ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 173-190 https://doi.org/10.26493/1855-3974.1341.5a3 (Also available at http://amc-journal.eu) Groups of symmetric crosscap number less than or equal to 17 Adrián Bacelo * Departamento de Algebra, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain Received 3 March 2017, accepted 21 March 2018, published online 13 June 2018 * Abstract Every finite group G acts on some non-orientable unbordered surfaces. The minimal topological genus of those surfaces is called the symmetric crosscap number of G. It is known that 3 is not the symmetric crosscap number of any group but it remains unknown whether there are other such values, called gaps. In this paper we obtain the groups with symmetric crosscap number less than or equal to 17. Also, we obtain six infinite families with symmetric crosscap number of the form 12k + 3. Keywords: Symmetric crosscap number, Klein surfaces. Math. Subj. Class.: 57M60, 20F05, 20H10, 30F50 1 Introduction A Klein surface X is a compact surface endowed with a dianalytic structure [1]. Klein surfaces may be seen as a generalization of Riemann surfaces including bordered and non-orientable surfaces. An orientable unbordered Klein surface is a Riemann surface. Given a Klein surface X of topological genus g with k boundary components the number p = rig + k — 1 is called the algebraic genus of X, where n = 2 if X is an orientable surface and n = 1 otherwise. In the study of Klein surfaces and their automorphism groups the non-euclidean crys-tallographic (NEC) groups play an essential role. An NEC group r is a discrete subgroup of G (the full group of isometries of the hyperbolic plane H) with compact quotient H/r. For a Klein surface X with p > 2 there exists an NEC group r, such that X = H/r, [27]. *The author wish to express his deep gratitude to the referees for their careful checking of the manuscript and for their very useful suggestions concerning both the style and the precision of arguments. The author is partially supported by UCM910444 and MTM2014-55565. E-mail address: abacelo@ucm.es (Adrian Bacelo) ©® This work is licensed under https://creativecommons.Org/licenses/by/3.0/ 174 Ars Math. Contemp. 15 (2018) 147-160 A finite group G of order N is a subgroup of the automorphism group of a Klein surface X = H/r if and only if there exists an NEC group A such that r is a normal subgroup of A with index N and G = A/r. Every finite group G acts as a subgroup of the automorphism group of some non-orientable surface without boundary, see [7]. The minimum topological genus of these surfaces is called the symmetric crosscap number of G and it is denoted by a(G). Such a surface of topological genus g > 3 has at most 84(g - 2) automorphisms. Hence, for each g there is a finite number of groups acting on surfaces of genus g. The systematic study of the symmetric crosscap number was begun by May in [23], although previous results from other authors are also to be noted, see for instance [7, 14, 19]. Four types of inter-related problems arise naturally when dealing with the symmetric crosscap number a(G). First of all, to obtain a(G) for any given group G, and for the groups belonging to a given infinite family. Second, to obtain a(G) for all groups G with o(G) < n for a given (small) value of n. Third, for a given value of g, to obtain all groups G such that a(G) = g. Evidently this question is feasible only for low values of g. Finally, to determine which values of g are in fact a(G) = g for a group G. The set of such values is called the symmetric crosscap spectrum and there exists a conjecture according to which g = 3 is the unique positive integer not belonging to the spectrum. In this paper we deal with the third question. We will study which groups have symmetric crosscap number less than or equal to 17. First, we will indicate all the results we know and then we will make a study of each group with symmetric crosscap number g < 17 that has not been studied in detail. Also, results on the spectrum are given. The contents of this paper form part of the doctoral thesis of the author, [3]. 