ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 15 (2018) 205-223
https://doi.org/10.26493/1855-3974.1419.3e9
(Also available at http://amc-journal.eu)
Enumerating regular graph coverings whose covering transformation groups are Z2-extensions of a cyclic group*
Jian-Bing Liu f
Mathematics, Beijing Jiaotong University, Beijing, 100044, P.R. China
Jaeun Lee
Mathematics, Yeungnam University, Kyongsan, 38541 Korea
Jin Ho Kwak
Mathematics, POSTECH, Pohang, 37673 Korea Mathematics, Beijing Jiaotong University, Beijing, 100044, P.R. China
Received 9 June 2017, accepted 12 October 2017, published online 20 June 2018
Several types of the isomorphism classes of graph coverings have been enumerated by many authors. In 1988, Hofmeister enumerated the double covers of a graph, and this work was extended to n-fold coverings of a graph by the second and third authors. For regular coverings of a graph, their isomorphism classes were enumerated when the covering transformation group is a finite abelian or dihedral group. In this paper, we enumerate the isomorphism classes of graph coverings when the covering transformation group is a Z2-extension of a cyclic group, including generalized quaternion and semi-dihedral groups.
Keywords: Graphs, regular coverings, voltage assignments, enumeration, Möbius functions (on a lattice), group extensions.
Math. Subj. Class.: 05C30, 20F28, 20K27
*The authors are grateful to anonymous referees for their valuable comments. The authors also would like to thank Young-Soo Kwon for illuminating discussions and remarks. This work was partially supported by the National Natural Science Foundation of China (11671030).
1 Corresponding author
E-mail addresses: jl0068@mix.wvu.edu (Jian-Bing Liu), julee@ynu.ac.kr (Jaeun Lee), jinkwak@postech.ac.kr (Jin Ho Kwak)
Abstract
©® This work is licensed under https://creativecommons.org/licenses/by/3.0/
206	Ars Math. Contemp. 15 (2018) 147-160
1 Introduction
Throughout this paper, all graphs and groups are assumed to be finite. Let G be a connected simple graph with vertex set V(G) and edge set E(G). The neighborhood of a vertex v € V(G), denoted by N(v), is the set of vertices adjacent to v. We use |X| for the cardinality of a set X. The number ft = |E(G)| - |V(G)| + 1 is equal to the number of independent cycles in G and it is referred to as the Betti number of G.
Two graphs G and H are isomorphic if there exists a one-to-one correspondence between their vertex sets which preserves adjacency, and such a correspondence is called an isomorphism between G and H. An automorphism of a graph G is an isomorphism of G onto itself. Thus, an automorphism of G is a permutation of the vertex set V(G) which preserves adjacency. Obviously, the automorphisms of G form a permutation group, Aut(G), under composition, which acts on the vertex set V (G).
A graph G is called a covering of G with projection p: G ^ G if there is a surjection p: V(G) ^ V(G) such that p|N: N(V) ^ N(v) is a bijection for any vertex v € V(G)
and 5 € p-1 (v). Also, we sometimes say that the projection p: G ^ G is a covering, and an n-fold covering if p is n-to-one. A covering p: G ^ G is said tobe regular (simply, A-covering) if there is a subgroup A of the automorphism group Aut(G) of G acting freely on G so that the graph G is isomorphic to the quotient graph G/A, say by h, and the quotient map G ^ G/A is the composition h o p of p and h. The fiber of an edge or a vertex is its preimage under p.
Two coverings^: Gj ^ G, i = 1, 2, are isomorphic if there exists a graph isomorphism $: G1 ^ G2 such that p2 o $ = p1, that is, the diagram
Gi ---► G2
G
commutes. Such a $ is called a covering isomorphism. A covering transformation is just a covering automorphism.
Every edge of a graph G gives rise to a pair of oppositely directed edges. By e-1 = vu, we mean the reverse edge to a directed edge e = uv. We denote the set of directed edges of G by D(G). Let A be a finite group. An ordinary voltage assignment (or, A-voltage assignment) of G is a function ^: D(G) ^ A with the property that ^(e-1) = ^(e)-1 for each e € D (G). The values of ^ are called voltages, and A is called the voltage group. The ordinary derived graph G x $ A derived from an ordinary voltage assignment ^: D(G) ^ A has as its vertex set V(G) x A, and as its edge set E(G) x A, so that an edge (e, g) of G x$ A joins a vertex (u, g) to (v,^(e)g) for e = uv € D(G) and g € A. In the (ordinary) derived graph G x$ A, a vertex (u, g) is denoted by ug and an edge (e, g) is denoted by eg. The first coordinate projection p$: G x$ A ^ G commutes with the left multiplication action of the ^(e) and the right multiplication action of A on the fibers, which is free and transitive, so that p$ is a regular |A|-fold covering, called simply an A-covering. Moreover, if the covering graph G x $ A is connected, then the group A becomes the covering transformation group of the A-covering.
For a group A, let C1 (G; A) denote the set of A-voltage assignments ^ of G. Choose a spanning tree T of G, and let
C1 (G; A) = € C 1(G; A) : ^(uv) is the identity for each uv € D(T)}.
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Gross and Tucker [4] showed that every A-covering G of a graph G can be derived from an A-voltage assignment ^ in C^ (G; A), say it T-reduced. From now on, let T denote a fixed spanning tree of a graph G, and we consider only an A-voltage assignment ^ in C^(G; A).
