An n⇥n matrix A is irreducible if for every i, j 2 [n] there exists a nonnegative integer t such that At(i, j) 6= 0. A non- negative matrix is irreducible if and only if it is strongly connected. It turns out that Perron eigenvalues and Perron eigenvectors of nonnegative matrices [36] have close relation with Laplacian spectrum and Laplacian eigenvectors. Theorem 3.1 (Perron-Frobenius Theorem [28, p. 10–4]). Let A be an irreducible nonneg- ative n⇥ n matrix with spectral radius r. Then the followings hold. (1) The number r is an eigenvalue of A. If n 2, we must have r > 0. (2) The eigenvalue r is simple. (3) The matrix A has a positive right eigenvector x and a positive left eigenvector y> associated to the eigenvalue r. (4) The only positive eigenvectors, left or right, are those associated to the eigenvalue r. (5) The eigenvalue r satisfies mini2[n] P j2[n] A(i, j)  r  maxi2[n] P j2[n] A(i, j). Y. Wu and D. Zhao: Three conjectures of Ostrander on digraph Laplacian eigenvectors 7 Corollary 3.2. Let M be a finite strongly connected weighted digraph. Then the following hold. (1) The number 0 is an eigenvalue of LM . (2) The eigenvalue 0 is simple. With the exception of the zero eigenvalue, all eigenvalues of M have a positive real part. (3) LM has a positive right eigenvector x = 1V(M) and a positive left eigenvector y> associated to the eigenvalue 0. (4) The only positive eigenvectors of LM , left or right, are those associated to the eigen- value 0. Proof. Let be the maximum out-degree of M . Note that LM = I (I+M DM ) while I +M DM is an irreducible nonnegative square matrix. It is clear that is an eigenvalue of LM if and only if is an eigenvalue of I + M DM . By the last claim in Theorem 3.1, ⇢(I +M DM ) = . We thus see that everything follows from Theorem 3.1. Let us fix M to be a finite weighted digraph throughout the remainder of this section. Let C be a strongly connected component of M , let H = C#M \ C, and let R = V(M) \ C#M . We can represent the matrix LM as follows: LM = 2 4 H C R H D|H WH,H WH,C WH,R C 0 D|C WC,C WC,R R 0 0 LR 3 5, (3.1) where LR is the Laplacian on the induced subgraph M(R,R), D|H and D|C are the H⇥H diagonal matrix and the C ⇥ C diagonal matrix whose v-th diagonal element is equal to deg+ M (v), and WP,Q records the weights of arcs from P to Q for P,Q 2 {H,C,R}. Note that WC,R = 0 and D|C WC,C coincides with the Laplacian of M(C,C) if C is a sink component of M . We will write D|C WC,C as LC in the case that C is a sink component of M . Observation 3.3. For each sink component C of M , there exists yC 2 R0V(M)\KerL>M with supp(yC) = C. Proof. By Corollary 3.2, there exists y 2 RC+ such that y>LC = 0>C . Let yC = 2 4 H 0 C y R 0 3 5. (3.2) It is easy to see that y> C LM = 0>. Observation 3.4 (Local minimum principle). Let M be a finite weighted digraph. Suppose f is harmonic at v with respect to M , namely f(v) is the weighted average of the values of f at the out-neighbors of v in M . If f takes local minimum value at v in the sense that f(v)  f(w) for every w 2 N+ M (v), then f takes the same value at its out-neighbors. 8 Art Discrete Appl. Math. 4 (2021) #P2.08 Observation 3.5. If H 6= ;, then Ker(D|H WH,H) = {0}. (3.3) In particular, D|H WH,H is of full rank. Proof. Suppose g 2 Ker(D|H WH,H). Define f 2 KV(M) by setting f(v) = ( g(v), if v 2 H; 0, otherwise. It is clear that f vanishes on all sink components of M and LM f = 0. By Observation 3.4, f must be the zero function and hence g = 0 follows. A perturbed Laplacian is the sum of a Laplacian and a nonzero nonnegative diago- nal matrix. Observation 3.5 essentially says that a perturbed Laplacian of any strongly connected digraph is nonsingular. This fact has been proved many times in the litera- ture. Caughman and Veerman [13, Lemma 2.4] as well as Mirzaev and Gunawardena [42, Lemma 2, Fig. 4] used the same trick of embedding a perturbed Laplacian into a Lapla- cian, which is somehow a disguised version of our deduction in Observation 3.5; Note that Gunawardena [25, p. 5] made use of the matrix-tree Theorem (indeed, Markov chain tree Theorem [2, 30]) to obtain the same result. For any digraph M , a basic open set in the Alexandrov topology induced by M is a set of the form S#M for some finite sink component S of M . Observation 3.6. For each sink component C of M , there exists xC 2 R0V(M) \ KerLM with supp(xC) = C#M . That is, each basic open set of M is the support of a nonnegative harmonic function on M . Proof. If H = ;, we can simply take xC :=  C 1C R 0R . Assume that H 6= ;. By Observation 3.5, D|H WH,H is nonsingular. This allows us set xC := 2 4 H (D|H WH,H)1WH,C1C C 1C R 0R 3 5. (3.4) It is straightforward to check that LM xC = 0. Note that xC , a real harmonic function with respect to M , is determined by its values at the sink component C. By Observation 3.4, the minimum value of xC is achieved inside R [ C and so xC is nonnegative. It then follows from Theorem 2.3 that supp(xC) is open. Since C ✓ supp(xC), we must have supp(xC) ✓ C [H = C#M ✓ supp(xC), as wanted. For each v 2 H and w 2 C, there is a path from v to w. This implies 1X `=0 ((D|H)1WH,H)`(D|H)1WH,C1C 2 RH+ . Y. Wu and D. Zhao: Three conjectures of Ostrander on digraph Laplacian eigenvectors 9 Therefore, another way to get Observation 3.6 is to show that (D|H WH,H)1 = (I D|1 H WH,H)1D|1H = P1 `=0((D|H)1WH,H)`(D|H)1. This is not as trivial as it may seem: Note that 13 = (1 4) 1 6= 1 + 4 + 42 + 43 + · · · . Anyway, we can derive it by appealing to the fact that ⇢(A) > ⇢(B) if B is an irreducible matrix and A B 0. If one thinks of C as one absorbing state, I (D|1 H )WH,H plays a role as the fundamental matrix in the study of Markov chains and the argument in the proof above parallels the usual deduction there [58]. The inverse of I (D|1 H )WH,H , which exposes lots of geometric information of the Laplacian, can be said to be a discrete analogue of Green’s function in analysis [4, 31, 35, 37, 59]. Remark 3.7. KerL> M governs the asymptotic behavior of the diffusion process given by LM while KerLM records all data about the asymptotic behavior of the consensus process given by LM [5, 57]. Theorem 3.8. Let M be a finite weighted digraph. Then the following hold. (1) A basis of KerLM is given by nonnegative vectors xS in Observation 3.6 where S ranges over all sink components of M . (2) A basis of KerL> M is given by nonnegative vectors yS in Observation 3.3 where S ranges over all sink components of M . (3) Both the algebraic multiplicity and the geometric multiplicity of the eigenvalue 0 of LM are equal to the number of sink components of M . Proof. In light of Observation 3.4, a harmonic function with respect to M is uniquely determined by its restriction on ⌅M . But for each f 2 KerLM and each sink component C of M , f |C is harmonic with respect to M(C,C) and so, by Observation 3.4 again, can only be a constant function. It then follows from Observation 3.6 that the first reading is valid. Since KerLM and KerL>M share the same dimension, the second reading follows from the first one and Observation 3.3. For the third reading, it remains to verify that the number of nonzero eigenvalues of LM plus the dimension of KerLM equals |V(M)|. When M has only one sink compo- nent, Corollary 3.2 together with Observation 3.5 proves the claim. In general, in view of Equation (3.1) and employing Theorem 3.2 and Observation 3.5, we can apply induction on the number of sink components to complete the proof. Proof of Theorem 2.7. Consider a nonnegative right eigenvector f of LM , say LM f = f . If supp(f) \ C 6= ; for some sink component C of M , then y> C f > 0, according to Observation 3.3. This gives 0 = 0f = y> C LM f = y > C f, and hence = 0. An application of the first part of Theorem 3.8 now concludes the proof. Proof of Theorem 2.8. Consider a nonnegative left eigenvector f> of LM , say f> LM = f >. From Observation 3.6 we derive that [ C:sink components of M supp(xC) = [ C:sink components of M C#M = V(M). 10 Art Discrete Appl. Math. 4 (2021) #P2.08 Therefore, there exists a sink component C of M such that supp(f) \ C#M 6= ;. It then follows 0 = f> LM xC = f > xC , (3.5) and hence = 0. By the second part of Theorem 3.8, the left null space of LM has a basis yS , where S runs through all sink components of M . This implies that f lies in the cone generated by this basis. Note that supp(yS) = S for any sink component S of M , and that any union of finitely many sink components of M is a closed set of M . We thus arrive at Theorem 2.8 (1) and Theorem 2.8 (3). If M is even transitive, we see that M takes a constant value on C ⇥ C for each strongly connected component C of M . Especially, for every sink component S of M , from Equation (3.2) we see that yS satisfies the condition asserted in the second reading. Since f lies in the cone generated by such yS , Theorem 2.8 (2) is proved. 4 Schrödinger operators on countable digraphs Trofimov [54] proved that every infinite locally finite vertex-transitive graph has a noncon- stant harmonic function, which grows at most exponentially with respect to the distance to a base point. Tointon gave a lovely treatment of a qualitative version of this result on general weighted graphs [53, Proposition 1.4]. The next example indicates a difference between graphs and digraphs. Example 4.1. Let M be the infinite directed path, namely V(M) = Z, M(i, i + 1) = 1 and M(i, j) = 0 for j 6= i+ 1. The digraph M is an infinite locally finite vertex-transitive digraph whose harmonic functions are all constant functions. Let M be an out-finite digraph and let ⇤ 2 RV(M). The Schrödinger operator on M with potential function ⇤, denoted by S M,⇤, is the linear map from KV(M) to itself such that (S M,⇤ f)(v) := (LM f)(v)+⇤(v)f(v) for all f 2 KV(M) [8, 10, 17, 20, 43, 51]. A Schrödinger operator with a nonnegative potential on a finite digraph is either a Laplacian matrix or a perturbed Laplacian matrix. We call any element from KerS M,⇤ a harmonic function with respect to S M,⇤. Here is an easy example to demonstrate the different behavirors of Schrödinger operators on finite and infinite digraphs. v0 v1,1 v1,2 · · · v2,1 v2,2 · · · Figure 4: A strongly connected out-finite digraph which is not in-finite. Example 4.2. Let (ai)1i=1 be a sequence of increasing positive numbers. We assign weights Y. Wu and D. Zhao: Three conjectures of Ostrander on digraph Laplacian eigenvectors 11 to the arcs in Figure 4 by M(x, y) = 8 >< >: 1 2 , if y 2 {v1,1, v2,1}, x = v0; aj aj+1 , if y = vi,j+1, x = vi,j , i 2 {1, 2}, j 2 N; 1 aj aj+1 , if y = v0, x = vi,j , i 2 {1, 2}, j 2 N. For the resulting weighted digraph M , and for every ⇤ 2 RV(M)0 , we have f 2 KerS M,⇤, where f(v) = ( 0, if v = v0; (1)iaj , if v = vi,j , i 2 {1, 2}, j 2 N. In particular, if we let ai = i for i 2 N, then we obtain an unbounded harmonic function; If we let ai = ii+1 for i 2 N, then we obtain a bounded harmonic function without maximum value. Moreover, we can add new vertices vi,j with i = 3, 4, . . . ,m, j 2 N, and set M(x, y) = 8 >< >: 1 m , if y 2 {v1,1, v2,1, . . . , vm,1}, x = v0; aj aj+1 , if y = vi,j+1, x = vi,j , i 2 [m], j 2 N; 1 aj aj+1 , if y = v0, x = vi,j , i 2 [m], j 2 N. For every ⇤ 2 RV(M)0 , the right null space of the Schrödinger operator S M,⇤ is of dimen- sion at least m 1. Indeed, for every pair of integers i, i0 such that 1  i < i0  m, we have fi,i0 2 KerS M,⇤, where fi,i0(v) = 8 >< >: aj , if v = vi,j , j 2 N; aj , if v = vi0,j , j 2 N; 0, otherwise. Let us try to seek some counterparts of results about finite Laplacians in the infinite setting. Here is an obvious parallel to Theorem 2.3. Lemma 4.3. Let M be a digraph and let ⇤ 2 RV(M)0 be a potential function on M . Let X be a subset of V(M) and let M 0 = M(X,X). Suppose f 2 RV(M)0 satisfy (S M,⇤ f)|X = f |X . Then supp(f)\X is open in the Alexandrov topology induced by M 0. In particular, if X = V(M), namely f is a nonnegative right eigenvector of S M,⇤, then supp(f) is open in the Alexandrov topology induced by M . Proof. Assume that M 0(u, v) > 0 for u 2 X and v 2 supp(f) \X . It follows that (deg+ M (u)+⇤(u))f(u) = X