2 Preliminaries An NEC group r is a discrete subgroup of isometries of the hyperbolic plane H, including orientation-reversing elements, with compact quotient X = H/r. Each NEC group r has associated a signature [22]: a(r) = (g, ±, [mi, .. ., mr], |(niji, .. ., nijSi), i = 1, ..., k}), (2.1) where g, k, r, m4, n,j are integers satisfying g, k, r > 0, > 2, n> 2. We will denote by [-], (-) and {-} the cases when r = 0, sj = 0 and k = 0, respectively. The signature determines a presentation of r, see [30], by generators xj (i = 1,..., r); e (i = 1,..., k); cj,j (i = 1,..., k; j = 0,..., Sj); aj, b (i = 1,..., g) if a has sign '+'; and dj (i = 1,..., g) if a has sign '-'. These generators satisfy the following relations: mi _ 1 2 _ 2 _ ( )ni,j _ 1- —1 _ 1 Xj 1; cj,j — 1 cj,j (cj,j—1cj,j) ' 1; ej cj,0ejcj,si 1 and nr=i xi nk=i ntiMiO—V) =1 if a has sign'+' nr=i xi nti ei ng=i % =1 ifa has sign -' The isometries xj are elliptic, ej, aj, bj are hyperbolic, Cj,j are reflections and dj are glide-reflections. A. Bacelo: Groups of symmetric crosscap number less than or equal to 17 175 Every NEC group r with signature (2.1) has associated a fundamental region whose area "(r), called area of the group, is: "(r) = 4g+k - 2+1 (l- + 1(l- ¿j))' <"> with n = 2 or 1 depending on the sign '+' or '-' in the signature. An NEC group with signature (2.1) actually exists if and only if the right-hand side of (2.2) is greater than 0. We denote by |r| the expression "(r)/2n and call it the reduced area of r. If r is a subgroup of an NEC group A of finite index N, then also r is an NEC group and the Riemann-Hurwitz formula holds, |r| = N|A|. Let X be a non-orientable Klein surface of topological genus g > 3 without boundary. Then by [28] there exists an NEC group r with signature: a(r) = (g, -, [-], {-}), (2.3) such that X = H/r. A group r with this signature is called a surface NEC group. If G acts as an automorphism group of X = H/r, then there exists another NEC group A such that G = A/r. From the Riemann-Hurwitz relation we have g - 2 = o(G)|A|, where o(G) denotes the order of G. Then ¿KG) < g = 2 + o(G)|A|, and so to obtain the symmetric crosscap number of G is equivalent to find a group A and an epimorphism 0: A ^ G, such that r = ker 0 is a surface NEC group (and so, without elements with finite order) and G = 0(A+), where A+ is the subgroup consisting of the orientation-preserving elements of A, see [28], and minimal |A|. The groups having symmetric crosscap numbers 1 and 2 have been classified by T. W. Tucker, [29]. The groups of symmetric crosscap number 1 are Cn, Dn, A4, S4 and A5. We have two families of groups of symmetric crosscap number 2, C2 x Cn, n > 2 even, and C2 x Dn, n even. It is known that there exists no group of symmetric crosscap number 3, [23]. The groups with symmetric crosscap number 4 and 5 were obtained in [8]. M. D. E. Conder at a conference in Castro-Urdiales in 2010 announced that using computing software, he had obtained the groups of symmetric crosscap number up to 65, in terms of their "SmallGroupLibrary" description. The result of this research is available in his webpage, [9]. The list contains the GAP reference of each group, its symmetric crosscap number and the corresponding