The enumeration problem of coverings became subject of investigation by many authors starting from the classical paper by Hurwitz published more then 100 years ago. In particular, enumeration of graph coverings became possible after the paper by Hall ([6]) published in 1949. In 1988, Hofmeister [8] counted double covers of graphs. Liskovets enumerated connected non-isomorphic coverings of the graph with a given Betti number, see [19, 20]. The number of connected and disconnected coverings were determined by Kwak and Lee in [15]. Later, Kwak, Lee and A. D. Mednykh counted cyclic and dihedral coverings over surfaces and graphs with prescribed topological characteristics, see [16,17].
Following notations in [14], let IsoR(G; n) denote the number of the isomorphism classes of regular (connected or disconnected) n-fold coverings of G, and use IsocR (G; n) for their connected ones. Similarly, let Iso(G; A) denote the number of the isomorphism classes of (connected or disconnected) A-coverings of G, and use Isoc(G; A) for their connected ones. By the properties of regularity of coverings, one can see that the number of the isomorphism classes of (connected or disconnected) n-fold regular coverings of a graph G is the sum of numbers of the isomorphism classes of connected d-fold regular coverings of G, where d runs over all divisors of n:
IsoR(G; n) = ^ IsocR(G; d).
d\n
Moreover, the number of the isomorphism classes of connected n-fold regular coverings of G is the sum of the numbers of the isomorphism classes of connected A-coverings of G, where A runs over all non-isomorphic groups of order n:
IsocR(G; n) = ^ Isoc(G; A).
A
Consequently, it just needs to determine the numbers Isoc(G; A) for every finite group A. Hong, Kwak and Lee [9] obtained an algebraic characterization of two isomorphic graph regular coverings given as follows.
Lemma 1.1. Let ^ G C1 (G; A) and ^ G C1 (G; B) be any two ordinary voltage assignments in G. If their derived (regular) coverings p<p: G x^ A ^ G and p^ : G x^ B ^ G are connected, then they are isomorphic if and only if there exists a group isomorphism a: A ^ B such that ^(uv) = a(^(uv)) for all uv G D(G) — D(T).
In particular, if two voltages ^ and ^ in C1 (G; A) derive connected coverings, then the derived coverings are isomorphic if and only if there exists a group automorphism a G Aut(A) such that
^(uw) = a(^(uv))
for all uv G D(G) — D(T).
With a linear ordering of the cotree edges of G, the set CT1(G; A) of T-reduced A-voltage assignments of G can be identified as
C1 (G; A) = A x • • • x A (^ times),
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that is, an A-voltage assignment ^ of G can be identified as a ,-tuple (gi,..., gp) of group elements gi G A. Moreover, such a ,-tuple of g's derives a connected covering if and only if it is transitive. It means by definition that the subgroup (g1,..., gp} generated by them acts transitively on the group A (under the left translation on A), or equivalently {g1, g2,...,gp} generates the whole group A.
Note that the automorphism group Aut( A) of A can act on the set of transitive ,-tuples of group elements gi G A coordinatewisely, and any two transitive ,-tuples of elements in A belong to the same orbit under the action if and only if they derive (connected) isomorphic A-coverings, by Lemma 1.1.
Clearly, the Aut(A)-action on the set of transitive ,-tuples of group elements gi G A is free (having no fixed element), and hence Burnside's counting Lemma gives a counting formula for Isoc(G; A) as follows.
Theorem 1.2 ([14]). For any finite group A,
Isoc(G; A) =	,
v ' 7 |Aut(A)|
where Q(A;,) = {(gi, g2,..., gp) G Ap | {gi, g2,..., gp} generates A}.
Note that the set Q(A;,) can be identified as the set of epimorphisms from the free group generated by , elements onto the group A.
To determine the number Isoc(G; A), we need to estimate |Aut(A)| and |Q(A;,)|. The number |Aut(A)| can certainly be determined for a few groups A. For example, one can refer to [14] for |Aut(A)| when A is abelian or dihedral groups. Also, one can see recent two papers [1], [7] for abelian case.
The other number |0(A; ,)| can be determined by a direct counting and it can also be determined in terms of the Mobius function defined on the subgroups lattice of A, as shown in [17]. The Mobius function assigns an integer ^(K) to each subgroup K of A by the recursive formula
E M(H)^1 if K = A, Zk W ) j0 if K < A.
Jones ([12, 13]) used the Mobius function to count the normal subgroups of a surface group or a crystallographic group, and applied it to count certain covering surfaces. We see that
|A|p = E |«(K;,)|.
K<A
It follows from the Mobius inversion that
|^(A;,)| = E M(K)|K|p.
K<A
The next theorem is deduced from Theorem 1.2. Theorem 1.3. For any finite group A,
1
Dl
K<A
Isoc(G; A) = |Aut(A)| E M(K)|K|p.
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Now, we have two ways of computing |0(A; ft) by a direct counting and by using the Mobius function on the subgroups lattice of A. For example, when A is cyclic or dihedral, Isoc(G; A) was determined by Kwak, Lee and Mednykh in [17] in terms of the Mobius function. However, it is not easy to determine the Mobius function on the subgroups lattice of any abelian group A. For an abelian group A, Isoc(G; A) was determined in [14] by a direct counting method.
This paper is organized as follows. In a coming section, we review an extension of a group, giving a classification of Z2-extensions of a cyclic p-group and a discussion on Z2-extensions of a cyclic group. In Sections 3 and 4, we determine the number Isoc(G; A) when A is a Z2-extension of a cyclic p-group, or Z2-extensions of any cyclic group, as main results in this paper. In Section 5, we try to extend our discussion to a Z2 -extension of an abelian group, by considering two special cases of them.