w2V(M) M(u,w)f(w) M(u, v)f(v) = M 0(u, v)f(v) > 0, and hence u 2 supp(f) \X . We now arrive at the main result of this paper. Theorem 4.4. Let M be a countable FON digraph and take ⇤ 2 RV(M)0 . Then exactly one of the following holds: 12 Art Discrete Appl. Math. 4 (2021) #P2.08 (1) There exists a basic open set of M which is the support of a nonnegative harmonic function for the Schrödinger operator S M,⇤. (2) The Schrödinger operator S M,⇤ is a surjective map from KV(M) to itself. If V(M) is finite, basically, Theorem 4.4 will follow from Observations 3.5 and 3.6. For the general case, we need the additional ingredient of taking inverse limit (projective limit). Parallel to the roles of Observations 3.5 and 3.6, we split the proof of Theorem 4.4 into two steps; See Theorems 4.8 and 4.13. For any X ✓ V(M), we denote by KV(M),X the functions in KV(M) which vanish outside of X . Note that KV(M),X is isomorphic to KX as a linear space. When V(M) is clear from the context, we often write the natural embedding map from KX to KV(M) as ◆X , namely for each f 2 KX it holds ◆X(f)(v) = ( f(v), if v 2 X; 0, if v 2 V(M) \X. Observation 4.5. Let M be a finite digraph and take ⇤ 2 RV(M)0 . Let M 0 be obtained from M by adding a new vertex r and setting M 0 V(M),V(M) = M , M 0(u, r) = ⇤(u) for u 2 V(M) and M 0(r, v) = 0 for all v 2 V(M 0). Then the embedding map ◆V(M) from KV(M) to KV(M 0) induces an isomorphism from KerS M,⇤ to the space of those harmonic functions on M 0 whose supports do not contain r. Proof. Take h 2 KerS M,⇤. Then, for all v 2 V(M) = V(M 0) \ {r} we have LM 0(◆V(M)(h))(r) = X w2V(M 0) M 0(r, w)(h(r) h(w)) = X w2V(M 0) 0⇥ (h(r) h(w)) = 0 and LM 0(◆V(M)(h))(v) = X w2V(M 0) M 0(v, w)(h(v) h(w)) = X w2V(M) M(v, w)(h(v) h(w)) +M(v, r)(h(v) 0) = X w2V(M) M(v, w)(h(v) h(w)) + ⇤(v)h(v) = S M,⇤(h) = 0. Y. Wu and D. Zhao: Three conjectures of Ostrander on digraph Laplacian eigenvectors 13 Suppose h 2 KerLM 0 and h(r) = 0. Then h = ◆V(M)(h|V(M)) and (S M,⇤(h|V(M)))(v) = X w2V(M) M(v, w)(h(v) h(w)) + ⇤(v)h(v) = X w2V(M) M 0(v, w)(h(v) h(w)) +M 0(v, r)(h(v) h(r)) = X w2V(M 0) M 0(v, w) h(v) h(w) = LM 0(h)(v) = 0 for all v 2 V(M). For any two complex numbers z1 and z2, the real inner product of z1 and z2, denoted by hz1, z2iR, is the real part of z1z2. If z1 = r1 exp p 1✓1 and z2 = r2 exp p 1✓2 , where r1, r2 2 R0 and ✓1, ✓2 2 R, then hz1, z2iR = r1r2 cos(✓2 ✓1). If |z2| |z1|, the Cauchy-Schwarz inequality says that hz2, z2 z1iR = hz2, z2iR hz2, z1iR hz2, z2iR p hz2, z2iRhz1, z1iR hz2, z2iR p hz2, z2iRhz2, z2iR = 0. Observation 4.6 (Local maximum modulus principle). Let M be an FOD digraph and let ⇤ 2 RV(M)0 . Suppose f 2 KerS M,⇤. If |f(v)| |f(w)| for every out-neighbor w of v in M , then f(v) = f(w) for w 2 N+ M (v) and ⇤(v)f(v) = 0. In particular, if |f(v)| |f(w)| for every w 2 v"M , then f(v) = f(w) for every w 2 v"M . Proof. By our remarks preceding Observation 4.6, hf(v), f(v) f(w)iR 0 for all w 2 N+ M (v), with equality if and only if f(v) = f(w). Observe that 0 = hf(v), 0iR = * f(v),⇤(v)f(v) + X w2N+M (v) M(v, w)(f(v) f(w)) + R = ⇤(v)|f(v)|2 + X w2N+M (v) M(v, w)hf(v), (f(v) f(w))iR ⇤(v)|f(v)|2. It then follows that ⇤(v) = 0 and f(v) = f(w) for all w 2 N+ M (v), as claimed. Lemma 4.7. Let M be an FOD digraph. Let ⇤ 2 RV(M)0 and let S be a finite sink component of M . If x 2 KerS M,⇤ and S \ supp(⇤) 6= ;, then supp(x) \ S = ;. Proof. By Observation 4.6, x takes a constant value B inside S. Take w 2 S \ supp(⇤). Then 0 = ⇤(w)x(w)+ P v2N+M (w) M(w, v) x(w)x(v) = ⇤(w)B, showing that B = 0. This gives supp(x) \ S = ;. The next result characterizes when a basic open set of a countable digraph M can be the support of a nonnegative element from the right null space of a Schrödinger operator on it. Theorem 4.8. Let M be a countable FOD digraph and take ⇤ 2 RV(M)0 . Suppose that S is a finite sink component of M . Then we can find hS 2 `1(V(M))\KerS M,⇤ such that hS 0 and supp(hS) = S#M if and only if S \ supp(⇤) = ;. 14 Art Discrete Appl. Math. 4 (2021) #P2.08 Proof. In view of Lemma 4.7, we only need to prove the backward direction. Fix an element in S, say p0. Let Q := {h 2 RV(M)0 : h(v)  h(p0), 8v 2 V(M)} ✓ ` 1(V(M)). Suppose that S \ supp(⇤) is nonempty and we want to find h 2 Q such that S M,⇤(h) = 0 and supp(h) = S#M . We enumerate the vertices in V(M) \ S as p1, p2, . . .. If V(M) \ S is a finite set of c elements, we use the convention that p0 = pc+1 = pc+2 = · · · . We put X0 := S and Xn+1 := Xn [ {pn+1} for all n 2 N0. For each n 2 N0, define Kn := {1Xnh : (S M,⇤ h)|Xn = 0, h 2 Q, supp(h) ✓ S#M}. Observations 3.6, 4.5 and 4.6 tell us that Kn ) {0}. Let K̂n := T1 m=n Km be the intersec- tion of Km for all m n. We claim that K̂n is not the trivial cone with a single point. Let Sn denote the unit sphere in the finite-dimensional inner product space KV(M),Xn . Since Sn is compact, the descending chain of nonempty closed sets Kn \ Sn ◆ un+1,n(Kn+1) \ Sn ◆ un+2,n(Kn+2) \ Sn ◆ · · · has a nonempty intersection Ŝn := T1 m=n um,n(Km) \ Sn = K̂n \ Sn. Therefore K̂n is not a trivial cone. Note that K̂n ✓ un+1,n(K̂n+1) for all n 2 N0. This allows us to find hn 2 K̂n for all nonnegative integers n so that h0 = 1S and hn = un+1,n(hn+1). We claim that the required function h can now be taken as the one satisfying h|Xn = hn for all n 2 N0. What remains to be verified is h(v) > 0 for all v 2 S#M . Indeed, we can find m such that there is a path Pv from v to a vertex in S such that Pv ✓ Xm. By the definition of Km, there exists eh 2 Q such that S M,⇤ eh|Xm = 0 and hm = 1Xmeh. Note that S ✓ supp(eh)\ Xm = supp(hm). Accordingly, Lemma 4.3 tells us that v 2 supp(eh)\Xm = supp(hm), and hence h(v) = hm(v) > 0, as desired. Example 4.9. Let M be an infinite directed path with a single sink vertex, namely V(M) = N0, M(i + 1, i) = 1 and M(j, i) = 0 if j 6= i + 1 for every i, j 2 N0. Then M is an FOD digraph and it has a single finite sink component. If h 2 KerLM and h(0) = c, then h(i) = c for every i 2 N0. This gives `1(V(M)) \ KerLM = {0}. We thus see that the analogue of Theorem 4.8 with FOD replaced by BOD does not hold. The next result is a generalization of Theorem 3.8 (1). Theorem 4.10. Let M be a countable FOD digraph and take ⇤ 2 RV(M) . Let S = {S : S is a sink component of M,S \ supp(⇤) = ;} be the collection of sink components with- out intersection with supp(⇤). Suppose |v"M | < 1 for every v 2 V(M). Then, for each S 2 S, the function hS as claimed in Theorem 4.8 is uniquely determined. Furthermore, the harmonic functions hS , S 2 S, are linearly independent, and they form a basis of KerS M,⇤ when |S| < 1. Proof. For each S 2 S, we fix one function hS as claimed in Theorem 4.8. To finish the proof, it suffices to show that every function h 2 KerS M,⇤ can be expressed as a unique linear combination of these hS for S 2 S. Since |v"M | < 1 for every v 2 V(M), each sink component of M is finite. What is more, M is actually an FON digraph. For every sink component S such that S\supp(⇤) 6= Y. Wu and D. Zhao: Three conjectures of Ostrander on digraph Laplacian eigenvectors 15 ;, Lemma 4.7 has shown that h vanishes on S. In view of Observation 3.4, we now find that, for every sink component S 2 S, the harmonic function h must take a constant value on S, say cS . Since the harmonic functions hS , where S 2 S, are linearly independent, we will complete the proof by verifying that h0 = h P S2S cShS is the zero function. Take v 2 V(M). Consider the induced finite digraph M 0 = M(v"M , v"M ). Note that h0|V(M 0) vanishes on all sink components of M 0 and that S M 0,⇤0 h0|V(M 0) = 0. By Theorem 3.8 and Observation 4.5, we get h0(v) = 0 and therefore we are done. Example 4.11. In Theorem 4.10 we cannot replace the assumption |v"M | < 1 by the condition that all sink components of M are finite. One may consider modifying Exam- ple 4.2 by adding a single sink vertex as an out-neighbour of v0 in Figure 4 and associating with the new arc an arbitrary positive weight. The modified digraph is a countable FOD digraph. It has a single sink component, but it bears at least two linearly independent harmonic functions. Lemma 4.12. Let M be a digraph and let ⇤ 2 RV(M)0 . If M has a finite sink component S such that supp(⇤)\S = ;, then S M,⇤ is not surjective (to the corresponding codomain). Proof. Note that the Laplacian LM(S,S) of the finite digraph M(S, S) is not surjective. But, considering that S is a sink component, for every g 2 RV(M) it holds (S M,⇤ g)|S = LM(S,S) g|S . This is the proof. It is known that a linear operator is surjective provided it has finite hopping range and satisfies the pointwise maximum principle [15, 33]. Our Theorem 4.13 is in the same vein. Theorem 4.13. Let M be a countable FON digraph and let ⇤ 2 RV(M)0 . Then S M,⇤ is a surjective map from KV(M) to itself if and only if it holds S \ supp(⇤) 6= ; for every finite sink component S of M . Proof. In view of Lemma 4.12, we only need to prove the backward direction. Suppose that supp(⇤) \ S is nonempty for every finite sink component S of M . Since M is countable, we may enumerate the vertices of M as v0, v1, v2, . . .. For each non- negative integer n, let Xn denote the set {v0, v1, . . . , vn} when n + 1  |V(M)| and let Xn = V(M) otherwise, let n = KV(M),Xn and let S n be the linear map from n to itself so that S n(h) = 1Xn S M,⇤(h) for all h 2 n. For any n 2 N0, let us show that S n is a surjective map on n. As dimn = n + 1 < 1, our task is to verify that S n is injective. Take h 2 KerS n. Suppose maxv2V(M) |h(v)| = |h(vi)| = B for some i 2 {0, . . . , n}. We intend to show that B = 0. By Lemma 4.6, we have h(w) = h(vi) for every w 2 vi"M . If vi"M contains a vertex x outside of Xn, then we have B = h(x) = 0. We thus turn to the case that vi"M ✓ Xn. Since Xn is finite, we find that vi"M contains a finite sink component of M , say S. By assumption, there exists w 2 supp(⇤) \ S. It follows 0 = ⇤(w)h(w) + P v2N+M (w) M(w, v) h(w) h(v) = ⇤(w)B, yielding that B = 0, as wanted. For every g 2 KV(M), we want to find f 2 KV(M) such that g = S M,⇤ f . For each n 2 N0, we set Hn = {1Xnh : (S M,⇤ h)|Xn = g|Xn , h 2 KV(M)}. It is nonempty since S 1 n (1Xng) 2 Hn. Note that for every n m 0 there is an affine map un,m from the affine space Hn to Hm given by un,m(h) = 1Xmh for all h 2 Hn. Observe that we have a descending chain of finite-dimensional affine subspaces of KV(M): Hm = Imum,m ◆ Imum+1,m ◆ Imum+2,m ◆ · · · 16 Art Discrete Appl. Math. 4 (2021) #P2.08 This sequence must stabilize after finitely many steps, namely there exists m0 m such that Imum0,m = Imum00,m for all m00 m0. We write Ĥm = Imum0,m for this nonempty affine subspace of Hm. We can verify that un+1,n maps Ĥn+1 onto Ĥn for all n 2 N0. It then follows that we can take (hn)n2N0 2 Q n2N0 Ĥn such that hn = un+1,n(hn+1). The required function f can now be taken as the one satisfying f |Xn = hn for all n 2 N0. Remark 4.14. Theorem 4.13 is not a special case of [33, Theorem 1] as the latter requires the digraph to be both out-finite and in-finite. For example, Figure 4 shows a strongly connected FON digraph M for which M> is not FON. For this digraph, Theorem 4.13 is applicable but [33, Theorem 1] is not. Remark 4.15. Let M be the infinite directed path described in Example 4.1. Then M is a countable weakly connected digraph without finite sink component, and it is BOD and hence FOD. For any g1 2 RV(M) with | supp(g1)| = 1, there is no h 2 `1(V(M)) such that LM h = g1. For g2 := 1V(M) 2 RV(M), there is no h 2 `1(V(M)) such that LM h = g2. These show that the analogues of Theorem 4.13 with FON replaced by FOD or BOD do not hold. It also tells us that we cannot change FON to be either FOD or BOD in Theorem 4.4. For a linear map, or for a map and its adjoint map, the relationship between injectivity and surjectivity is of lots of interest [14, 34]. Theorem 4.13 is about surjectivity. Let us also include a simple observation on injectivity. For each set V , let K[V ] = KVfin denote the linear space spanned by V , namely the set of functions on V with a finite support. Note that dimKV > dimK[V ] if and only if V is infinite. Theorem 