NEC group A. However, this list gives information neither on the algebraic structure of the groups nor on the epimorphism 0 which determines the action of the NEC group A over the group G. Throughout the paper, we use extensively this fundamental work by Conder, in order to study which are the concerned groups. For each group G we have described its algebraic structure, its presentation and the corresponding epimorphism, but here we will only show the algebraic structure and its presentation. In the most complicated cases, we will show also the epimorphism. In the presentations we skip the abelian relations. The full details are to be found in [2] and [3]. For groups of order 32 and 64 we use the notation given by Hall and Senior in [20]. The algebraic identification allows us to know the subgroups structure of the involved groups, and this is essential to determine all the groups that act on a surface of a given genus. Along the article Cn, Dn, DCn and QAn denote, respectively, the cyclic, dihedral, dicyclic and quasiabelian groups, for more details see [12, 13]. 176 Ars Math. Contemp. 15 (2018) 147-160 3 Groups of symmetric crosscap number 6 to 9 In symmetric crosscap number 6 some groups stand out: 1. The group [80,46]: Coxeter described this group of order 80 in [12], where he named it as (2, 5, 5; 2) with presentation and algebraic structure as shown in the table. 2. The group [160, 234]: This group contains the previous one of order 80. In [12] it is denoted as (4, 5 | 2,4). Table 1: Groups of symmetric crosscap number 6. GAP G Relations [+ Generators] Reference [8, 4] DC2 ~ Q a4 a2b2,b !aba [23] [16, 3] (4, 4 | 2, 2) a4 b4, (ab)2, (a-1b)2 [15] [16, 6] QA4 a8 b2, baba3 [15] [16, 8] ¿4 a8 b2 , baba5 [15] [16, 13] (2, 2, 2)2 a2 b2, c2, abcacb, abcbac, bcabac [15] [16, 14] C2 x C2 x C2 x C2 a2 b2,c2,d2 [19] [32, 27] T4ai a2 b2, c2, d2, e2, cecae, dedbe [32, 43] Tea! a8 b2, c2, (ab)2, aca3c [80, 46] (2, 5, 5; 2) a2 b5, (ab)5, (a-1b-1ab)2 [120, 35] C2 x A5 a2 [+ (1 2 34 5), (1 2 3)] [l60, 234] (4, 5 | 2,4) a4 b5, (ab)2, (a-1b)4 Attending to symmetric crosscap number 7, we must analyze the group [72,15], which contains the group of order 36 that appears in the table below (see [26]) and so that the algebraic structure is ((C2 x C2) x C9) x C2. In this case, we are going to give the epimorphism. This group has a presentation given by generators a, b, c and relations a4 = b9 = c2 = (ac)2 = (cb)2 = (ab)2 = cb-iab-ia-2 = 1. An associated NEC group is A with signature (0; +; [-]; {(2,4,9)}) and reduced area 752 and an epimorphism 0: A ^ G is 0(ci,o) = cb, 0(ci,i) = ac, 0(ci^) = c, 0(ci^) = cb. The image of ci;icij2 is the generator a, the image of cij2cij3 is the generator b, and finally, c is the image of the element (ciiicij2)2cij2cij3(cijicij2)3cij2ci 3. So we have the generators as images of orientation-preserving elements, and so that the group acts on a non-orientable surface. For symmetric crosscap number 8 we just have to emphasize the group of order 504, that is PSL(2,8), whose symmetric crosscap number was firstly studied in detail by Wendy Hall in [21]. To end this section, we comment some groups with symmetric crosscap number 9, where we find: 1. The group [42,1], which we call (7, 6,5), according to the Coxeter-Moser notation in [13]. It contains G2i, which is also a group of this symmetric crosscap number, and so that its algebraic structure is G2i x C2. Its presentation can be expressed in terms of permutations taking a =(1 23456 7) and b =(1 54 6 2 3). A. Bacelo: Groups of symmetric crosscap number less than or equal to 17 177 Table 2: Groups of symmetric crosscap number 7. GAP G Relations [+ Generators] Reference [12,1] DC3 a6, a3b2, b-1aba [23] [24, 8] (4, 6 | 2, 2) a4,b6, (ab)2, (a-1b)2 [15] [36, 3] (C2 x C2) x C9 a2, b2, c9, [a, b], c-1acb, c-1bcba [72,15] ((C2 x C2) X C9) x C2 a4, b9, c2, (ac)2, (cb)2, (ab)2, cb-1ab-1a-2 Table 3: Groups of symmetric crosscap number 8. GAP G Relations [+ Generators] Reference [24, 5] C4 x D3 a4, b2, c2,(bc)3 [16] [24,10] C3 x D4 a3, b2, c2,(bc)4 [16] [48, 38] D3 x D4 a2, b2, c2, d2, (ab)3, (cd)4 [17] [56,11] (C2 x C2 x C2) x C7 a7, b2, c2, d2, badca-1, caba-1, daca-1 [504,156] PSL(2, 8) a2, b3, (ab)7, ([a, b]4b)2 [21] 2. The group [168,42] is PSL(2, 7). In this case, the presentation given in the table can be expressed by permutations b = (234)(576) and a = (1 2 3)(4 5 6) and relations a3 = b3 = (ab)4 = (a-1b)4 = 1, see [12]. Two more presentations for this group are useful: (a) R4,S4, (RS)2, (R-1S)3 (b) R2, S3, (RS)7, (R-1S-1RS)4 Studying this group, there are actions given by NEC groups with two different signatures: (i) For an NEC group A with signature (0; +; [-]; {(3, 3,4)}) and reduced area 24, we take the presentation given by permutations. So an associated epimorphism 0: A ^ G is: 0(c1jO) = (baba2)2, 0(cM) = (a2b)2, 0(c1j2) = (ba2)2, 0(c1j3) = (baba2)2 Consider the image of c1,0c1,1 and the image of c1,1c1,2. Then the image of the element (c1j0c1;1 )2c1j1c1i2c1i0c1i1(c1i1c1j2)2 is (1 5 4 3 6 2 7), a permutation of order 7. This element, together with the elements of order 3 and order 4, 0(c1,Oc1,1) and 0(c1j2c1j3), generate a group of order 84 at least, but PSL(2, 7) is simple, so it is the full group. So the group G is generated by images of orientation-preserving elements and the group acts on a non-orientable surface. (ii) For an NEC group A with signature (0; +; [3]; {(4)}) and reduced area 24, we use the presentation (b). An associated epimorphism 0: A ^ G is: 0(xi) = S, 0(ei) = S2, 0(c1jO) = R, 0(01,1) = SRS-1 178 Ars Math. Contemp. 15 (2018) 147-160 It is clear that 9 is an epimorphism. The element ci o^i is orientation-reversing, its seventh power is also orientation-reversing and the image of (c1j0x1)7 is the identity element, so the group acts on a non-orientable surface. 3. The group [336,208] has order 336 = 168 • 2. Then we can guess its algebraic structure is PSL(2, 7) x C2. We can find a presentation of this group in [10], and an epimorphism 9 does exist. Hence this is the group we are looking for. Table 4: Groups of symmetric crosscap number 9. GAP G Relations [+ Generators] Reference [21,1] G21 a7, b3, aba3b-1 [15] [30,1] C5 x D3 a5, b2, c2, (bc)3 [16] [30, 2] C3 x D5 a3, b2, c2, (bc)5 [16] [42,1] (7, 6, 5) a7,b6,b-1aba2 [60, 8] D3 x D5 a2, b2, c2, d2, (ab)3, (cd)5 [17] [168,42] PSL(2, 7) a3, b3, (ab)4, (a-1b)4 [336, 208] PSL(2,7) x C2 a3, b8, c2, (ac)2, (cb)2, (ab)2, cb-1(ab-2)3a-1 4 Groups of symmetric crosscap number 10 to 17 Firstly, we analyze the groups with symmetric crosscap number 10, where we can find 30 different groups, most of them of order 32, 48 and 64. We just emphasize: 1. For the group [48, 29] we use two presentations, the one given in the table (generators a, b, c and relations a2,b3, c3, (bc)4, (ab)2, (ac)2, [b, c](bc)2) and another one given by generators R, S and relations R8, S3, (RS)2, R4SR4S-1. For this case, three signatures of NEC groups are given: (i) For an NEC group A with signature (0; +; [-]; {(2, 2, 3,3)}) and reduced area 6 we take the presentation given in the table and an epimorphism 9: A ^ G given by 9(ci,o) = ac, 9(ci,i) = (bc)2, 9(ci^) = ba, 9(ci^) = a, 9(ci^) = ac The group acts on a non-orientable surface, because the image of the element cij2cij3 is the generator b, the image of the element ci 3ci 4 is the generator c and the