2 Review on extensions of groups
We review briefly an extension of a group with some recent results to use it in this paper.
Let N and Q be two groups. A group A is an extension of N by Q (or a Q-extension of N) if N is a normal subgroup of A and the quotient group A/N = Q. Or equivalently, a sequence
1 anaAA Q a1
is exact. The extension is split if N has a complement in A. By a complement of N in A, we mean a subgroup H satisfying A = NH and NnH = 1. Otherwise, the extension is
nonsplit.
Let us assume that the given extension is split. For a complement H of N in A, one has H = A/N. So we can view Q as a subgroup of A. A trivial case is an (internal) direct product of two groups N and Q: A = NQ with NnQ = 1 and a trivial commutator [N, Q] = 1. For all nontrivial cases, it holds A = NQ with Nn Q = 1, but the commutator [N, Q] is not trivial and the multiplication in N is twisted by an action of the elements of Q, that is, for n e N and qj e Q with i, j e {1, 2},
(niqi)(n292) = n1(q1U2q-1)qiq2 = n1n'^(qi) qiq2,
where a: Q a Aut(N) is a homomorphism defined by a(q)_i = Inn(q-1). The semidi-rectproduct A = N x a Q of N by Q with respect to a is defined on the set
A = {(n, q) | n G N,q G Q}
with a multiplication
(ni, qi)(n2, q2) = (nin^^ , qiq2).
The semidirect product A = N xa Q is in fact a group with (n, q)_i = (n-a(q), q_i). If we identify Q and N with {(1, q) | q e Q} and {(n, 1) | n e N}, respectively, then A = NQ = QN and NnQ = 1. So a semidirect product A = N xa Q is a split extension of N by Q. Consequently an extension of N by Q is split if and only if A is a semidirect product of N by Q.
A (split or nonsplit) extension of a cyclic group by another cyclic group is called a metacyclic group. The next two lemmas are famous in finite group theory, see [11] and [10], respectively.
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Lemma 2.1 (Holder). Let A be a metacyclic group which is an extension of a cyclic group of order n by a cyclic group of order m. Then A has the following presentation
A = (a,b | an = l,bm = a*, b-1ab = ar),	(2.1)
where n, m, t and r satisfy
rm = 1 (mod n), t(r - 1) = 0 (mod n).	(2.2)
Conversely, for any parameters n, m, t, r satisfying Equation (2.2), the relations in Equation (2.1) define a metacyclic group which is an extension of a cyclic group of order n by a cyclic group of order m.
A subgroup N of A is a Hall subgroup if |N| is coprime to |A : N|.
Lemma 2.2 (Schur-Zassenhaus). Let N be a normal Hall subgroup of A. Then
(1)	N has a complement in A.
(2)	If H and K are two complements of N in A, then there is an element n e N such that n-1Hn = K.
By Lemmas 2.1 and 2.2, one can show that a Z2 -extension of a cyclic p-group with odd prime p is a cyclic or a dihedral group. Now, let p = 2.
The following theorems in this section come from an unpublished manuscript [18] Chapter 3 by Kwak and Xu. Since the authors cannot find these theorems in any other sources, we add their proofs in this paper.
Theorem 2.3. Let A be a Z2-extension of a cyclic 2-group Z2n-i with n > 4. Then A is isomorphic to one of following six groups.
(1)	(the cyclic group)
Z2n = (b | a2"-1 = 1,b2 = a),
(2)	(the non-cyclic abelian group)
Z2n-i x Z2 = (a,b | a?n = 1, b2 = 1, b-lab = a),
(3)	(the dihedral group)
D2n = (a,b | a2"-1 = 1, b2 = 1, b-lab = a-1),
(4)	(the generalized quaternion group)
Q2n = (a,b | a2"-1 = 1, b2 = a2"-2 ,b-1 ab = a-1),
(5)	(the ordinary metacyclic group)
M2" = (a,b | a2"-1 = 1, b2 = 1, b-1ab = a1+2"-2),
(6)	(the semidihedral group)
SD2n = (a,b | a2"-1 = 1, b2 = 1, b-1 ab = a-1+2"-2).
All the six groups are not isomorphic one another.
Proof. Since (1) and (2) are trivial cases, we assume that A is not abelian. By Lemma 2.1, A has the following presentation:
A = (a,b | a2"-1 = 1, b2 = a*, b-1ab = ar),
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where t and r satisfy
r2 = 1 (mod 2n-1), t(r - 1) = 0 (mod 2n-1).
By A non-abelian, one has r = —1 or ±1 + 2n-2 (mod 2n-1), the latter two cases can happen only when n > 4. If r = —1 or — 1 + 2n-2 (mod 2n-1), then 2n-1 | 2t and hence 2n-2 | t, it follows that t = 0 or 2n-2 (mod 2n-1). Now we consider the three cases separately.
(i)	r = — 1 (mod 2n-1). In this case we get the dihedral group (3) and the generalized quaternion group (4) depending on t = 0 or 2n-2 (mod 2n-1), respectively. These two groups are not isomorphic. Note that the following cases (ii) and (iii) happen only when n > 4. So, when n = 3 we have only the above two groups.