4.16. Let M be an infinite FOD digraph. Let ⇤ 2 RV(M)0 . If S \ supp(⇤) 6= ; for every finite sink component S of M , then KerS M,⇤ \`p(V(M)) = {0} for every 0 < p < 1 and KerS M,⇤ \KV(M)fin = {0}. Proof. Let 0 6= g 2 KerS M,⇤. There are two cases to consider. Suppose the maximum of |g| over V(M) can be achieved at some vertex v. Observa- tion 4.6 then ensures g(w) = g(v) 6= 0 for every w 2 v"M . By Lemma 4.7, the harmonic function g must vanish on each finite sink component of M . Therefore, v"M contains in- finitely many vertices and |supp(g)| = 1. It also says that the summation P w2v" |g(w)| p diverges for every p > 0. Suppose the maximum of |g| over V(M) cannot be achieved anywhere. This means that there exists a sequence of vertices v1, v2, . . . such that |g(v1)| < |g(v2)| < · · · . Henceforth, |supp(g)| = 1, and the summation P1 i=1 |g(vi)| p diverges for every p > 0. We now conclude that KerS M,⇤ \`p(V(M)) = {0} and KerS M,⇤ \KV(M)fin = {0}, as wanted. Proof of Theorem 4.4. Combine Theorems 4.8 and 4.13. Question 4.17. There is a countability assumption in the statements of Theorems 4.4, 4.8, 4.10 and 4.13. Can this be relaxed? Y. Wu and D. Zhao: Three conjectures of Ostrander on digraph Laplacian eigenvectors 17 5 Final remarks Starting from the conjectures of Ostrander, we have tried to link the Alexandrov topology of a countable digraph and the support of nonnegative harmonic functions of a Schrödinger operator on that digraph. It may be interesting to go forward to the measurable framework [5, 6, 7, 19, 38], like the existing theory on Dirichlet forms, graphons, Borel equivalence relations, etc. To get stronger results, one may need to focus on some natural and im- portant situations, say models built on Bratteli diagrams. In some sense our work in this paper is based on the traditional homogeneous Markov chain theory. There is a beauti- ful decomposition-separation Theorem for finite nonhomogeneous Markov chains [49]. It looks interesting if we can also adapt that research to the study of countable weighted di- graphs. In the more algebraic direction, one can consider harmonic functions with values in a field of positive characteristic [60]. We conclude the paper with two simple examples about harmonic functions of finite Laplacians and invite the readers to see if it is possible to find a generalization to some infinite settings. It is well-known that Laplacian is related with energy minimization [11, 19]. The fol- lowing result of Harper provides one such example. Example 5.1 (Minimum mean-square error [23, 26]). The n-cube Hn has vertex set {0, 1}n and two vertices are adjacent if and only if their Hamming distance is one, i.e., they dif- fer exactly in one coordinate. How to design a bijection f from V(M) to [2n] so thatP uv2E(Hn)(g(u) g(v)) 2 attains the minimum value? For each i 2 [n], denote by Fi the function with Fi(a1 · · · an) = ai for all a1 · · · an 2 {0, 1}n = V(Hn). We consider the canonical bijection g = f + 1 where the function f is given by f = P n t=1 2 t1 Ft. To see that this function g really has minimum energy, we note that P uv2E(Hn)(g(u) g(v)) 2 =P uv2E(Hn)(f(u) f(v)) 2 = hf,LHn fi.The minimum eigenvalue of LHn is 0 and the second smallest eigenvalue of LHn is 2. The eigenspace of 0 is the space of constant functions and the eigenspace of 2 is spanned by those functions Fi for i 2 [n]. In Section 3, we let a matrix act both from right and from left on some vectors, namely consider an operator and its adjoint. This trick sometimes connects deterministic side and stochastic side, as the next example will show. Note that if we take a digraph M to be the probability transition matrix of a Markov chain, its Laplacian is simply I M and so the invariant measure lies in the left null space of LM . Example 5.2. Let M be an n ⇥ n row stochastic matrix. For each i 2 [n], let Mi be the matrix obtained from the identity matrix of order n by replacing its i-th row with the i-th row of M . Assume that the set of sink vertices of M is S and that every vertex in M has a path leading into S. Note that Mi is the identity matrix if and only if i 2 S. Let f 2 [n]N0 such that f1(i) is an infinite set for all i 2 [n]. When multiplying these matrices Mi on column vectors, we are modelling the process of opinion dynamics. If we apply Mf(t) at time t, the limiting opinion profile is determined in [58, Theorem 1.1]. Basically, this is equivalent to determining the limit of finite products Mf(0)Mf(1) · · ·Mf(t). However, if we apply Mf(0)Mf(1) · · ·Mf(t) on the right of row vectors for all t, we see that it is simply iteratively applying the matrix M on row spaces and so we know its limit [58, Theorem 3.3] immediately from the theory of absorbing Markov chains. ORCID iDs Yaokun Wu https://orcid.org/0000-0002-6811-7067 18 Art Discrete Appl. Math. 4 (2021) #P2.08 Da Zhao https://orcid.org/0000-0002-9582-0778 References [1] R. Agaev and P. 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Koptyug av. 4, 630090, Novosibirsk, Russia Received 22 February 2021, accepted 1 September 2021, published online 9 November 2021 Abstract In this work we present a survey of results on the problem of finding the minimum cardinality of the support of eigenfunctions of graphs. Keywords: Eigenfunction, eigenfunctions of graphs, eigenspace, minimum support, trade, bitrade, 1-perfect bitrade, weight distribution bound. Math. Subj. Class.: 05C50, 05E30, 05B30, 15A18 1 Introduction The eigenvalues of a graph are closely related to its structural properties and invariants (see the monographs [23, 32, 33]). Eigenfunctions (equivalently, eigenvectors) of graphs, in contrast to their eigenvalues, have received only sporadic attention of researchers. In particular, basic properties of eigenfunctions of graphs can be found in the work of Merris [68]. Among the most famous results we can recall the theory around Perron-Frobenius vector [7, 42, 74] with its applications to a variety of problems including ranking, pop- ulation growth models, Markov chains behavior and many other [52, 65, 73, 82]; and the results about Fiedler vector [29, 34, 38] and its connection to the problems of spectral graph partitioning and clustering [72, 79], graph coloring [3], graph drawing [59] and other (for example, [83, 84]). In addition, it is worth noting a series of works [16, 17, 18, 30, 35, 36, 41, 80, 95] devoted to various discrete versions of Courant’s nodal domain theorem. We refer the reader to [19], [31, Chapter 9] and [81] for more details about eigenfunctions of graphs. *The study was carried out within the framework of the state contract of the Sobolev Institute of Mathematics (project no. 0314-2019-0016). The authors are grateful to Evgeny Bespalov, Denis Krotov, Vladimir Potapov and Konstantin Vorob’ev for helpful discussions. †Corresponding author. The author is supported by RFBR according to the research project 20-51-53023 E-mail addresses: ev.v.sotnikova@gmail.com (Ev Sotnikova), graphkiper@mail.ru (Alexandr Valyuzhenich) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 2 Art Discrete Appl. Math. 4 (2021) #P2.09 In this work we consider undirected graphs without loops and multiple edges. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. Let G = (V,E) be a graph with vertex set V = {v1, . . . , vn} and let be an eigenvalue of G. The set of neighbors of a vertex x is denoted by N(x). A function f : V ! R is called a - eigenfunction of G if f 6⌘ 0 and the equality · f(x) = X y2N(x) f(y) (1.1) holds for any vertex x 2 V . Note that if f is a -eigenfunction of G, then A ! f = ! f , where A is the adjacency matrix of G and ! f = (f(v1), . . . , f(vn))T , i.e. ! f is an eigen- vector of the matrix A with eigenvalue . The set of functions f : V ! R satisfying (1.1) for any vertex x 2 V is called a -eigenspace of G. Denote by U(G) the -eigenspace of G. The support of a function f : V ! R is the set S(f) = {x 2 V | f(x) 6= 0}. A -eigenfunction of G is called optimal if it has the minimum cardinality of the support among all -eigenfunctions of G. In this work we focus on the following extremal problem for eigenfunctions of graphs. Problem 1.1 (the MS-problem). Let G be a graph and let be an eigenvalue of G. Find the minimum cardinality of the support of a -eigenfunction of G. In what follows, in this work we will use the abbreviation the MS-problem instead of Problem 1.1. Now we discuss the deep connection between the MS-problem and the intersection problem of two combinatorial objects and the problem of finding the minimum size of trades. Many combinatorial objects (equitable partitions, completely regular codes, Steiner systems S(k 1, k, n), 1-perfect codes, etc.) can be defined as eigenfunctions of graphs with some discrete restrictions. The study of such objects often leads to the problem of finding the minimum possible difference between two objects from the same class (for ex- ample, see [37, 40, 48, 76, 77]). Since the symmetric difference of such two objects is also an eigenfunction of the corresponding graph, this problem is directly related to the MS-problem. Trades of different types are used for constructing and studying the structure of differ- ent combinatorial objects (combinatorial t-designs, codes, Latin squares, etc.). Trades are also studied independently as some natural generalization of objects of the corresponding type (trades can exist even if the corresponding complete objects do not exist). Roughly speaking, trades reflect possible differences between two combinatorial objects from the same class: if C 0 and C 00 are two combinatorial objects with the same parameters, then the pair (C 0\C 00, C 00\C 0) is a trade (for more information on trades see [15, 24, 45, 64]). Many types of trades (T (k 1, k, v) Steiner trades, q-ary Tq(k 1, k, v) Steiner trades, 1-perfect trades, extended 1-perfect trades, latin trades, etc.) can be represented as eigenfunctions of the corresponding graphs with some additional discrete restrictions (for example, see [63, Section 2.4]). So, for such trades the problem of finding the minimum size can be reduced to the MS-problem for the corresponding graphs (see, for example, [64, 92, 98]). In particular, the MS-problem has appeared as a natural generalization of the following results. • Let C1 and C2 be two distinct binary perfect codes of length n = 2m 1. In [37] Etzion and Vardy proved that the maximum possible cardinality of their intersection E. Sotnikova and A. Valyuzhenich: Minimum supports of eigenfunctions of graphs: a survey 3 C1\C2 is 2nm2 n1 2 . Equivalently, they found the minimum possible cardinality of their symmetric difference C14C2. This result can be proved by applying the so- called weight distribution bound for the Hamming graph H(n, 2) and its eigenvalue 1 (see Subsection 2.1 and Section 4). • In [48] Hwang proved that the minimum size of a T (t, k, v) trade is 2t+1 and ob- tained a characterization of T (t, k, v) trades of size 2t+1. In particular, the minimum size of a T (t, k, v) Steiner trade was found in [48]. For t = k 1 this result can be proved by applying the weight distribution bound for the Johnson graph J(v, k) and its eigenvalue k (see Subsection 2.2 and Section 4). It is interesting that Frankl and Pach [40] also found the minimum size of a T (t, k, v) trade. They formulated their results in terms of null t-designs. In Section 8 we will meet null designs again during our discussion about optimal eigenfunctions of the Grassmann graph. The MS-problem was first formulated by Krotov and Vorob’ev [98] in 2014 (they con- sidered the MS-problem for the Hamming graph). During the last six years, the MS- problem has been actively studied for various families of distance-regular graphs [8, 44, 62, 64, 88, 89, 91, 92, 93, 98, 96] and Cayley graphs on the symmetric group [51]. In particular, the MS-problem is completely solved for all eigenvalues of the Hamming graph [92, 93] and asymptotically solved for all eigenvalues of the Johnson graph [96]. Note that for eigenfunctions of distance-regular graphs a lower bound for its support cardinality is known. This bound is called the weight distribution bound and we will discuss it in details in Section 4. In this work we give a survey of results on the MS-problem. We also discuss constructions of optimal eigenfunctions and the main ideas of the proofs of the results. Now we would like to consider the following problem. Problem 1.2. Let G = (V,E) be a graph and let be an eigenvalue of G. Find min f2U(G),f 6⌘0 |{x 2 V | f(x) 0}|. Note, that the statements of the MS-problem and Problem 1.2 are similar. An analogue of Problem 1.2 for association schemes was first formulated in 1984 by Bier [10]. Later, Bier and Delsarte [13] and Bier [11, 12] studied the same problem for eigenvectors belong- ing to the direct sum of several eigenspaces of an association scheme. Bier and Manickam [14], Manickam and Miklós [66] and Manickam and Singhi [67] initiated the study of Prob- lem 1.2 for the second largest eigenvalue of Johnson and Grassmann graphs. In particular, the following two conjectures were formulated in 1988. Conjecture 1.3 (Manickam, Miklós and Singhi [66, 67]). Let x1, . . . , xn be real numbers such that x1 + . . .