image of the element cij3ciji(cij2ci 4)2 is the generator a, so these three images generate the group, and they are images of orientation-preserving elements. (ii) For an NEC group A with signature (0; +; [3]; {(2, 2)}) and reduced area i we take the second presentation and so an associated epimorphism is 9: A ^ G given by 9(xi) = S, 9(ei) = S-\ 9(ci0) = RS, 9(cM) = R4, 9(cij2) = SR A. Bacelo: Groups of symmetric crosscap number less than or equal to 17 179 The images of the elements (ciiicij0ei)5 and x1 are the generators R and S respectively and both are orientation-preserving elements, so it is a group acting on a non-orientable surface. 6(x1) = RS, 0(x2) = S, 6(e1) = SR-1, 0(cM) = R4 The quotient gives a non-orientable surface because the images of the elements x1x2 and x2 are the generators R and S respectively and both elements are orientation-preserving. 2. The group [96,70] can be expressed in terms of permutations, by means of the generators a = (1 2)(3 4)(5 8)(6 7) and b = (1 5)(2 8 3 64 7). 3. We can find the group [96,193] in [24], called G|8, but in the presentation given there, one relation is missing. We have added it, as can be seen in Table 5. In symmetric crosscap number 11 we have to stand out two things: One is that the presentation of group [108,15] can be expressed in terms of permutations of S18 as a = (4 7)(5 8)(6 9)(13 16)(14 17)(15 18) and b = (1 17 5 14 2 18 6 15 3 16 4 13) (7 12 9 118 10); and the other is that the group [108,17] is G3'6'6 in the notation of [12]. For symmetric crosscap number 12 and 13, we have nothing to remark. In symmetric crosscap number 14 we find several groups of order 48, and the following groups stand out: 1. The presentation of the group [72,43] has been deduced from its algebraic structure (C3 x A4) x C2. We have taken d as the generator of C2 and we have determined how d acts on the other generators. 2. The same argument has been applied to the group [96, 89], where its algebraic structure (D2 x D6) x C2 determine its presentation. In this case, e is the generator to add. The presentation is given by generators a, b, c, d, e and relations a2, b2, c2, d2, e2, (ab)2, (cd)6, eabea, ecdec. Let A be an associated NEC group with signature (0; +; [-]; {(2, 2, 2,4)}) and reduced area |, so an epimorphism is 0: A ^ G given by 8 0(c1,o) = e, 0(c1,1 ) = b, 0(c1,2) = a, 0(c^) = c, 0(cM) = e The elements c1,0, c1,2, c1,3, c1,1 and (c1,4c1,3)2 have as images the generators e, a, c, b, d respectively and generate the group. On the other hand the element (c1'0c1,2)2 c1,1 has as image the identity element and it is orientation-reversing. Thus, the group acts on a non-orientable surface. 3. The same happens for [96,115] and its algebraic structure is (C2 x D12) x C2, where d is the generator of C2 and so that we have to determine its relations with the other generators. In symmetric crosscap number 15, we just note that the group [1092, 25] was obtained in [21] by Wendy Hall, who proved that PSL(2,13) is a group of 84(g - 2) automorphisms of a surface of genus g, and so g = 15. Nothing stands out in symmetric crosscap number 16. But in symmetric crosscap number 17 we have again the same situation that in symmetric crosscap number 14. For the group [72,23] we have deduced the presentation from its algebraic structure (C6xD3)xC2, taking d as the generator of C2 and obtaining its action on the other generators. 180 Ars Math. Contemp. 