(ii)	r = — 1 + 2n-2 (mod 2n-1). In this case t = 0 (mod 2n-2). Thus b2 = 1 or a2" .If b2 = a2" , letting b1 = ba, then
1.2/1. \2 7,2/7-1 IN 7.2 -1+2"-2	2"-2 2"-2 -i
b2 = (ba)2 = b2(b 1 ab)a = b2a 1+2 a = a2 a2 =1.
Thus we get the group (6).
(iii)	r = 1 + 2n-2 (mod 2n-1). In this case, one has t • 2n-2 = 0 (mod 2n-1) which implies that t is even. Let t = 2s. Since n > 4, there is a j satisfying j(1 + 2n-3) + s = 0 (mod 2n-2). Let b1 = baj. Then
b2 = b2(b-1aj b)aj = b2aj(2+2"-2) = a2(j(1+2"-3)+s) = 1. Now the generators a, b1 satisfy the relations in the group (5), with b instead of b1.
Finally, we shall show that the mentioned four non-abelian groups are not isomorphic, and we assume that n > 4. It is easy to see that in these four cases the derived group A' = ([a, bj). We calculate the commutator [a, b] and get
{a-2	for the groups (3) and (4),
a2"	for the group (5),
a-2+2" 2 for the group (6).
So, one has |A'| =2 for (5), and |A'| = 2n-2 for the others. It follows that the group (5) is not isomorphic to any one of the rest. To prove the rest three groups are not isomorphic, we calculate the square of the elements of the form ba® outside (a). We have
{1	for the group (3),
a2"-2 for the group (4), a®2" 2 for the group (6).
This shows that the subgroup of order 2n-1 in A is unique, and outside this subgroup (a), all elements are of order 2 in the group (3), order 4 in the group (4), and some are of order 2 and the others are of order 4 in the group (6). Therefore, all the four groups are not isomorphic to one another.	□
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Let A be a Z2-extension of a cyclic group Zn, where n = p^0Pi1 • • • is the prime decomposition with p0 = 2. First, we consider the case that n is odd, that is, a0 = 0.
Theorem 2.4. Let A be a Z2-extension of a cyclic group Zn = Zp^i x • • • x Zpa with n odd. Then, A has a presentation
A = (ai, .. ., as, b | aPi = b2 = 1, [aj, aj] = 1, b-1ajb = a^ for all i, j), where r2 = 1 (mod p"i) for all i. There are 2s non-isomorphic such extended groups.
Proof. By Lemma 2.2, A is split. Since A is a metacyclic group, by Lemma 2.1, A has the presentation
A = (a, b | an = b2 = 1, b-1ab = ar),
with r2 = 1 (mod n). The action of b on each element of Zn by conjugacy is an automorphism of Zn of order at most 2. Since Aut(Zn) = Aut(Zpai) x • • • x Aut(Zpj=), the b-conjugation on Zn corresponds to an s-tuple (r1,..., rs) with r = ±1 (mod p"i) for i € {1,..., s}. Thus the s-tuple (r1,..., rs) has 2s choices and A is presented by

A = (a1,...,as,b | ap = b = 1, [ai,aj] = 1,b ajb = a^ for all i,j).
To finish the proof, it suffices to show that different s-tuples (r1,..., rs) give non-isomor-phic groups. It is easy to see that Zpa is a subgroup of the center of A if and only if r = 1. Hence the groups with different s-tuples (r 1,..., rs) have different center of A. Therefore, there are 2s non-isomorphic Z2-extensions of Zn.	□
Next we consider the case of even n. Let A be a Z2-extension of a cyclic group Zn = Zpao x Zpai x • • • x Zpj= with p0 = 2. We deal with three cases a0 = 1,2 or a0 > 3 in the next theorem. First we determine the Sylow 2-subgroup S0 of A which is a Z2-extension of Z2ao = (a0). This has been done by Theorem 2.3. Namely,
S0 = (a0, b0 | a0 0 = 1, b§ = a0°, b-1 a0b0 = a0°),
where t0 = 0,1 or 2a-1, r0 = ±1 or ±1 + 2a°-1 depending on the types of S0 in Theorem 2.3. Next, take b = b0. Thus each Sylow 2-subgroup and each element of order at most 2 in Aut(Zp«i) x • • • x Aut(Zp?s) gives a unique Z2-extension of Zn.
Theorem 2.5. Let A be a Z2-extension of a cyclic group Zn = Zpa° x Zpai x • • • x Zpa with p0 = 2.
(1) If a0 = 1, then A has the following presentations

(i) A = (a0, a1, .. ., as, b | aPi = b2 = 1, [aj, aj] = 1, b 1ajb = a^ for all i, j)
(S0 = Z2 x Z2).
(ii) A = (a0, a1, ..., as, b | aPi = 1, b2 = a0, [aj, aj] = 1, b-1ajb = a^ for all
i, j), (S0 = Z4).
There are 2s+1 non-isomorphic groups. (2) If a0 = 2, then A has the following presentations
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(i)	A = (ao, ai, ... ,as,b | aPi	= b2 = 1, [ai ,aj ] = 1, b 1ajb = arii for all i, j), (S0 = Z4 x Z2 or D8).
(ii)	A = (ao, ai,. .., as, b | aPi = 1, b2 = ao, [aj, aj] = 1, b-1ajb = ai®for all i,j), (So = Z8).
(iii)	A = (ao, ai,. .., as, b | aPi = 1, b2 = a0, [aj, aj] = 1, b-1ajb = ai*for all
i,j), (So =
There are 2s+1 non-isomorphic groups. (3) If ao > 3, then A has the following presentations

(i)	A = (ao, ai, ... ,as,b | aPi = b2 = 1, [a^ aj ] = 1, b 1ajb = ai* for all i, j),
(So = Z2ao X Z2, D2a0 + 1 , SD2a0 + 1 , or M2a0 + 1 ).