+ xn = 0. If n 4k, then there are at least n1 k1 k-element subsets of the set {x1, . . . , xn} with nonnegative sum. The second conjecture is an analogue of Conjecture 1.3 for vector spaces. Let V be an n-dimensional vector space over a finite field Fq . Let ⇥V k ⇤ q denote the family of all k- dimensional subspaces of V and let ⇥n k ⇤ q denote the q-Gaussian binomial coefficient. For each 1-dimensional subspace v 2 ⇥V 1 ⇤ q , assign a real-valued weight f(v) 2 R so that the sum of all weights is zero. For a general subspace S ⇢ V , define its weight f(S) to be the sum of the weights of all the 1-dimensional subspaces it contains. 4 Art Discrete Appl. Math. 4 (2021) #P2.09 Conjecture 1.4 (Manickam and Singhi [67]). Let V be an n-dimensional vector space over Fq and let f : ⇥V 1 ⇤ q ! R be a weighting of the 1-dimensional subspaces such that P v2[V1 ]q f(v) = 0. If n 4k, then there are at least ⇥n1 k1 ⇤ q k-dimensional subspaces with nonnegative weight. Conjecture 1.3 is still open. However, there are several relatively recent works [2, 28, 39, 75] with polynomial bounds. In particular, Alon, Huang and Sudakov [2] verified Conjecture 1.3 for n 33k2. A linear bound n 1046k was obtained by Pokrovskiy [75]. In 2014 Chowdhury, Sarkis and Shahriari [28] and Huang and Sudakov [47] independently showed that Conjecture 1.4 holds for n 3k. Using the technique of the work [28], Ihringer [49] proved that Conjecture 1.4 is true for n 2k and large q. Some new results on Problem 1.2 for the third largest eigenvalue of the Johnson graph can be found in [71]. It seems very intriguing to establish the interconnection between Problem 1.2 and the MS- problem. The paper is organized as follows. In Section 2, we give two examples of combinatorial problems that are closely related to the MS-problem. In Section 3, we introduce basic definitions and notations. In Section 4, we discuss what the weight distribution bound is and how it can be calculated from the intersection arrays of the distance-regular graphs. We complete this section with several intuitive examples and some results for the special cases when the bound is achieved. In Sections 5-11, we give a survey of results on the MS- problem for the Hamming graph, the Doob graph, the Johnson graph, the Grassmann graph, the bilinear forms graph, the Paley graph and the Star graph respectively. In Section 12, we present some observations on optimal eigenfunctions of graphs. In Section 13, we formulate several open problems. 2 Eigenfunctions in combinatorial configurations and the MS-problem In this section, we recall that equitable 2-partitions, 1-perfect codes and T (k 1, k, v) Steiner trades can be defined as eigenfunctions of graphs with some discrete restrictions. We also discuss the connections of the MS-problem with the intersection problem of two 1-perfect codes of a given graph and the problem of finding the minimum size of Steiner trades. 2.1 Equitable partitions and 1-perfect codes Let G = (V,E) be a graph. An ordered r-partition (C1, . . . , Cr) of V is called equitable if for any i, j 2 {1, . . . , r} there is Si,j such that any vertex of Ci has exactly Si,j neighbors in Cj . The matrix S = (Si,j)i,j2{1,...,r} is called the quotient matrix of the equitable partition. A set C ✓ V is called a 1-perfect code in G if every ball of radius 1 contains one vertex from C. For more information on equitable partitions and perfect codes we refer the reader to [9], [43, Chapter 5] and [1, 46, 86, 87]. Let G be a k-regular graph and let (C1, C2) be an equitable 2-partition of G with the quotient matrix S = ✓ a b c d ◆ . The eigenvalues of S are k and a c. We define the function f(C1,C2) on the vertices of G E. Sotnikova and A. Valyuzhenich: Minimum supports of eigenfunctions of graphs: a survey 5 by the following rule: f(C1,C2)(x) = ( b, if x 2 C1; c, if x 2 C2. One can verify that f(C1,C2) is an (a c)-eigenfunction of G. So, any equitable 2-partition can be represented as an eigenfunction of the corresponding graph. Suppose that C is a 1-perfect code in G. Then the partition (C,C) is equitable with the quotient matrix ✓ 0 k 1 k 1 ◆ . Therefore, the function f(C,C) is a (1)-eigenfunction of G. So, if C1 and C2 are 1-perfect codes in G, then the function f = f(C1,C1) f(C2,C2) is also a (1)-eigenfunction of G. Moreover, we have the equality |S(f)| = |C14C2|. Thus, the problem of finding the minimum cardinality of the symmetric difference of two distinct 1-perfect codes of a regular graph can be reduced to the MS-problem for this graph and eigenvalue 1. 2.2 T (k 1, k, v) Steiner trades Let v, k, t be positive integers such that v > k > t and let X be a set of size v. A pair (T0, T1) of disjoint collections of k-subsets (blocks) of X is called a T (t, k, v) trade if every t-subset of X is included in the same number of blocks of T0 and T1. The size of a T (t, k, v) trade (T0, T1) is |T0|+ |T1|. A T (t, k, v) trade is called Steiner if every t-subset of X is included in at most one block of T0 (T1). For further details on T (t, k, v) trades we refer the reader to [15, 45, 55]. Suppose that (T0, T1) is a T (k 1, k, v) Steiner trade. The Johnson graph J(v, k) can be defined as follows. The vertices of J(v, k) are k-subsets of X , and two vertices are adjacent if they have exactly k 1 common elements. We define the function f(T0,T1) on the vertices of J(v, k) by the following rule: f(T0,T1)(x) = 8 >< >: 1, if x 2 T0; 1, if x 2 T1; 0, otherwise. For a (k 1)-subset A of X denote by C(A) the set of vertices of J(v, k) containing the set A (these vertices form a clique of size v k + 1 in J(v, k)). We note that C(A) either contains one element from T0 and one element from T1 or does not contain elements from T0 [ T1. Using this fact, one can easily check that f(T0,T1) is a (k)-eigenfunction of J(v, k). Moreover, we have the equality |S(f(T0,T1))| = |T0|+ |T1|. Thus, the problem of finding the minimum size of T (k 1, k, v) Steiner trades can be reduced to the MS-problem for the Johnson graph J(v, k) and its eigenvalue k. 6 Art Discrete Appl. Math. 4 (2021) #P2.09 3 Basic definitions Recall that a distance dG(v, u) = d(u, v) between two vertices v and u in a graph G = (V, E) is the length of the shortest path that connects them. The largest distance between any pairs of vertices is called the diameter D. A connected graph G = (V, E) is called distance-regular if it is regular of degree k and for any two vertices v, u 2 V at distance i = d(v, u) there are precisely ci neighbors of u which are at distance i1 from v and precisely bi neighbors of u which are at distance i+ 1 from v; where ci and bi do not depend on the choice of vertices u and v but depend only on d(u, v). Numbers bi, ci, ai = k bi ci are called the intersection numbers and a set {b0, b1, . . . , bD1; c1, . . . , cD} is called an intersection array of a distance-regular graph G. For more details about distance-regular graphs, the reader is referred to a classical monograph [22] and a recent survey [94]. Let G1 = (V1, E1) and G2 = (V2, E2) be simple graphs. The Cartesian product G1⇤G2 of graphs G1 and G2 is defined as follows. The vertex set of G1⇤G2 is V1 ⇥ V2; and any two vertices (x1, y1) and (x2, y2) are adjacent if and only if either x1 = x2 and y1 is adjacent to y2 in G2, or y1 = y2 and x1 is adjacent to x2 in G1. Suppose G1 = (V1, E1) and G2 = (V2, E2) are two graphs. Let f1 : V1 ! R and f2 : V2 ! R. Denote G = G1⇤G2. We define the tensor product f1 · f2 on the vertices of G by the following rule: (f1 · f2)(x, y) = f1(x)f2(y) for (x, y) 2 V (G) = V1 ⇥ V2. We will use the tensor product of functions for constructing optimal eigenfunctions of the Hamming and Doob graphs in Subsection 5.1 and Section 6. Let Sym(X) denote the symmetric group on a finite set X and let Symn denote the symmetric group on the set {1, . . . , n}. Let ⌃q = {0, 1, . . . , q 1}. Let f(x1, . . . , xn) be a function defined on the set ⌃nq , let ⇡ 2 Symn and let 1, . . . ,n 2 Sym(⌃q). We define the functions f⇡ and f⇡,1,...,n as follows: f⇡(x1, . . . , xn) = f(x⇡(1), . . . , x⇡(n)) and f⇡,1,...,n(x1, . . . , xn) = f(1(x⇡(1)), . . . ,n(x⇡(n))). We will use the functions f⇡ and f⇡,1,...,n in Subsections 5.1 and 5.2. Let G = (V,E) be a graph. A set C ✓ V is called a completely regular code in G if the partition (C(0), . . . , C(⇢)) is equitable, where C(d) is the set of vertices at distance d from C and ⇢ (the covering radius of C) is the maximum d for which C(d) is nonempty. In other words, a subset of V is a completely regular code in G if the distance partition with respect to the subset is equitable. For more information on completely regular codes see [21], [43, Chapter 11.7] and [57, 58]. We will use completely regular codes in Section 12. 4 The weight distribution bound In this section we recall what the weight distribution bound is and how it can be used as a lower bound for the MS-problem in case of distance-regular graphs. The weight distri- bution bound is well known and has appeared in several papers under different disguise (for more details see [61, 64]). In order for this survey to be self-contained we would like to provide the full proof and equip the reader with several intuitive examples. We also highlight the connections with related theoretical frameworks. E. Sotnikova and A. Valyuzhenich: Minimum supports of eigenfunctions of graphs: a survey 7 In Subsection 4.1, we give a detailed proof of the weight distribution bound and dis- cuss related results. In Subsection 4.2, we provide several intuitive examples to illustrate how the weight distribution bound works. In Subsection 4.3, we give a survey of results on the existence problem of eigenfunctions of distance-regular graphs meeting the weight distribution bound. 