15 (2018) 147-160 Table 5: Groups of symmetric crosscap number 10. GAP G Relations [+ Generators] Ref. [16, 2] C4 x C4 a4,b4 [19] [16,4] C4 X C4 a4, b4, b-1aba [15] [16, 9] dc4 a8, a4b2, b-1aba [23] [16,10] C4 x C2 x C2 a4,b2,c2 [19] [24, 3] (2, 3, 3) a3, abab-1a-1b-1 [15] [32, 5] a2, b8, c2, bcb-1ac [32, 6] r7a-i a2 ,b2, c2, d4, bdbad-1, cdcbad-1 [32, 7] r7a,2 a8, b2, c2, aba3b, aca-1bc [32, 9] rsai a2, b8, c2, bcbac [32,11] rse a4, b4, c2, bcba-1c [32,17] r2k ai6,b2,aba7b [32,19] r$a2 ai6,b2,abagb [32, 28] r4bi a2,b2,c4,d2, bdbad, (cd)2 [32, 34] r4a2 a4, b4, c2, (ac)2, (bc)2 [32, 42] r3b a8, b2, c2, (ac)2,bcba4c [32,46] C2 x C2 x D4 ~ r2ai a2,b2,c2,d2, (ab)2, (cd)4 [17] [32, 49] r5ai a4, b2a2,c2a2, d2a2, abab~i, cdcd-i [48, 29] GL(2, 3) a2, b3, c3, (bc)4, (ab)2, (ac)2, [b, c](bc)2 [48, 31] C4 x A4 a4, [+ (1 4)(3 2), (1 2 3)] [18] [48, 33] SL(2, 3) X C2 a2, b3, c3, (bc)4, abac, [b,c]2(bc)2 [48, 50] (C2 xC2 xC2 xC2) XC3 a2,b3,c3, (cb)2, (ab-i)3, c-ib~iabca, cbab-ic-ia [64,128] ri5ai a2,b2, c2, e2, f 2,d2f, [a, b]fd~i, [a, c]e, [a, d]f, [b, d]f [64,134] r26ai a2,b2, c2, e2, f 2,d2f, [a, b]fd-i, [a c]e, [a, d]f, [b, d]f, [b, e]f, [c d]f [64,138] r25a'i a2, b2, c2, d2, e2, f2, [a, b]d, [a, c]e, [b, e]f, [c, d]f [64,190] rigai a2,b2, c2, f 2,d2fe-i,e2f, [a, b]e-1d-1, [a, c]f, [a, d]fe-1, [a, e]f, [b,d]fe-1, [b, e]f [96, 70] ((C2 x C2 x C2 x C2) X C3) X C2 a2,b6, (bab-1 a)2, (b-2a)3 [96,187] (C2 x S4) X C2 a4, b12, c2, (ab)2, (cb)2, (ac)2, cb-1ab-1a-2 [96,193] GL(2, 3) X C2 a2, b8, c3, (bc)2, (ac)2, (ab)2, b4cb4c-1 [96, 227] ((C2 x C2 x C2 x C2) X C3) X C2 a2, b3, c2, d2, e2, f2, (ba)2, cada, cbdb-1, daca, dbdcb-1, eafa, ebfeb-1, faea, fbeb-1 [192, 955] (((C2 x C2 x C2 x C2) X C3) X C2) X C2 a4, b6, c2, (ab)2, (cb)2, (ac)2, (ab-1)4, cb-1ab2a-1b3a-1 A. Bacelo: Groups of symmetric crosscap number less than or equal to 17 181 Table 6: Groups of symmetric crosscap number 11. GAP G Relations [+ Generators] Ref. [18, 5] Ce x C3 ae,b3 [19] [27, 3] (3, 3 | 3, 3) a3, b3, (ba)3, (b-1a)3 [15] [36,13] C2 x ((C3 x C3) X C2) a2, b3, c3, d2, (ba)2, (ca)2 [54, 5] (2, 3, 6; 3) a3, be, (ab)2, (ba-1b)3 [54, 8] ((C3 x C3) X C3) X C2 a2, b3, c2, (b-1a)2, (ca)3, (b-1c)2(bc)2, (ab-1c)2bac [108,15] ((C3 x C3) X C3) X C4 a2, (b-2a)3,b-1ab4ab-3, b-1abab-2 abab-1a [108,17] G3,e,e a2, b2, c2, (ab)2, (ac)3, (bc)e, (abc)e [216, 87] (((C3 x C3) X C3) a4, be, c2, (ab)2, (cb)2, (ac)2, x C4 ) x C2 c(b-1a)3a Table 7: Groups of symmetric crosscap number 12. GAP G Relations [+ Generators] Reference [20,1] DC5 a10, a5b2, b-1aba [23] [40, 5] C4 x D5 a4, b2, c2, (bc)5 [16] [40, 8] (C10 x C2) X C2 a10, b2, (aba)2, (a-1b)2(ab)2 [40,10] C5 x D4 a5, b2, c2, (bc)4 [16] [40,12] C2 x (5,4, 2) a5, b4, bab-1a3 [80, 39] D5 x D4 a2, b2, c2, d2, (ab)5, (cd)4 [17] [240,189] C2 x S5 a2, [+ (1 2 34 5), (1 2)] Table 8: Groups of symmetric crosscap number 13. GAP G Relations [+ Generators] Reference [42, 3] C7 x S3 ar, [+ (1 2 3), (1 2)] [16] [42, 4] C3 x Dr a3, b2, c2,(bc)r [16] [52, 3] C13 X C4 a4, b13, baba-1 [60, 9] C5 x A4 a5, [+(1 2 3), (1 4)(2 3)] [18] [84, 8] D3 x Dr a2, b2, c2, d2, (ab)3, (cd)r [17] [120, 38] (C5 x A4) X C2 a4, b15, c2, (ab)2, (cb)2, (ac)2, cb-1ab-1a2 182 Ars Math. Contemp. 15 (2018) 147-160 Table 9: Groups of symmetric crosscap number 14. GAP G Relations [+ Generators] Ref. [16,12] C2 x Q a4, a2b2, c2, b-1aba [23] [24,4] C2 x DC3 a3, b4, c2, bab-1 a [18] [24, 7] DCe a12,aeb2,b-1aba [23] [24,15] Ce x C2 x C2 ae,b2,c2 [19] [32,48] T2& a4, b2, c2, d2,bcba2c [36,11] C3 x A4 a3, [+ (1 2)(3 4), (1 2 3)] [18] [36,12] Ce x D3 ae,b2,c2, (bc)3 [16] [48, 6] C24 X C2 a24, b2, baba13 [48,14] (C12 x C2) X C2 a3, b4, c4, (bc)2, (b-1c)2, c-1aca [48, 21] C3 x (4,4 | 2, 2) a3, b4, c4, (bc)2, (b-1c)2 [48, 24] C3 x QA4 a8, b2, c3, baba3 [48, 37] (C12 x C2) X C2 a3, b2, c2, d2, dcbcdb, dcbdbc, bdcdbc, (ba)2, (da)2 [48,43] C2 x ((Ce x C2) x C2) a4, be, c2, (ab)2, (a-1b)2 [48,49] C2 x C2 x A4 a2, b2, [+ (1 2)(3 4), (1 2 3)] [48, 51] D2 