(ii)	A = (ao, ai,. .., as, b | aPi = 1, b2 = ao, [aj, aj] = 1, b-1ajb = arii for all i,j), (So = Z2ao + 1 ).
ai	a — 1
(iii)	A = (ao, ai ,...,as,b | ap = 1, b2 = ao 0 , [aj,aj- ] = 1,b-1ajb = a^ for all i,j), (So = Q2"0+1).
There are 6 • 2s non-isomorphic groups.
For each extension group A appeared so far, the number Isoc(G; A) shall be determined in the next section.
3 In cases of Z2-extensions of a cyclic p-group
For each group A in the classification of Z2 -extensions of a cyclic p-group listed in the previous section, we aim to determine the number Isoc(G; A) in this section. However, for an abelian or a dihedral group A, it has already been done in [14]. Hence, we need to do it only for each group A listed in the last three cases of Theorem 2.3. For a Z2-extension A of a finite group H, we call an element x normal type if x e H and quotient type otherwise. Note that H is normal in A, and a product of any two normal type elements is normal type. For any two quotient type elements ab, a'b, their product is aba'b = ab2b-1a'b, and hence a product of any two quotient type elements is normal type. A word in {x1,..., xs} is any expression of the form yj1 • • • yj,fc where y1,... ,yk e {x1,... ,xs} and i1,... ,ik e {1, -1}, denoted by w(x1,..., xs). The number k is known as the length of the word. When writing words, it is common to use exponential notation as an abbreviation.
Lemma 3.1. Let A be a Z2-extension of a finite group H. For a subset I of S = {1, ...,P}, let
Qz (A; ft) = {(x1,..., xp) e 0(A; ft) : xj is quotient type for exactly indices i e I}. Then |Q(A;ft)| = (2p - 1)|Q{i}(A; ft)|. Proof. Recall that
Q(A; ft) = {(x i,..., x p) e Ap : (xi,...,xp) = A}.
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For each tuple (xi,..., x^) g O(A; P), at least one of entries xj should be quotient type to generate the whole group A. Then,
Q(A; P) = y Ox (A; P), disjoint union,
0=ICS
and
|^(A; P)| = £ (A; P)|.
0=ICS
For any non-empty subset I of S, choose an index j0 G I and define a map ^: (A; P) ^ n{j0}(A; P) by replacing all quotient type entries xj for i G I by xj0xj except xj0. Then one can see that ^ is well-defined and bijective. It follows (A; P) | = (2^ - 1)|O{j0}(A; P)|. One can assume that j0 = 1 for convenience.	□
Lemma 3.2. Let A be a Z2-extension of a finite group H. If each xj is normal type except xi, then (x1,..., x^} = A if and only if (xi, x2,...,x^, x-1x2x1,..., x-1x^ x1) = H.
Proof. Assume (x2, x2,..., x^, x-1 x2x1,... x-1x^x1) = H and each xj is normal type except x1. Then (x1,..., x^} = (x1, x2, x2,..., x^, x-1x2x1,... x-1x^x1) = (x1, H) = A. Now assume that (x1,..., x^} = A. For any g G A, g can be expressed by a word w(x1,..., x^). For odd k, xjxf = x1 • (x-1xjx1) • (x1)(k-1)/2 and for even k, xjxk = xj • (x^/2). Rewrite g, one has
g = w(x1,..., x^) = x1w(x2, x2,..., x^, x-1x2x1,..., x-1x^x1), i = 0,1.
It follows that g is normal type if and only if i = 0. Therefore, ( x2,x2,...,x^ , x- x2x1, ...,x-1x^ x1} = H.	□
Corollary 3.3. Let A be a Z2-extension of a cyclic group Zn. If each xj is a normal type element except x1, then (x1,..., x^} = A if and only if (x2, x2,..., x^} = Zn.
We determine |O(A; P) | and | Aut(A) | for each group A listed in the last three cases of Theorem 2.3 in the following.
Lemma 3.4. Let A be a Z2-extension of a cyclic group Z2n-i and let A be non-abelian. Then |^(A; P)| = 2("-2)^+1(2^ - 1)(2^-1 - 1).
Proof. By Lemma 3.1, it just needs to determine ^{1}(A;P). By Corollary 3.3,
(x2, x2,..., xs} = Z2n-i if and only if (x1,..., xs) G ^{1}(A; P). By the last three cases of Theorem 2.5, one can assume b2 = a4 with t = 0 or 2n-2 for a generator a of Z2n-i. Note that x1 is quotient type, say x1 = baj. Then x2 = b2 • b-1ajb • aj = b2aj(1+r) = at+j(1+r) with r G {-1, ±1 + 2n-2}. Suppose x1 generates Z2n-i. Then t + i(1 + r) = 1 (mod 2). But it is impossible by checking case by case. So (x2,... ,xs} = Z2n-i. By |O(Z2n-i; P - 1)| = 2(n-2)(^-1)(2^-1 - 1), which was shown by Kwak et al. in [14], it follows |^(A;P)| = (2^- 1)2n-1 |^(Z2n-i;p-1)| = 2("-2)^+1(2^-1)(2^-1 -1). □
Lemma 3.5. For n > 4,
(1)	|Aut(Z2n)| =2n-1,
(2)	|Aut(Z2n-i x Z2)| = 2n,
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.2n-3
2n-3
(3)	|Aut(D2n )| = 2'
(4)	|Aut(Q2n )| = 2'
(5)	|Aut(M2n )| = 2n,
(6)	|Aut(SD2n )| = 22n-4.