4.1 The proof of the weight distribution bound In this subsection we discuss the proof of the weight distribution bound and related results. Let A be the adjacency matrix of some distance-regular graph G = (V, E). Now consider the distance-i graph Gi = (V, Ei) defined as follows: two vertices x and y are adjacent in Gi if and only if they are at distance i in G. In other words, {x, y} 2 Ei () dG(x, y) = i. By Ai we denote the adjacency matrix of Gi. For a vertex x of G and 0  i  D denote Ni(x) = {y 2 V | dG(y, x) = i}. Considering the combinatorial definition of distance regularity from the matrix point of view, we obtain the following recurrence (see, for example, equation (1) in [94]): AiA = aiAi + bi1Ai1 + ci+1Ai+1, (4.1) for i = 0, 1, . . . , D where b1A1 = cD+1AD+1 = 0. From the above we can show that there exist polynomials Pi of degree i such that: Ai = Pi(A), i = 0, 1, . . . , D. It is well known that in case of Hamming graphs these polynomials are actually Kravchuk polynomials (up to some linear change of variables) and those are Eberlein polynomials in case of Johnson graphs. But how can we make use of it in finding the lower bound for our MS-problem? Sup- pose f is a -eigenfunction of our graph G. Since Ak ! f = k ! f , we get the following equations: Ai ! f = Pi(A) ! f = Pi() ! f . In other words, f is a Pi()-eigenfunction of Gi. As an immediate consequence we obtain: Pi() · f(x) = X y2Ni(x) f(y). (4.2) In other words, in distance-regular graphs the sum of the eigenfunction values on the vertices at distance i from a fixed vertex x depends only on f(x) and the corre- sponding eigenvalue. Without lost of generality we can consider f(x) = 1. The array [1, P1(), . . . , PD()] is called the weight distribution of a -eigenfunction. Thus, from (4.1) we can write the following recurrence: P0() = 1, P1() = , Pi() = Pi1() bi2Pi2() ai1Pi1() ci , where i = 2, . . . , D. 8 Art Discrete Appl. Math. 4 (2021) #P2.09 Now we are just one step away from obtaining the lower bound we are looking for. Let z be a vertex of G such that |f(z)| = max y2V |f(y)|. Let us prove the inequality |S(f) \Ni(z)| |Pi()| (4.3) for any 1  i  D. Using (4.2), we obtain |Pi() · f(z)| = | X y2Ni(z) f(y)| = | X y2S(f)\Ni(z) f(y)|  X y2S(f)\Ni(z) |f(y)|   |S(f) \Ni(z)| · |f(z)|. (4.4) Then we have |S(f)| = DX i=0 |S(f) \Ni(z)| DX i=0 |Pi()|. So, we prove the next lemma. Lemma 4.1 ([64, Corollary 1]). Let f be a -eigenfunction for a distance-regular graph G of diameter D, then the following bound takes place: |S(f)| DX i=0 |Pi()|. In case of irrational eigenvalues this bound can be refined as follows: Lemma 4.2. Let f be a -eigenfunction for a distance-regular graph G of diameter D, then the following bound takes place: |S(f)| DX i=0 d|Pi()|e. In addition, eigenfunctions of distance-regular graphs meeting the weight distribution bound have the following interesting properties. Lemma 4.3. Let f be a -eigenfunction of a distance-regular graph G and let f meet the weight distribution bound. Then the following statements hold: 1. There is a real positive number c such that f(x) 2 {c, 0, c} for any vertex x of G. 2. For any vertex x 2 S(f) and any 1  i  D the function f is either non-negative or non-positive on the set Ni(x). Proof. 1. Let us analyze the proof of Lemma 4.1 more carefully. Since f meets the weight distribution bound, we have the equalities in (4.3) and (4.4) for any 1  i  D. Therefore |f(y)| = |f(z)| for any y 2 S(f) \ Ni(z). So, we have |f(y)| = |f(z)| for any vertex y 2 S(f) and we can take c = |f(z)|. 2. Let us consider a vertex x 2 S(f) and i 2 {1, . . . , D}. By the first case of this lemma we obtain |f(x)| = max y2V |f(y)|. So, the inequality (4.4) holds for z = x. Moreover, we have equality in (4.4) for z = x. Consequently, all vertices from S(f) \ Ni(x) take either positive values or negative values. E. Sotnikova and A. Valyuzhenich: Minimum supports of eigenfunctions of graphs: a survey 9 Lemma 4.4. Let f be a -eigenfunction of a distance-regular graph G, where < 0, and let f meet the weight distribution bound. If x and y are two distinct vertices from S(f) and x and y are adjacent in G, then f(x)f(y) < 0. Proof. Let x be a vertex from S(f). Without loss of generality, we can assume that f(x) > 0. Using Lemma 4.3 and the definition of an eigenfuction, we see that all vertices from the set S(f) \ N1(x) take negative values. So, x does not have neighbors with positive values. 4.2 Examples In this subsection, we would like to illustrate Lemmas 4.1 and 4.2 on some well-known graphs (see [88] for details). We start with the Petersen graph. The Petersen graph is a cubi- cal distance-regular graph on 10 vertices. Its intersection array is {3, 2; 1, 1} and its eigen- values are {2(4), 1(5), 3(1)}. Calculating the weight distribution we obtain [1,,2 3]. • For = 1 it gives us the lower bound 4. An optimal 1-eigenfunction achieves this bound. A subgraph induced on non-zero vertices can be described as two non- incident edges. An example is presented below (Figure 1). Figure 1: Optimal 1-eigenfunction of the Petersen graph. • For = 2 the lower bound is the same. But this case is different because this bound cannot be achieved. Optimal (2)-eigenfunction has a support of cardinality 6 and the corresponding induced subgraph is either a cycle on six vertices, or H-graph. See Figure 2 and Figure 3. As a quick illustration of bound refinement, let us consider the Heawood graph, a distance-regular graph on 14 vertices. Its intersection array is {3, 2, 2; 1, 1, 3} and its spec- trum is {±3(1),± p 2 (6)}. The weight distribution is [1,,2 3, 13 ( 3 5)]. Thus for = ± p 2 the exact weight distribution bound is 2 + 2 p 2, while the refined bound is 6. Figure 4 presents an example of an optimal p 2-eigenfunction. More examples can be found in [88], where the MS-problem is solved together with a characterisation of such functions for 10 out of 13 cubical distance-regular graphs for all their eigenvalues. Thus for any distance-regular graph a lower bound on the cardinality of a -eigenfunction support can be calculated directly from the intersection array of a graph with respect to the corresponding eigenvalue . However this bound is not necessary feasible. 10 Art Discrete Appl. Math. 4 (2021) #P2.09 Figure 2: Optimal (2)-eigenfunction of the Petersen graph — cycle Figure 3: Optimal (2)-eigenfunction of the Petersen graph — H-graph Figure 4: Optimal p 2-eigenfunction of the Heawood graph. 4.3 Eigenfunctions of distance-regular graphs meeting the weight distribution bound In this subsection we discuss results on the existence problem of eigenfunctions distance- regular graphs meeting the weight distribution bound. We also give examples of distance- regular graphs when the weight distribution bound is achieved. Firstly, we need several definitions. Let G be a connected k-regular graph. Suppose that S is a set of (s + 1)-cliques in G such that every edge of G is included in exactly m cliques from S. The pair (G,S) is called a (k, s,m) pair. A couple (T0, T1) of mutually disjoint nonempty sets of vertices is called a clique bitrade if every clique from S either intersects with each of T0 and T1 in exactly one vertex or does not intersect with both of them. A (k, s,m) pair (G,S) is called a Delsarte pair if the graph G is distance-regular and S consists of Delsarte cliques. Recall that a clique in a distance-regular graph of degree k and diameter D is called a Delsarte clique if it consists of exactly 1 k/D vertices, where D is the smallest eigenvalue of the graph. For a function f : V ! R denote S+(f) = {x 2 V | f(x) > 0} and S(f) = {x 2 V | f(x) < 0}. Let G be a distance-regular graph of diameter D admitting a Delsarte pair (G,S). Suppose f is a D-eigenfunction of G meeting the weight distribution bound. Let us show that (S+(f), S(f)) is a clique bitrade in G. Indeed, by Theorem 2 from [64] for every Delsarte clique C from S it holds P x2C f(x) = 0. Then Lemmas 4.3 and 4.4 imply that E. Sotnikova and A. Valyuzhenich: Minimum supports of eigenfunctions of graphs: a survey 11 any clique from S either does not contain vertices from S(f) or contains one vertex from S+(f) and one vertex from S(f). Consequently, (S+(f), S(f)) is a clique bitrade in G. Thus, we prove the following result. Lemma 4.5. Let G be a distance-regular graph of diameter D admitting a Delsarte pair. A function f is a D-eigenfunction of G meeting the weight distribution bound if and only if (S+(f), S(f)) is a clique bitrade in G meeting the weight distribution bound. Using Lemma 4.5 and Theorem 3 from [64], we see that for every distance-regular graph admitting a Delsarte pair the existence of a D-eigenfunction achieving the weight distribution bound is equivalent to the existence of a regular bipartite isometric subgraph of degree D. Taking into account this observation, it was shown in [44, 64, 89] that the weight distribution bound is achieved for the smallest eigenvalue of the following graphs: • Hamming graph; • Johnson graph; • Grassmann graph; • Paley graph of square order (since this graph is self-complementary of diameter 2, this property also holds for the second non-principal eigenvalue); • strongly regular bilinear forms graph over a prime field. For a deeper dive into the theory of clique bitrades the interested reader is referred to Sections 2 and 3 of [64]. Finally, we note that eigenfunctions of distance-regular graphs meeting the weight dis- tribution bound exist not only in the case of the smallest eigenvalue. For example, the weight distribution bound is achieved for: • n-dimensional hypercube, where n is odd, and its eigenvalue 1; • n-dimensional hypercube, where n is even, and its eigenvalue 0; • the Hamming graph H(2, q) and its eigenvalue q 2. 5 Hamming graph In this section, we give a survey of results on the MS-problem and its generalizations for the Hamming graph. The Hamming graph H(n, q) is defined as follows. Let ⌃q = {0, 1, . . . , q 1}. The vertex set of H(n, q) is ⌃nq , and two vertices are adjacent if they differ in exactly one position. This graph is a distance-regular graph. The Hamming graph H(n, q) has n + 1 distinct eigenvalues i(n, q) = n(q 1) q · i, where 0  i  n. Denote by Ui(n, q) the i(n, q)-eigenspace of H(n, q). The direct sum of subspaces Ui(n, q) Ui+1(n, q) . . . Uj(n, q) for 0  i  j  n is denoted by U[i,j](n, q). We say that a function f 2 U[i,j](n, q), where f 6⌘ 0, is optimal in the space U[i,j](n, q) if |S(f)|  |S(g)| for any function g 2 U[i,j](n, q), g 6⌘ 0. Firstly, we briefly discuss all results on the MS-problem for the Hamming graph. After that, we will consider the more general Problem 5.1 for functions from the space U[i,j](n, q). In [62] Krotov based on the approach of work [78] proved that the minimum 12 Art Discrete Appl. Math. 4 (2021) #P2.09 cardinality of the support of a i(n, 2)-eigenfunction of H(n, 2) is max(2i, 2ni). In [98] Krotov and Vorob’ev showed that the cardinality of the support of a i(n, q)-eigenfunction of H(n, q) is at least 2i · (q 2)ni for iq 2 2n(q1) > 2 and q n · ( 1 q 1) i/2 · ( i n i ) i/2 · (1 i n )n/2 for iq 2 2n(q1)  2. In [91] Valyuzhenich for q 3 proved that the minimum cardinality of the support of a 1(n, q)-eigenfunction of H(n, q) is 2 · (q 1) · qn2 and obtained a characterization of optimal 1(n, q)-eigenfunctions. Later in [92, 93] the following gener- alization of the MS-problem for the Hamming graph was considered. Problem 5.1. Let n 1, q 2 and 0  i  j  n. Find the minimum cardinality of the support of functions from the space U[i,j](n, q). In [93] Valyuzhenich and Vorob’ev found the minimum cardinality of the support of a function from the space U[i,j](n, q) for arbitrary q 3 except the case when q = 3 and i + j > n. Moreover, in [93] a characterization of functions that are optimal in the space U[i,j](n, q) was obtained for q 3, i + j  n and q 5, i = j, i > n2 . In [92] Valyuzhenich found the minimum cardinality of the support of a function from the space U[i,j](n, q) for q = 2 and q = 3, i + j > n. Thus, Problem 5.1 is completely solved for all n 1 and q 2. As a consequence, the MS-problem for the Hamming graph is also solved for all eigenvalues. In what follows, in this section we will consider in detail Problem 5.1. In Subsec- tion 5.1, we present constructions of functions that are optimal in the space U[i,j](n, q). In Subsection 5.2, we give a survey of results on Problem 5.1 and discuss the main ideas of the proof of these results. In particular, we carefully explore Lemma 5.3 which is a key tool for solving Problem 5.1. In Subsection 5.3, we focus on a connection between Problem 5.1 and the problem of finding the minimum size of 1-perfect bitrades in the Hamming graph. 