x De a2, b2, c2, d2,(ab)2, (cd)e [17] [72, 42] C3 x S4 a3, [+(1 2), (1 2 3 4)] [72, 43] (C3 x A4) X C2 a3, b2, c3, d2, (da)2, (dc)2, [+ b = (1 2)(3 4), c =(12 3)] [72, 44] A4 x S3 [+ (1 2)(3 4), (1 2 3), (5 6 7), (5 6)] [72,46] D3 x De a2, b2, c2, d2,(ab)3, (cd)e [17] [96, 89] (D2 x De) x C2 a2, b2, c2, d2, e2, (ab)2, (cd)e, eabea, ecdec [96,115] (C2 x D12) X C2 a2, b2, c2, d2, (bc)12, dcbdc [96, 226] C2 x C2 x S4 a2, b2, [+ (1 2 3 4), (1 2)] [144,183] S3 x S4 [+(1 2 3), (1 2), (4 5 6 7), (4 5)] [180,19] A5 x C3 a3, [+(1 3 24 5), (2 4 3), (2 4)(1 3)] [360,121] A5 x D3 a3, b10, c2, (ab)2, (cb)2, (ac)2, b-2ab3a-1b-4a-1c Table 10: Groups of symmetric crosscap number 15. GAP G Relations [+ Generators] Ref. [24,1] (-2, 2, 3) a8, b8, (a3b)3, a2be, a2(b-1a-1)3, b2(b-1a-1 )3 [15] [39,1] C13 X C3 a3,b13,bab10a-1 [48,15] (C3 x D4) X C2 a2, b8, c3, (ab)2, (ac)2, b-1cbc [78,1] (C13 X C3) X C2 a2, b3, c13, (ca)2,cbc10b-1 [1092, 25] PSL(2,13) a3, b7, c2, (ab)2, (cb)2, (ac)2, b-1(ab-2)ea-1c [21] A. Bacelo: Groups of symmetric crosscap number less than or equal to 17 183 Table 11: Groups of symmetric crosscap number 16. GAP G Relations [+ Generators] Reference [28,1] dc7 a14, a7b2, b-1aba [23] [56,4] C4 x D7 a4, b2, c2, (bc)7 [16] [56, 7] (C14 x C2) * C2 a2, b14, (bab)2, (b-1a)2(ba)2 [56, 9] C7 x D4 a7, b2, c2, (bc)4 [16] [72,16] C2 x ((C2 x C2) * Cg) ag, b2, c2, d2, bacba-1, caba-1 [112, 31] D7 x D4 a2, b2, c2, d2, (ab)7, (cd)4 [17] [144,109] (C2 x ((C2 x C2) a4, b18, c2, (ab)2, (cb)2, (ac)2, * Cg)) * C2 cb-1ab-1a-2 Table 12: Groups of symmetric crosscap number 17. GAP G Relations [+ Generators] Ref. [25, 2] C5 x C5 a5, b5 [14] [27, 2] Cg x C3 ag,b3 [14] [27,4] Cg * C3 a3, bg, bab5a-1 [15] [36, 6] C3 x DC3 a12, b3, baba-1 [18] [50, 3] C5 x D5 a5, b2, c2, (bc)5 [16] [50,4] (C5 x C5) * C2 a2, b5, c5, (ba)2, (ca)2 [54, 3] C3 x Dg a3, b2, c2, (bc)g [16] [54,4] Cg x D3 ag,b2,c2, (bc)3 [16] [54, 6] (Cg * C3) * C2 a2, bg, c3, (ba)2,cb7c-1b-1 [54, 7] (Cg x C3) * C2 a2, b3, cg, (ba)2, (ca)2 [68, 3] C17 * C4 a4, b17, bab4a-1 [72, 23] (Ce x D3) * C2 ae, b2, c2, d2, (bc)3, bdcbd, dada3 [72, 39] (C3 x C3) * C8 a8, b3, c3, baca-1, cab-1a-1 [100,12] (C5 x C5) * C4 a4, b5, c5, bab3a-1, cac3a-1 [100,13] D5 x D5 a2, b2, c2, d2, (ab)5, (cd)5 [17] [108,16] D3 x Dg a2, b2, c2, d2, (ab)3, (cd)g [17] [200, 43] (D5 x D5) * C2 a4, b10, c2, (ab)2, (cb)2, (ac)2, [ab, ba], cb-1(ab-3)2a-2 [360,118] Ae [+(1 4 2 3 5), (3 5 4), (1 2 4 3)(5 6)] [720, 764] Ae * C2 a3, b8, c2, (ab)2, (cb)2, (ac)2, cb-1ab3ab-2a-1ba-1b-3a-1 184 Ars Math. Contemp. 15 (2018) 147-160 5 Groups with symmetric crosscap number 12 k + 3 Firstly, the strong symmetric genus is the minimum genus of any Riemann surface on which G acts, preserving orientation. For this parameter, there is a group of every strong symmetric genus, [25]. The symmetric genus is the smallest non-negative integer g such that the group G acts faithfully on a closed orientable surface of genus g (not necessarily preserving orientation). For this parameter, the spectrum includes every non-negative integer g ^ 8 or 14 (mod 18), and moreover, if a gap occurs at some g = 8 or 14 (mod 18), then the prime-power factorization of g — 1 includes some factor pe = 5 (mod 6), [11]. In the study of the spectrum of the symmetric crosscap number, the groups with symmetric crosscap number of the form 12k + 3 are very interesting. It is known that for all n = 12k + 3, there is a finite group with symmetric crosscap number n, see [6]. Conversely, for some values n = 12k + 3, it is not known whether there exists a group with symmetric crosscap number n. So that, we can enunciate some theorems whereby we find infinite families of groups whose