Proof. Since the first three cases have been shown in [14], we only need to show the last three cases. To do this separately, let A be a Z2 -extension of a cyclic group Z2n-1 and let an automorphism a G Aut(A) be of the form a ^ aibk ,b ^ a? b£ with 0 < i, j < 2n-1 - 1 and 0 < k,i < 1.
(4)	Since the identity (aib)2 = bb-1albalb = b2 gives the orders o(aib) = 4 and o(a) = 2n-1 = 4 for n > 4, the image a(a4) should be of the form a1 with (i, 2n-1) = 1. The surjectivity of a implies that the choices of a(b) are a? b with j = 0,..., 2n-1 -1. Moreover, all of such possible choices a(a) and a(b) satisfy the defining relations of Q2n. Hence |Aut(Q2n)| = 22n-3 by counting the choices of a(a) and a(b), that is, the choices of i, j, k, i.
(5)	If k = 0, then a(a) = a1 for some i with (i, 2n-1) = 1. If k = 1, then a(a) = alb for some i with (i, 2n-1) = 1, because the order preserving condition says o(a®b) = o(a) = 2n-1, and (aib)m = bmai(1+ •+rm-1) for all m > 1, where r = 1 + 2n-2. Next, we determine the possible values of a(b). If i = 0, then j should be 2n-2. In this case, all possible values a(a) and a(b) do not satisfy the defining relations of M2n. Thus it should be i = 1. Now the order condition o(akb) = o(b) = 2 implies j = 2n-2 or 0. Consequently, a has four different forms.
(i)	a ^ ai,b ^ b,
(ii)	a ^ ai,b ^ a2" b,
(iii)	a ^ aib, b ^ b,
(iv)	a ^ aib,b ^ a2" b.
In these four cases, a(a) and a(b) satisfy the defining relations of M2n. Therefore, the four different cases give | Aut(M2n) | = 2n.
(6)	Since (aib)2 = ai2" 2, one gets o(aib) = 2 for even i and o(aib) = 4 for odd i. Hence a should be of the form a ^ ai, b ^ a?b with (i, 2n-1) = 1 and j even. Moreover, all such possible values a(a) and a(b) satisfy the defining relations of
SD2n .So |Aut(SD2n )| = 22n-4.	□
As a special case, |Aut(Q8)| = 24 which is not included in the above lemma. From Theorem 1.2 and Lemmas 3.4 and 3.5, one can get the following theorem.
Theorem 3.6. For a Z2-extension of A a cyclic group Z2n-i for n > 2,
(2(^-1)(n-1)(2^ - 1)	if A is Z2" ,
2(^-2)(n-2) + (n-3)(2^ - 1)(2^-1 - 1)	if A is Z2"-1 X Z2,
2(^-2)(n-2)(2^ - 1)(2^-1 - 1)	if A is D2" for n > 3,
Isoc(G; A) ={ 2(l3-2)(2l3 - 1)(2^-1 - 1)/3	if A is <Q,
2(^-2)(n-2)(2^ - 1)(2^-1 - 1)	if A is Q2" for n > 4,
2(^-1)(n-2)-1(2^ - 1)(2^-1 - 1)	if A is M2n for n > 4,
^2(^-2)(n-2)+1(2^ - 1)(2^-1 - 1)	if A is SD2" for n > 4,
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where the first three cases were shown in [14].
By using the Mobius function, Isoc(G; A) can also be determined. For example, for a generalized quaternion group Q2n, a proper subgroup S of Q2n is isomorphic to Z2m or Qm, where Z2m = (a2" m 1} and = (a2" m, a®6) for m € {1,..., n - 1} and i € {0,..., 2n-m - 1}. From the subgroups lattice of Q2n, see Figure 1, one has
1 ifS = Q2n ,
-i if s = z2n-i, q2un-i or ,
.......... . Q(°)
)= ,
' 2 if S = Z2n-2 0 otherwise.
Q2„
Z q40) Q« Q42) ' ■ ■ ■ Qi2"-2-2) Qi2"-2-1)
1
Figure 1: The subgroup lattice of Q2n It follows from Theorem 1.3
Isoc(G; Q2n) = > 3
l (23^-3 - 3 • 22^-3 + 2^-2)

which coincides with the formula given in Theorem 3.6.
if n = 3, if n > 3,
If A = M2n, then every proper subgroup S of M2n is isomorphic to Zm or M2'
(i)
for m G {2,..., n — 1} and i G {0,1}, where Z2m = (c

and M.
(1)
(1)
l>, M2m = (
(a2
„n-2
b). If m = 1, then S is isomorphic to Z2
(0)
(an-2>
or Z2 ; = (an 2b). Now from the subgroups lattice of M2n illustrated in Figure 2 and
|Aut(M2n
2n, one can have
Isoc(G;M2n) = — (2n^ — 3 • 2(n-1)^ + 2(n-2)^+1),
2
a
a

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Zo
Z2
M.

M

2n-2
M
2"-2
Z4
M
(0)
M
(1)
Figure 2: The subgroup lattice of M2n.
which coincides exactly with the result in Theorem 3.6. Also, by using the Mobius function, one can show that
Isoc(G; SD2n) =	- 3 • 2(n-1^ + 2(n-2^+1).