5.1 Constructions of functions with the minimum cardinality of the support In this subsection, we discuss constructions of functions that are optimal in the space U[i,j](n, q). It is interesting that in all cases such functions are constructed as a tensor product of several elementary optimal functions defined on the vertices of the Hamming graph of diameter not greater than three. Firstly, we define five sets of elementary optimal functions. For k,m 2 ⌃q we define the function aq,k,m on the vertices of the Hamming graph H(2, q) by the following rule: aq,k,m(x, y) = 8 >< >: 1, if x = k and y 6= m; 1, if y = m and x 6= k; 0, otherwise. The function a3,1,1 is shown in Figure 5. We note that aq,k,m is optimal in the space U1(2, q) for any k,m 2 ⌃q . Denote Aq = {aq,k,m | k,m 2 ⌃q}. E. Sotnikova and A. Valyuzhenich: Minimum supports of eigenfunctions of graphs: a survey 13 We define the function '1 on the vertices of the Hamming graph H(2, 3) by the fol- lowing rule: '1(x, y) = 8 >< >: 1, if x = y = 0; 1, if x = 1 and y = 2; 0, otherwise. For a, b 2 ⌃3 denote by a b the sum of a and b modulo 3. We define the function ' on the vertices of the Hamming graph H(3, 3) by the following rule: '(x, y, z) = 8 >< >: '1(x, y), if z = 0; '1(x 1, y 1), if z = 1; '1(x 2, y 2), if z = 2. The function ' is shown in Figure 6. We note that ' is optimal in the space U2(3, 3). Denote B = {'⇡,1,2,3 | ⇡ 2 Sym3,1,2,3 2 Sym(⌃3)}. For k,m 2 ⌃q and k 6= m we define the function cq,k,m on the vertices of the Hamming graph H(1, q) by the following rule: cq,k,m(x) = 8 >< >: 1, if x = k; 1, if x = m; 0, otherwise. The function c4,0,1 is shown in Figure 7. We note that cq,k,m is optimal in the space U1(1, q) for any k,m 2 ⌃q and k 6= m. Denote Cq = {cq,k,m | k,m 2 ⌃q, k 6= m}. For k 2 ⌃q we define the function dq,k on the vertices of the Hamming graph H(1, q) by the following rule: dq,k(x) = ( 1, if x = k; 0, otherwise. The function d4,0 is shown in Figure 7. We note that dq,k is optimal in the space U[0,1](1, q) for any k 2 ⌃q . Denote Dq = {dq,k | k 2 ⌃q}. Let eq : ⌃q ! R and eq ⌘ 1. The function e4 is shown in Figure 7. We note that eq is optimal in the space U0(1, q). Denote Eq = {eq}. Now, we define four classes of functions that are optimal in the space U[i,j](n, q) for the corresponding cases. Let i + j  n. We say that a function f defined on the vertices of H(n, q) belongs to the class F1(n, q, i, j) if f = c · iY k=1 gk · nijY k=1 hk · jiY k=1 vk, where c is a real non-zero constant, gk 2 Aq for k 2 [1, i], hk 2 Eq for k 2 [1, n i j] and vk 2 Dq for k 2 [1, j i]. 14 Art Discrete Appl. Math. 4 (2021) #P2.09 Figure 5: Function a3,1,1 in H(2, 3). Figure 6: Function '(x, y, z) in H(3, 3). E. Sotnikova and A. Valyuzhenich: Minimum supports of eigenfunctions of graphs: a survey 15 Figure 7: Functions c4,0,1, d4,0 and e4 in H(1, 4). Let i + j > n. We say that a function f defined on the vertices of H(n, q) belongs to the class F2(n, q, i, j) if f = c · njY k=1 gk · i+jnY k=1 hk · jiY k=1 vk, where c is a real non-zero constant, gk 2 Aq for k 2 [1, n j], hk 2 Cq for k 2 [1, i+ j n] and vk 2 Dq for k 2 [1, j i]. Let i2 + j  n and i + j > n. We say that a function f defined on the vertices of H(n, 3) belongs to the class F3(n, i, j) if f = c · 2ni2jY k=1 gk · i+jnY k=1 hk · jiY k=1 vk, where c is a real non-zero constant, gk 2 A3 for k 2 [1, 2n i 2j], hk 2 B for k 2 [1, i+ j n] and vk 2 D3 for k 2 [1, j i]. Let i2 + j > n. We say that a function f defined on the vertices of H(n, 3) belongs to the class F4(n, i, j) if f = c · njY k=1 gk · i+2j2nY k=1 hk · jiY k=1 vk, where c is a real non-zero constant, gk 2 B for k 2 [1, n j], hk 2 C3 for k 2 [1, i+ 2j 2n] and vk 2 D3 for k 2 [1, j i]. We note that functions from F1(n, q, i, j) and F2(n, q, i, j) are optimal in the space U[i,j](n, q) for q 2, i + j  n and q 2 (q 6= 3), i + j > n respectively. We also note that functions from F3(n, i, j) and F4(n, i, j) are optimal in the space U[i,j](n, 3) for i 2 + j  n, i+ j > n and i 2 + j > n respectively. 16 Art Discrete Appl. Math. 4 (2021) #P2.09 5.2 Problem 5.1 In this subsection, we discuss Problem 5.1. The following theorem is a combination of the results proved in [92, 93] (see [93, Theorems 1 and 3] and [92, Theorems 3-6]). Theorem 5.2. 1. Let f 2 U[i,j](n, q), where q 2, i+ j  n and f 6⌘ 0. Then |S(f)| 2i · (q 1)i · qnij and this bound is sharp. Moreover, for q 3 the equality |S(f)| = 2i · (q 1)i · qnij holds if and only if f⇡ 2 F1(n, q, i, j) for some permutation ⇡ 2 Symn. 2. Let f 2 U[i,j](n, q), where q 2, q 6= 3, i+ j > n and f 6⌘ 0. Then |S(f)| 2i · (q 1)nj and this bound is sharp. Moreover, for i = j and q 5 the equality |S(f)| = 2i ·(q1)ni holds if and only if f⇡ 2 F2(n, q, i, i) for some permutation ⇡ 2 Symn. 3. Let f 2 U[i,j](n, 3), where i2 + j  n, i+ j > n and f 6⌘ 0. Then |S(f)| 23(nj)i · 3i+jn and this bound is sharp. 4. Let f 2 U[i,j](n, 3), where i2 + j > n and f 6⌘ 0. Then |S(f)| 2i+jn · 3nj and this bound is sharp. Now, we discuss the main ideas of the proof of Theorem 5.2. Let f be a real-valued function defined on the vertices of the Hamming graph H(n, q) and let k 2 ⌃q , r 2 {1, . . . , n}. We define a function frk on the vertices of H(n 1, q) as follows: for any vertex y = (y1, . . . , yr1, yr+1, . . . , yn) of H(n 1, q) f r k (y) = f(y1, . . . , yr1, k, yr+1, . . . , yn). One of the important points in the proof of Theorem 5.2 is the following. Lemma 5.3 ([93, Lemma 4]). Let f 2 U[i,j](n, q) and r 2 {1, 2, . . . , n}. Then the follow- ing statements are true: 1. frk frm 2 U[i1,j1](n 1, q) for k,m 2 ⌃q . 2. Pq1 k=0 f r k 2 U[i,j](n 1, q). 3. frk 2 U[i1,j](n 1, q) for k 2 ⌃q . Lemma 5.3 is a very useful tool for studying of eigenfunctions of the Hamming graph. It shows the connection between eigenspaces of the Hamming graphs H(n, q) and H(n 1, q). In particular, this lemma allows to apply induction on n, i and j (we can use the induction assumption for the functions frk frm, Pq1 k=0 f r k and f r k ). Moreover, we suppose that Lemma 5.3 can be useful not only for the MS-problem but also for other problems. For example, recently in [69] Mogilnykh and Valyuzhenich used Lemma 5.3 for investigation of equitable 2-partitions of the Hamming graph with the eigenvalue 2(n, q). One interest- ing generalization of Lemma 5.3 for the products of graphs can be found in [90, Theorem 3.11]. E. Sotnikova and A. Valyuzhenich: Minimum supports of eigenfunctions of graphs: a survey 17 5.3 Minimum 1-perfect bitrades in the Hamming graph In this subsection, we discuss one interesting application of Theorem 5.2 for the problem of finding the minimum size of 1-perfect bitrades in the Hamming graph. Let us recall some definitions. Let G = (V,E) be a graph. For a vertex x 2 V denote B(x) = N(x) [ {x}. Let T0 and T1 be two disjoint nonempty subsets of V . The ordered pair (T0, T1) is called a 1-perfect bitrade in G if for any vertex x 2 V the set B(x) either contains one element from T0 and one element from T1 or does not contain elements from T0 [ T1. The size of a 1-perfect bitrade (T0, T1) is |T0|+ |T1|. Example 5.4. Let T0 = {000, 111} and T1 = {001, 110}. Then (T0, T1) is a 1-perfect bitrade of size 4 in H(3, 2) (see Figure 8). Figure 8: 1-perfect bitrade in H(3, 2). Example 5.5. Let G = (V,E) be a graph. Suppose C1 and C2 be two distinct 1-perfect codes in G. Then (C1 \ C2, C2 \ C1) is a 1-perfect bitrade in G. Let (T0, T1) be a 1-perfect bitrade in a graph G = (V,E). We define the function f(T0,T1) : V ! {1, 0, 1} by the following rule: f(T0,T1)(x) = 8 >< >: 1, if x 2 T0; 1, if x 2 T1; 0, otherwise. In what follows, in this subsection we will consider the following problem. Problem 5.6. Let n 3 and q 2. Find the minimum size of a 1-perfect bitrade in H(n, q). For q = 2 Problem 5.6 was essentially solved by Etzion and Vardy [37] and Solov’eva [85] (the results were formulated for more special cases of 1-perfect bitrades embedded into perfect binary codes, but both proofs work in the general case). In [70] Mogilnykh and Solov’eva for arbitrary q 2 proved that the minimum size of a 1-perfect bitrade in H(q + 1, q) is 2 · q!. Now, using Theorem 5.2, we give a short solution of Problem 5.6 for q = 3 and q = 4. Firstly, we need the following result. 18 Art Discrete Appl. Math. 4 (2021) #P2.09 Lemma 5.7 ([92, Lemma 6]). Let (T0, T1) be a 1-perfect bitrade in a graph G. Then f(T0,T1) is a (1)-eigenfunction of G. Lemma 5.7 implies that we can consider Problem 5.6 only for n = qm + 1, where m 1 (because 1 is an eigenvalue of H(n, q)). Suppose that (T0, T1) is a 1-perfect bitrade in H(qm + 1, q). By Lemma 5.7 we have that f(T0,T1) is a (1)-eigenfunction of H(qm+1, q). We note that 1 = (q1)m+1(qm+ 1, q). Applying Theorem 5.2 for n = qm+ 1 and i = j = (q 1)m+ 1, we obtain that |S(f(T0,T1))| 2 (q1)m+1 · (q 1)m for q 4 and |S(f(T0,T1))| 2 m+1 · 3m for q = 3. Consequently, we have |T0|+ |T1| = |S(f(T0,T1))| 2 (q1)m+1 · (q 1)m (5.1) for q 4 and |T0|+ |T1| = |S(f(T0,T1))| 2 m+1 · 3m (5.2) for q = 3. On the other hand, in [70] Mogilnykh and Solov’eva for arbitrary q 2 showed the existence of 1-perfect bitrades in H(qm+1, q) of size 2 · (q!)m. Thus, the bounds (5.1) and (5.2) are sharp for q = 4 and q = 3 respectively, and we obtain a solution of Problem 5.6 for q 2 {3, 4}. Finally, we note that Theorem 5.2 implies that for q 5 optimal (1)- eigenfunctions of the Hamming graph H(qm + 1, q) do not correspond to its 1-perfect bitrades (in this case we have a characterization of all optimal (1)-eigenfunctions). So, this approach does not work for q 5. 6 Doob graph In this section, we give a survey of results on the MS-problem for the Doob graph. The Shrikhande graph Sh is the Cayley graph on the group Z24 with the generating set S, where S = {±(0, 1),±(1, 0),±(1, 1)}. Figure 9: The Shrikhande graph. The Doob graph D(m,n), where m > 0, is the Cartesian product of m copies of the Shrikhande graph and n copies of the complete graph K4. In other words, we have D(m,n) = Shm⇤Kn4 . This graph is a distance-regular graph with the same parameters as the Hamming graph H(2m+n, 4). The Doob graph D(m,n) has 2m+n+1 distinct eigen- values i(m,n) = 6m+ 3n 4i, where 0  i  2m+ n. In [8] Bespalov proved that the E. Sotnikova and A. Valyuzhenich: Minimum supports of eigenfunctions of graphs: a survey 19 minimum cardinality of the support of a 1(m,n)-eigenfunction of D(m,n) is 6·42m+n2 and obtained a characterization of optimal 1(m,n)-eigenfunctions. He also showed that the minimum cardinality of the support of a 2m+n(m,n)-eigenfunction of D(m,n) is 22m+n and obtained a characterization of optimal 2m+n(m,n)-eigenfunctions. In what follows, in this section we will consider the results obtained in [8]. Now, we discuss constructions of optimal 1(m,n)-eigenfunctions and 2m+n(m,n)- eigenfunctions. It is interesting that as in the case of the Hamming graph such functions are constructed as a tensor product of several elementary optimal eigenfunctions. Firstly, we define two sets of elementary optimal eigenfunctions. For a 2 Z24 we define the function pa on the vertices of the Shrikhande graph by the following rule: pa(x) = 8 >< >: 1, if x 2 {a+ (3, 1), a+ (3, 2), a+ (2, 1)}; 1, if x 2 {a+ (2, 3), a+ (1, 2), a+ (1, 3)}; 0, otherwise. We note that the support of pa consists of two disjoint copies of the complete graph K3. The function p(0,3) is shown in Figure 10. Denote P = {pa | a 2 Z24}. For a 2 Z24 and b 2 {(0, 1), (1, 0), (1, 1)} we define the function ra,b on the vertices of the Shrikhande graph by the following rule: ra,b(x) = 8 >< >: 1, if x 2 {a, a+ 2b}; 1, if x 2 {a+ b, a+ 3b}; 0, otherwise. We note that the vertices from the support of ra,b form a cycle of length 4. The function r(0,0),(0,1) is shown in Figure 10. Denote R = {ra,b | a 2 Z24, b 2 {(0, 1), (1, 0), (1, 1)}}. We will also use the sets of functions A4 and C4 defined in Section 5. Figure 10: Functions p(0,3) and r(0,0),(0,1). For m > 0 denote by Im,n the function that is defined on the vertices of D(m,n) and is identically equal to 1. For n 1 denote by In the function that is defined on the vertices of H(n, 4) and is identically equal to 1. Now, we define two classes of optimal 1(m,n)-eigenfunctions and one class of opti- mal 2m+n(m,n)-eigenfunctions. We say that a function f defined on the vertices of D(m,n) belongs to the class G1(m,n) if f = c · g1 . . . gm · In, where c is a real non-zero constant, gi 2 P for some i 2 {1, . . . ,m} and gj = I1,0 for any j 2 {1, . . . ,m} \ i. 20 Art Discrete Appl. Math. 4 (2021) #P2.09 Let n 2. We say that a function f defined on the vertices of D(m,n) belongs to the class G2(m,n) if f = c · Im,0 · h1 . . . hn1, where c is a real non-zero constant, hi 2 A4 for some i 2 {1, . . . , n 1} and hj = I1 for any j 2 {1, . . . , n 1} \ i. We say that a function f defined on the vertices of D(m,n) belongs to the class G3(m,n) if f = c · g1 . . . gm · h1 . . . hn, where c is a real non-zero constant, gi 2 R for any i 2 {1, . . . ,m} and hj 2 C4 for any j 2 {1, . . . , n}. The main results proved in [8] are the following. Theorem 6.1 ([8, Theorem 1]). Let f be a 1(m,n)-eigenfunction of D(m,n), where m > 0. Then |S(f)| 6 · 42m+n2. Moreover, if |S(f)| = 6 · 42m+n2, then the following statements hold: 1. If n 2, then f 2 G1(m,n) or f 2 G2(m,n). 