symmetric crosscap number is of the type 12k + 3. The symmetric crosscap numbers obtained in Theorems 5.1 to 5.5, although of 12k + 3 form, were already obtained for other groups, as we can see in the proofs. In the case of Theorem 5.6, also these numbers n were already covered, since the group C7(12fc+7) x C3, in the terms of the statement, has symmetric crosscap number 84k + 51, see [6]. But they are important because they give more examples of groups of this type of n, helping us to see how these groups act. Theorem 5.1. Let n = 12k + 3 be such that n — 2 has all its prime factors congruent to 1 (mod 3). Then Ci2fc+i x C3 and (Cm+i x C3) x C2 have symmetric crosscap number n. Proof. Firstly we have that C12k+1 x C3 has a presentation given by generators a, b such that a3 = b12k+1 = (ab)3 = 1. Now let A be an NEC group with signature (1; —; [3,3]; {—}), whose reduced area is 3. We can define an epimorphism 0: A ^ G given by 0(x1 ) = a-1, 0(x2) = ab, 0(d1) = b6k We have that the images of x1 and x1x2 are the generators a-1 and b respectively, and both are preserving-orientation elements, then we have that it is a group that acts on a non-orientable surface. Besides, the NEC group area is minimal ([6]), and so the symmetric crosscap number of C12k+1 x C3 is n. Now we have (C12k+1 x C3) x C2 that has a presentation given by generators a, b, c and relations a3 = b12fc+1 = c2 = (ab)3 = 1, ca = ac and bc = cb-1. Now let A be an NEC group with signature (0; +; [2,3]; {(—)}), whose reduced area is 6. Therefore, if we define an epimorphism from this NEC group, (C12k+1 x C3) x C2 will have symmetric crosscap number less or equal to n. We can define an epimorphism 0: A ^ G given by 0(x1) = cb, 0(x2) = b-1a-1, 0(e1) = ac, 0(c1,o) = c We have that the element c1,0, the element e1c1,0 and the element c1,0x1 have as images the generators c, a and b respectively. Besides the element (e1c1,0)3 has as image the identity element and it is orientation-reversing, so we have just proved that the group acts on a non-orientable surface. Because of this epimorphism we can say that (C12k+1 x C3) x C2 A. Bacelo: Groups of symmetric crosscap number less than or equal to 17 185 has symmetric crosscap number at most n. But since it contains C12k+1 x C3, that has symmetric crosscap number n, a((C12fc+1 x C3) x C2) = n. □ Theorem 5.2. Let n = 12k + 3 be such that n — 2 = m2 is a square. Then: (i) (3,3 | 3, m) has symmetric crosscap number n. (ii) There are two groups with algebraic structure (3,3 | 3, m)xC2, namely (2,3,2m; 3) and (2, 3, 6; m), that have symmetric crosscap number n. Proof. Firstly we have that the group (3, 3 | 3, m) of order 3m2 has a presentation given by generators a, b and relations a3 = b3 = (ab)3 = (a-1b)m = 1. From [15], we know that this group has symmetric crosscap number m2 + 2. Now we have two groups with algebraic structure (3,3 | 3, m) x C2: (i) The first one, that is the group (2,3,2m; 3) in the notation of [12], of order 6m2, has a presentation given by generators a, b, c and relations a3 = b3 = c2 = (ab)3 = (a-1b)m = 1, ca = a2c and cb = b2c. Take an NEC group A with signature (0; +; [2]; {(3,3)}), that has reduced area 6. We define an epimorphism 0: A ^ G given by 0(x1) = c, 0(e1) = c, 0(c1,o) = ac, 0(^,1) = cb, 0(c1,2) = a-1 c We have that 0(x1) = c, 0(c1,0x1) = a and 0(x1c1,1) = b, and the element (e1c1,0)3 has as image the identity element and it is orientation-reversing. Thereby we have proved that the group acts on a non-orientable surface. Thereupon we have that this group has symmetric crosscap number at most m2 + 2, but as it contains (3, 3 | 3, m) that has that symmetric crosscap number n, then we have proved that