22n 4
For some small ft and n, the numbers Isoc(G; A) are tabulated in Table 1. Table 1: The number Isoc for small ft and n.
(ft, n)	Wj2n	Z^n-1 X Z2	D2n	Isoc Q2n	M2n	SD2n	A
(2, 3)	12	3	3	1	0	0	22
(2, 4)	24	6	3	6	24	6	69
(2, 4)	48	12	3	6	48	6	123
(3, 3)	112	42	42	56	0	0	252
(3, 4)	448	168	84	168	672	168	1708
(3, 5)	1792	672	168	336	2688	336	5992
(4, 3)	960	420	420	560	0	0	2360
(4, 4)	7680	3360	1680	3360	13440	3360	32880
(4, 5)	61440	26880	6720	13440	107520	13440	229440
2
n
n-1
1
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4 In cases of Z2-extensions of any cyclic groups
In this section we determine Isoc(G; A) for a Z2-extension A of a cyclic group Zn (of any order n, not necessarily to be a p-group). Again, let A be a Z2-extension of a cyclic group Zn = Zpao x Zpai x • • • x Zp?= and let n = p^0Pi1 • • • P?s be the prime decomposition with p0 = 2. Let the b-conjugation on Zn correspond to an (s + 1)-tuple (r0, ri,..., rs), where r0 G {±1, ±1 + 2ao-1} and r = ±1 for i G {1,..., s} with —1 in exactly t entries ¿4,..., ¿t. Let n = 2ao n1n2 with n1 = nj=1 P^3. Then A is isomorphic to Bx Zn2 where B is a Z2-extension of Zni since any element of Zn2 commutes with each element of A. Since (|B|, |Zni |) = 1, one has Isoc(G; B x Zni) = Isoc(G; B) • Isoc(G; Zni), as shown in [14]. Because Isoc(G; Zni) has already been determined, we just need to determine Isoc(G; B).
Lemma 4.1. Let B be a Z2-extension of a cyclic group Zn = Zp^o x • • • x Zpa with p0 = 2 and s > 1, and let n = 2ao m. Let the b-conjugation on Zn correspond to an (s + 1)-tuple (r0, r1,..., rs), where r0 G {±1, ±1 + 2ao-1} and all other ri's are —1. Then
MB' P)l= — 1)m2ao^lQ(Zm; P — 1)1 if 2ao+1 I o(b),
1 ( ; P)I \(2g — 1)m2ao |^(Z2^om; P — 1)| otherwise.
Proof. By Theorem 2.5, one can assume b2 = a0 with t G {0,1, 2ao-1}. By Lemma 3.1, it just needs to determine |Q{1}(B; P)|. Take (x1,..., xg) G Q{1}(A; P). Note that x1 is a quotient type element and other Xj s are all normal type. Since
Zn = Zpao X Zpai X • • • X Z„«s ,
po	pi	ps
any element of Zn can be presented gjhj with gj G Zp^o and h G f]f=1 Zpa. So x1 can be presented by g1h1b and other xj's can be preseioted by gjhj. By Corollary 3.3,
(x1,..., xg} = B if and only if (x1, x2,..., xg} = Zn. By x2 = (g1 h1b)2 = b2g1+r°, one has (b2g1+ro ,g2h2,... ,gg hg} = Zn = Zp«o x^ • •xZ^s. Recall that b2 = a0 G Zp«o withp0 = 2, then (b2g1+ro, g2,..., gg} = Zp«o and (h2,..., hg} = Zp«i x • • • x Zpj=. So (b2g1+ro, g2,..., gg) G Q(Z2^o; P), (h2,...,hg) G ^(Zp^i x •••ix Zp?s ; p — 1). To count the choice of (x1,..., xg), equivalently to count the number of (g1,..., gg) and (h1,..., hg). When computing x2 = b2g1+ro, h1 can be any element of g=1 Zpa, and
it follows h1 has m choices by m = f]g=1 p"i. The number of choices of (h2,..., hg) is
equal to |0(Zm; p — 1)|. Hence number of choices of (h1,..., hg) is m|^(Zm; p — 1)|. Now we determine the number of choices of (g1,..., gg) in the following.
Assume that 2ao+1 | o(b) and it follows t = 1. Then the Sylow 2-subgroup of B is
Z2ao+i. By Theorem 2.5, (b2} = (ao} = Z2«o and ro = 1. Since g1 G Z2 ao, one has (b2g1+ro} = (b2g2} = (b2} = Z2ao. By (b2 g1+ro ,g2,...,gg} = (b2} = Z2^o, (g1,.. .,gg) has 2aog choices. So |fy1}(B; p)| = 2ao g m|^(Zm; p — 1)|, and it follows |fi(B; P)| = (2g — 1)m2aog|fi(Zm; P — 1)|.