2. If n 2 {0, 1}, then f 2 G1(m,n). Theorem 6.2 ([8, Theorem 2]). Let f be a 2m+n(m,n)-eigenfunction of D(m,n), where m > 0. Then |S(f)| 22m+n. Moreover, if |S(f)| = 22m+n, then f 2 G3(m,n). Remark 6.3. We note that the bound proved in Theorem 6.2 can also be obtained by applying the weight distribution bound for the smallest eigenvalue of the Doob graph. 7 Johnson graph In this section, we give a survey of results on the MS-problem for the Johnson graph. The Johnson graph J(n,!) is defined as follows. The vertices of J(n,!) are the binary vectors of length n with ! ones; and two vertices are adjacent if they have exactly ! 1 common ones. The Johnson graph J(n,!) has ! + 1 distinct eigenvalues i(n,!) = (! i)(n ! i) i, where 0  i  !. In [96] Vorob’ev et al. showed that for a fixed ! and sufficiently large n the minimum cardinality of the support of a i(n,!)-eigenfunction of J(n,!) is 2i · n2i !i and obtained a characterization of optimal i(n,!)-eigenfunctions. Thus, the MS-problem for the Johnson graph is asymptotically solved for all eigenvalues. Now we discuss the main results obtained in [96]. Firstly, we define the function f i,!,n on the vertices of the Johnson graph J(n,!) by the following rule: f i,!,n(x1, . . . , xn) = 8 >>>>< >>>>: 1, if x2k1 + x2k = 1 for any 1  k  i and x1 + x3 + . . .+ x2i1 is even; 1, if x2k1 + x2k = 1 for any 1  k  i and x1 + x3 + . . .+ x2i1 is odd; 0, otherwise. So, the support of f i,!,n consists of binary vectors (x1, . . . , xn) of weight ! such that the product (x1 x2) · . . . · (x2i1 x2i) is not equal to zero. In [96, Proposition 1] it was shown that f i,!,n is a i(n,!)-eigenfunction of J(n,!) and |S(f i,!,n)| = 2i · n2i !i . The main result proved in [96] is the following. Theorem 7.1 ([96, Theorem 4]). Let i and ! be positive integers, ! i. There is n0(i,!) such that for all n n0(i,!) and any i(n,!)-eigenfunction f of J(n,!) the following holds: |S(f)| 2i · ✓ n 2i ! i ◆ , (7.1) E. Sotnikova and A. Valyuzhenich: Minimum supports of eigenfunctions of graphs: a survey 21 and any function that attains the bound (7.1) is equivalent to f i,!,n up to a permutation of coordinate positions and the multiplication by a scalar. Remark 7.2. We note that the bound (7.1) for i = ! and arbitrary n can also be obtained by applying the weight distribution bound for the smallest eigenvalue of the Johnson graph. Now, we discuss the main ideas of the proof of Theorem 7.1. Let f be a real-valued function defined on the vertices of the Johnson graph J(n,!) and let j1, j2 2 {1, 2, . . . , n}, j1 < j2. We define a function fj1,j2 on the vertices of J(n 2,!1) as follows: for any vertex y = (y1, y2, . . . , yj11, yj1+1, . . . , yj21, yj2+1, . . . , yn) of J(n 2,! 1) fj1,j2(y) = f(y1, y2, . . . , yj11, 1, yj1+1, . . . , yj21, 0, yj2+1, . . . , yn) f(y1, y2, . . . , yj11, 0, yj1+1, . . . , yj21, 1, yj2+1, . . . , yn). One of the important ingredients in the proof of Theorem 7.1 is the following. Lemma 7.3 ([96, Lemma 1]). Let f be a i(n,!)-eigenfunction of J(n,!), where j1, j2 2 {1, 2, . . . , n} and j1 < j2. Then fj1,j2 is a i1(n 2,! 1)-eigenfunction of J(n 2,! 1) or the all-zero function. Lemma 7.3 is a very useful tool for studying of eigenfunctions of the Johnson graph. In particular, this lemma allows to apply induction on n, ! and i (we can use the induction assumption for the function fj1,j2 ). Moreover, we suppose that Lemma 7.3 can be useful not only for the MS-problem but also for other problems. For example, recently in [97] Vorob’ev applied Lemma 7.3 for characterization of equitable 2-partitions of the Johnson graph with the eigenvalue 2(n,!). Finally, we note that Lemma 7.3 is an analogue of Lemma 5.3 (see Subsection 5.2). Let v = (v1, . . . , vn) be a real non-zero vector such that v1 + . . .+ vn = 0. We define the function fv on the vertices of the Johnson graph J(n,!) by the following rule: f v(x1, . . . , xn) = X 1in : xi=1 vi. For k 2 {1, . . . , n 1} denote v k = (1, . . . , 1| {z } k , k n k , . . . , k n k| {z } nk ). In [71] Mogilnykh et al. proved the following improvement of Theorem 7.1 for 1(n,!)- eigenfunctions of J(n,!). Theorem 7.4 ([71, Theorem 1]). Let f be an optimal 1(n,!)-eigenfunction of J(n,!), where n 2! and ! 2. Then f is f1,!,n or fvk for some k 2 {2, . . . , n 2} such that k! n 2 N up to a permutation of coordinate positions and the multiplication by a scalar. 8 Grassmann graph In this section, we give a survey of results on the MS-problem for the Grassmann graph. The Grassmann graph Jq(N,m) is a distance-regular graph with the vertex set consisting 22 Art Discrete Appl. Math. 4 (2021) #P2.09 of all m-dimensional subspaces of a vector space of dimension N over a finite field Fq . Two vertices are adjacent whenever the corresponding subspaces intersect in a (m 1)- dimensional subspace. The MS-problem for the minimum eigenvalue of the Grassmann graph was studied in [64]. But it is interesting that this problem can be tracked earlier to the works [26, 27, 50] where it was considered in terms of finding the minimum null t-designs of the lattices of subspaces over a finite field. In [50] G. D. James made a conjecture about the minimum support size of non-zero null t-designs of the lattices of subspaces over a finite field. S. Cho confirms the conjecture in [27] and in [26] characterizes all the null t-designs with minimum supports in terms of maximal isotropic spaces of some bilinear form. Coming back to the Grassmann graph, we obtain the following theorem that gives us the characterization of optimal D-eigenfunctions for the Grassmann graph (compare with Theorems 1,2 from [26] and Theorem 5 from [64]). For more details about null t-designs and totally isotropic spaces the reader is referred to [26] and Chapter 18 of [50]. Theorem 8.1. Suppose f is an optimal D-eigenfunction of the Grassmann graph Jq(N,m), where N 2m and D is its minimum eigenvalue. Then the cardinality of its support is DP i=0  m i q · qi(i1)/2 which is also equal to the value of the weight distribution bound and the non-zeros of the function f correspond to the maximal totally isotropic subspaces of a 2m-dimensional space, equipped with a bilinear form B with a Gram matrix ✓ 0 Em Em 0 ◆ up to the equivalence (or, equivalently, with respect to a non-degenerate quadratic form Q). Thus for the minimum eigenvalue of the Grassmann graph Jq(N,m) the MS-problem is solved and the weight distribution bound is achieved. 9 Bilinear forms graph In this section, we give a survey of results on the MS-problem for bilinear forms graph. More details can be found [89]. The bilinear forms graph Bilq(n,m) is a distance-regular graph with the vertex set V consisting of all n ⇥ m matrices over a finite field Fq and two vertices being adjacent when their matrix difference has a rank 1. For the sake of convenience, we will further suppose that m  n. Thus the diameter D of the bilinear forms graph Bilq(n,m) is equal to m. Here as well as in the previous section we consider the MS-problem only for the case of minimum eigenvalue D. In this case we have the following lower bound for the minimum support cardinality: mX i=0  m i q · qi(i1)/2 It is interesting that the weight distribution for bilinear forms graph coincides with that of the Grassmann graph. Later we will see the importance of this connection. The key idea here is that bilinear forms graph belongs to a family of so-called Delsarte cliques graphs (for more details about Delsarte cliques graphs, the reader is referred to [6]). This property leads to the following observations: • Theorem 2 from [64] implies that for a Delsarte cliques graph G a function f is a D- eigenfunction of G if and only if for every Delsarte clique C it holds P v2C f(v) = 0. E. Sotnikova and A. Valyuzhenich: Minimum supports of eigenfunctions of graphs: a survey 23 • Theorem 3 from [64] tells us that for a Delsarte clique graph G in case of D = 2 if the weight distribution bound is achieved then non-zeros of optimal D-eigenfunction induce a complete bipartite graph. Note that for bilinear forms graph Bilq(2, 2) we have D = q1, thus non-zeros of optimal D-eigenfunction achieving the weight distribution bound induce a complete bipartite graph Kq+1,q+1 if such a function exists. It appears that in case of strongly regular bilinear forms graphs Bilp(2, 2) (those with D = 2) the weight distribution bound can be achieved. An explicit construction of an optimal D-eigenfunction can be found in [89]. Below are the statements that summarize this construction, but first let us introduce additional notation. Suppose a1 is a generating element of the multiplicative group F⇤p. Denote a0 = 0, a2 = a 2 1, . . . , ap2 = a p2 1 , ap1 = a p1 1 = 1 e⇤ = [0, 1], e0 = [1, 0], e1 = [1, a1], . . . , ep1 = [1, ap1] Theorem 9.1 ([89, Theorem 3]). Let Bilp(2, 2) be a bilinear forms graph over a prime field Fp. For any ⌫ 2 Fp, such that ⌫ 6= ⇠2 for all ⇠ 2 Fp, and bi = 1a2i ⌫+1 the independent set N = n0 1 e⇤, b0  1 a0⌫ e0, . . . , bp1  1 ap1⌫ ep1 o together with P = n0 0 e⇤; b0  1 a0⌫ e0 + b0  a0 1 e⇤; . . . ; bp1  1 ap1⌫ e0 + bp1  ap1 1 e⇤ o form non-zeros of D-eigenfunction f as two parts of a complete bipartite graph Kp+1, p+1 and f(v) = 8 >< >: c, for v 2 P, c, for v 2 N , 0, else for some constant c 6= 0. Let us illustrate this theorem with some small example. Consider a bilinear forms graph Bil3(2, 2). Using the construction above we obtain the following sets: N = n0 0 0 1 ,  1 0 0 0 ,  2 1 1 2 ,  2 2 2 2 o , P = n0 0 0 0 ,  1 0 0 1 ,  2 2 1 2 ,  2 1 2 2 o . Here under the notation above a0 = 0, a1 = 2, a2 = 1, e⇤ = [0, 1], e0 = [1, 0], e1 = [1, 2], e2 = [1, 1], ⌫ = 1, b0 = 1, b1 = 2, b2 = 2. Thus we proved that there exists a family of optimal D-eigenfunctions of the bilinear forms graph Bilp(2, 2) over a prime field Fp that achieve the lower bound. However the construction described above does not provide the full characterization of all optimal D- eigenfunctions. 24 Art Discrete Appl. Math. 4 (2021) #P2.09 What happens if we look at bilinear forms graphs of larger diameter? It appears that the weight distribution bound cannot be achieved. And for proving this the connection between bilinear graphs and the Grasssmann graphs comes in handy. The bilinear forms graph Bilq(n,m) with m  n can be considered as a subgraph of the Grassman graph Jq(n+m,m) as follows: given a fixed subspace W of dimension n, all m-spaces U such that U \ W = 0 are the vertices of Bilq(n,m). This embedding leads to the following result about the Delsarte cliques of these graphs (see Lemma 8 from [89]): Lemma 9.2. Delsarte cliques of bilinear forms graph Bilq(n,m) are embedded in Delsarte cliques of a Grassmann graph Jq(n +m,m) in the sense that for any Delsarte cliques C and bC of a bilinear forms graph and the Grassmann graph correspondingly, either C ⇢ bC or C \ bC = ;. Since for any D-eigenfunction the sum of its values over a Delsarte clique is zero, from the previous Lemma we immediately obtain the following Corollary which simply tells us that we can extend eigenfunctions of bilinear forms graph to those of the Grassmann graph: Corollary 9.3. Suppose f is a D-eigenfunction of a bilinear forms graph Bilq(n,m). Then bf is an eigenfunction of the Grassmann graph Jq(n+m,m), where bf(M) = ( f(M), if M 2 V (Bilq(n,m)) 0, else This corollary is crucial for the final result: Theorem 9.4 ([89, Theorem 7]). Let Bilq(n,m) be a bilinear forms graph of diameter D 3. Then the minimum support of an eigenfunction corresponding to the minimum eigenvalue does not achieve the weight distribution bound. The main idea behind the proof of this theorem can be described as follows. Suppose the opposite holds and f is an optimal D-eigenfunction that achieves the weight distribu- tion bound. Under the notation of Corollary 9.3, bf is an optimal D-eigenfunction of the Grassmann graph Jq(n + m, m). According to the Theorem 8.1 characterizing optimal eigenfunctions of the Grassmann graphs, the non-zeros of bf correspond to the maximal totally isotropic spaces of a non-degenerate quadratic form Q. Now we recall the graphs embedding construction: there exists a subspace W of dimension n that trivially inter- sects with all the maximal totally isotropic subspaces. A well-known corollary from the Chevalley theorem states that any non-degenerate quadratic form is isotropic on a vector space of dimension not less that 3 over the finite field Fq (here the diameter of a graph plays its role). Thus there exists a non-zero vector w 2 W such that Q(w) = 0, therefore is a 1-dimensional totally isotropic space and, hence, is contained in a maximal to- tally isotropic subspace. This contradicts the trivial intersection of W with all the maximal totally isotropic subspaces. According to this theorem optimal D-eigenfunctions of Bilq(n, m) do not satisfy the weight distribution bound. This lead to an open the MS-problem for bilinear forms graphs of diameter D 3. E. Sotnikova and A. Valyuzhenich: Minimum supports of eigenfunctions of graphs: a survey 25 10 Paley graph In this section, we give a survey of results on the MS-problem for the Paley graph. Let q be an odd prime power, where q ⌘ 1(4). The Paley graph P (q) is the Cayley graph on the ad- ditive group F+q of the finite field Fq with the generating set of all squares in the multiplica- tive group F⇤q . This graph is a strongly regular with parameters (q, q12 , q5 4 , q1 4 ). The eigenvalues of P (q) are 0 = q12 , 1 = 1+pq 2 and 2 = 1pq 2 . In [44] Goryainov et al. for i 2 {1, 2} proved that the minimum cardinality of the support of a i-eigenfunction of P (q2), where q is an odd prime power, is q + 1. In what follows, in this section we will discuss the results obtained in [44]. Let q be an odd prime power and let be a primitive element of the finite field Fq2 . Denote ! = q1, Q0 = h!2i and Q1 = !h!2i. We define the function f on the vertices of the Paley graph P (q2) by the following rule: f(x) = 8 >< >: 1, if x 2 Q0; 1, if x 2 Q1; 0, otherwise. One of the main results proved in [44] is the following. Theorem 10.1 ([44, Theorem 2]). Let q be an odd prime power and let be a primitive element of the finite field Fq2 . Then the following statements hold: 1. If q ⌘ 1(4), then f is a 2-eigenfunction of P (q2) and |S(f)| = q + 1. 2. If q ⌘ 3(4), then f is a 1-eigenfunction of P (q2) and |S(f)| = q + 1. Since the Paley graph P (q2) is self-complementary, Theorem 10.1 implies that for any i 2 {1, 2} P (q2) has i-eigenfunction f such that |S(f)| = q + 1. On the other hand, by the weight distribution bound we obtain that a 2-eigenfunction of P (q2) has at least q + 1 non-zero values. Since P (q2) is self-complementary, the same bound holds for a 1-eigenfunction of P (q2). Thus, the minimum cardinality of the support of a i- eigenfunction of P (q2), where i 2 {1, 2}, is q + 1. Now we discuss one interesting connection between the sets Q0 and Q1 and maximal cliques of the Paley graph P (q2). The maximum possible size of a clique of P (q2) is q (all cliques of such size are Delsarte cliques). Blokhuis [20] determined all cliques and all cocliques of size q in P (q2) and showed that they are affine images of the subfield Fq . Baker et al. [5] found maximal cliques of order q+12 and q+3 2 for q ⌘ 1(4) and q ⌘ 3(4) respectively, but these cliques are not the only cliques of such size. Moreover, there are no known maximal cliques whose size belongs to the gap from q+12 (from q+3 2 , respectively) to q. Kiermaier and Kurz [56] studied maximal integral point sets in affine planes over finite fields and found maximal cliques of size q+32 in P (q 2) for q ⌘ 3(4). Using the sets Q0 and Q1 defined above, Goryainov et al. [44] constructed new maximal cliques of size q+1 2 and q+3 2 for q ⌘ 1(4) and q ⌘ 3(4) respectively in P (q 2). Theorem 10.2 ([44, Theorem 1]). Let q be an odd prime power and let be a primitive element of the finite field Fq2 . Then the following statements hold: 1. If q ⌘ 1(4), then Q0 and Q1 are maximal cocliques of size q+12 in the graph P (q 2). 2. If q ⌘ 3(4), then Q0 [ {0} and Q1 [ {0} are maximal cliques of size q+32 in the graph P (q2). 26 Art Discrete Appl. Math. 4 (2021) #P2.09 11 Star graph In this section, we give a survey of results on the MS-problem for the Star graph. The Star graph Sn, n 3, is the Cayley graph on the symmetric group Symn with the generating set {(1 i) | i 2 {2, . . . , n}}. This graph is not distance-regular. The spectrum of the Star graph is integral [25, 60]. For n 4, the eigenvalues of Sn are ±(n k), where 1  k  n; and the eigenvalues of S3 are {2,1, 1, 2}. The multiplicities of eigenvalues of the Star graph were studied in [4, 53, 54]. In particular, explicit formulas for calculating multiplicities of eigenvalues ±(n k), where 2  k  12, were found. In [51] Kabanov et al. found the minimum cardinality of the support of an (n 2)-eigenfunction of Sn and obtained a characterization of optimal (n 2)-eigenfunctions for n 8 and n = 3. In what follows, in this section we will consider the results obtained in [51]. Now, we discuss one construction of optimal (n 2)-eigenfunctions of the Star graph. Let i 2 {1, . . . , n} and j, k 2 {2, . . . , n}, where j 6= k. We define the function f j,ki on the vertices of the Star graph Sn by the following rule: f j,k i (⇡) = 8 >< >: 1, if ⇡(j) = i; 1, if ⇡(k) = i; 0, otherwise. In [51, Lemma 2] it was shown that f j,ki is an (n2)-eigenfunction of Sn and |S(f j,k i )| = 2(n 1)!. Denote F = {f j,ki | i 2 {1, . . . , n}, j, k 2 {2, . . . , n}, j 6= k}. The main result proved in [51] is the following. Theorem 11.1 ([51, Theorem 20]). Let f be an (n 2)-eigenfunction of Sn, where n 8 or n = 3. Then |S(f)| 2(n 1)!. Moreover, |S(f)| = 2(n 1)! if and only if f = c · f̃ , where c is a real non-zero constant and f̃ 2 F . Now, we discuss the main ideas of the proof of Theorem 11.1. Firstly, we need some definitions. Let M = (mi,j) be a real n⇥ n matrix. We say that M is special if M is non-zero and the following conditions hold: 1. mi,1 = 0 for any i 2 {1, . . . , n}. 2. m1,j = 0 for any j 2 {1, . . . , n}. 3. Pn j=1 mi,j = 0 for any i 2 {1, . . . , n}. For a real n⇥ n matrix M = (mi,j) denote gM (n) = |{⇡ 2 Symn | nX i=1 mi,⇡(i) 6= 0}|. The key point of the proof of Theorem 11.1 is the following. For an arbitrary (n 2)- eigenfunction f of Sn we can construct some special n ⇥ n matrix M(f) and match the permutations from Symn with diagonals of M(f) in such a way that the value of f on a E. Sotnikova and A. Valyuzhenich: Minimum supports of eigenfunctions of graphs: a survey 27 permutation ⇡ is the sum of elements of the corresponding diagonal of M(f). In other words, we have the equality |S(f)| = gM(f)(n) (11.1) for any (n 2)-eigenfunction f of Sn. This observation allows us to reduce the MS- problem for the Star graph Sn and its eigenvalue n 2 to the following extremal problem on the set of all special n⇥ n matrices. Problem 11.2. Given a positive integer n, to find the minimum value of gM (n) for the class of special n⇥ n matrices M . In [51, Theorem 19] for n 8 and n = 3 it was proved that gM (n) 2(n1)! for any special n⇥n matrix M . Moreover, in [51, Theorem 19] a classification of special matrices in the equality case was obtained. Using these results and the equality (11.1), we can finish the proof of Theorem 11.1. 12 Some remarks on optimal eigenfunctions of graphs In this section, we give some observations on optimal eigenfunctions of graphs. Recall that the MS-problem is formulated for arbitrary real-valued functions from the corresponding eigenspace. Surprisingly, in many cases optimal eigenfunctions take only three distinct values (for example, see Theorems 5.2, 6.1, 6.2, 7.1, 11.1). But, in general case it is not true. For example, there are optimal (2)-eigenfunctions of the Petersen graph that take five distinct values (see Figure 11). Figure 11: Optimal (2)-eigenfunction of the Petersen graph. There is an interesting connection between optimal eigenfunctions corresponding to the second largest eigenvalue of a given graph and completely regular codes in this graph. In particular, an arbitrary optimal 1(n, q)-eigenfunction (1(n,!)-eigenfunction) of the Hamming graph H(n, q) (the Johnson graph J(n,!)) is the difference of the characteristic functions of two completely regular codes of covering radius 1 (see [91, Theorem 3] and [96, Theorem 4]). The Star graph Sn does not have completely regular codes of covering radius 1 with the eigenvalue n 2. However, an arbitrary optimal (n 2)-eigenfunction of Sn is the difference of the characteristic functions of two completely regular codes of covering radius 2 (see [51, Lemma 22]). 28 Art Discrete Appl. Math. 4 (2021) #P2.09 13 Open problems In this section, we briefly recall the main results on the MS-problem and formulate several open problems. Recall that Problem 5.1 is completely solved for all n 1 and q 2. In particular, the MS-problem for the Hamming graph H(n, q) is solved for all eigenvalues. Moreover, a characterization of functions that are optimal in the space U[i,j](n, q) was obtained for q 3, i+ j  n and q 5, i = j, i > n2 . Taking into account these results, we formulate the following two problems for the Hamming graph. Problem 13.1. Characterize functions that are optimal in the space U[i,j](n, q) for the cases q = 2 and q 3, i+ j > n (in this problem we assume that i < j). Problem 13.2. Characterize optimal i(n, q)-eigenfunctions of the Hamming graph H(n, q) for q 2 {3, 4} and i > n2 . The MS-problem for the Doob graph D(m,n) is solved for the second largest eigen- value 1(m,n) and the smallest eigenvalue 2m+n(m,n). So, it seems very interesting to consider the following question. Problem 13.3. Solve the MS-problem for the third largest eigenvalue 2(m,n) of the Doob graph D(m,n). The MS-problem for the bilinear forms graph Bilq(n,m) is solved for the smallest eigenvalue D in case n = m = 2 and q is prime. For bilinear forms graphs of larger diameters over the arbitrary field it is proved that the weight distribution bound cannot be attained. This leads to the following interesting questions: Problem 13.4. For the bilinear forms graph Bilq(n,m) of diameter D: • Characterize optimal D-eigenfunctions in case of D = 2 for a prime q (including the case of n 6= m). • Solve the MS-problem for the smallest eigenvalue D in case of D = 2 and arbitrary q. • Solve the MS-problem for the smallest eigenvalue D in case of D 3 and arbitrary q. The MS-problem for the Grassmann graph Jq(N,m) is solved for the smallest eigen- value D. Since the Grassmann graph can be considered as a q-analogue of the Johnson graph it may be interesting to consider the following question: Problem 13.5. Solve the MS-problem for the second largest eigenvalue of the Grassmann graph Jq(N,m). The MS-problem for the Paley graph P (q2) is solved for both non-principal eigenval- ues. We formulate the following problem for optimal eigenfunctions. Problem 13.6. Characterize optimal 1-eigenfunctions and 2-eigenfunctions of the Paley graph P (q2). The MS-problem for the Star graph Sn is solved only for the second largest eigenvalue. So, the following question is very natural. E. Sotnikova and A. Valyuzhenich: Minimum supports of eigenfunctions of graphs: a survey 29 Problem 13.7. Solve the MS-problem for the third largest eigenvalue of the Star graph Sn. At the end of this section we also would like to bring the attention of the reader to the following problems: Problem 13.8. For distance-regular graphs find the conditions for the weight distribution bound to be achieved. Problem 13.9. For distance-regular graphs find a sharper lower bound on the cardinality of a graph eigenfunction support than the weight distribution bound. Problem 13.10. 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