If 2ao+1 does not divide o(b), then, by Theorem 2.5, b2 = a0 with t G {0, 2ao-1}. If t = 0, then b2 = 1 and r0 G {±1, ±1 + 2ao-1}. It follows b2g1+ro = 1, g2, g2ao_i or g2+2ao~i. So g1+ro can not be the generator of Z2ao .If t = 2ao-1, then r0 = —1. Then b2g 1+ro = a2ao , again, b2g1+ro can not generate Z2ao. So (b2g1+ro,g2,...,gg} = Z2ao if and only if (g2,..., gg} = Z2ao. It follows (g2,..., gg) G ^(Z2ao; P — 1) and g1
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219
is any element in Z2«o. Then (g^ ..., gg) has 2ao |n(Z2«o; P —1)| choices. So |Q(B; P)| =
(2g — 1)m2ao |n(Z2a,m; P — 1)|.	□
Lemma 4.2. Let B be a Z2-extension of a cyclic group Zn = Zp°=o x Zpai x • • • x Zpa with p0 = 2 and s > 1, and let n = 2ao m. Let the b-conjugation on Zn correspond to an (s + 1)-tuple (r0, ri,..., rs), where r0 G {±1, ±1 + 2ao-1} and all other r's are —1.
(1)	If r0 = 1, then
|Aut(B)| ^2aom^(n) f 2ao+1 | o(b), l2my>(n) otherwise.
(2)	If r0 = —1, then |Aut(B)| = 2ao my(n).
(3)	If r0 = 1 + 2ao-i, then |Aut(B)| = 2my(n).
(4)	If r0 = —1 + 2ao-i, then |Aut(B)| = 2a°-1m^(n).
Proof. Again, one can assume that b2 = a0 with t G {0,1,2ao-1}. For an automorphism a of B, a(ak) should be of the form a^ with (ik,pak) = 1 for k G {1,..., s} since a is order-preserving. Suppose that a(a0) is quotient type, then b-1akb = ak since a(a0) commutes with a(ak) for each k. So r = 1 for each i G {1,..., s}, which is a contradiction. Then a(a0) is normal type, say a(a0) = a0o with (i0,2ao) = 1. Assume a(b) = aUoa^1 • • • aUUsb. We need to count the number of choices of u0,..., ug. By computing, (a^a?1 • • • a"= b)2 = b2auo(1+ro) • • • auf(1+r^) = b2auo(1+ro). Note that o(a(b)) = o(b) and o(b) is even. By hypothesis, r1 = • • • = rg = —1, and it follows (a(b))2 = b2aUo(1+ro). Then u can be any element of Zfor i G {1,..., s}. Now it needs to determine the number of choices of u0.	*
(1)	If r0 = 1 and o(b) = 2ao+1, then (a(b))2 = b2a0Mo. By Theorem 2.5, b2 = a0 in this case. Then o(b2a2wo) = o(b2) = 2ao, and it follows u0 has 2ao choices. Hence |Aut(B)| = 2aomy(n). If r0 = 1 and o(b) = 2, then u0 can be 0 or 2ao-1. So |Aut(B)| = 2m^(n).
(2)	If r0 = —1, then (a(b))2 = b2 and o(b) is 2 or 4, by Theorem 2.5. So u0 can be any element of Z2ao and has 2ao choices. It follows |Aut(B)| = 2aom^(n).
(3)	If r0 = 1 + 2«o-1, then o(b) = 2. So (a(b))2 = b^2^^ = a^2^^ = 1. If follows u0(2 + 2ao-1) = 0 (mod 2ao). Then u0 has 2 choices: 0 or 2ao-1. Hence |Aut(B)| = 2my(n).
(4)	If r0 = —1 + 2ao-1,then o(b) = 2. So (a(b))2 = ago2ao-1 = 1. Then u02ao-1 = 0
(mod 2ao), and it follows u0 has 2aR0-1 choices. Hence |Aut(B)| = 2ao-1m^(n).
□
The next lemma follows from Theorem 1.2 and Lemmas 4.1 and 4.2.
Lemma 4.3. Let B be a Z2-extension of a cyclic group Zn = Zp^o x Zp^1 x • • • x Z and let n = paop^1 • • • be the prime decomposition with p0 = 2. Let the b-conjugation on Zn correspond to an (s + 1)-tuple (r0, r1,..., rs), where r0 G {±1, ±1 + 2ao-1} and all other r,'sare — 1.
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Ars Math. Contemp. 15 (2018) 147-160
(1) If ro = 1,
Isoc(G; B) =
i (2f - i)2«o^-«o ^ p(ai-1)(f-1)(pf-1 - 1) if 2ao+1 | o(b),
v(n) v(n)
i=i
i (2f - i)2°o-1 n p(ai-1)(f-1)(pf-1 -1)
otherwise.
i=0
(2) If ro = -1, then
Isoc(G; B) = — (2f - 1) n
(*i-1)(p-1)( f-1
(pf-1 - 1).
(3) If ro = 1 + 2ao-1, then
Isoc(G; B) = ^2«o-1(2f - 1) np,(a<-1)(f-1)^f-1 - 1).
(4) If r0 = -1 + 2ao-1, then
Isoc(G; B)
<^(n)
2(2f -1) n p
(Oi-1)(f-1^ 8-1
(pf-1 -1).
i=o
Now one can get main theorem of this section.
Theorem 4.4. Let A be a Z2-extension of a cyclic group Zn = Zp°=o x Zpai x • • • x Zpa and let n = p^p^1 • • • be the prime decomposition with p0 = 2. Let the b-conjugation on Zn correspond to an (s + 1)-tuple (r0, r1;..., rs), where r0 G {±1, ±1 + 2ao-1} and ri = ±1 for i G {1,..., s} with -1 in exactly t entries ..., £t. Let J = {^1,..., £t}, K = {1,..., s} - J and
N
1
^(n)
(2f -1) n p(ai-1)(f-1)(pf-1 -1) n p(ai-1)f (pf -1).
ieJ
ie/c
Then Isoc(G; A) = TN, where
2«of - ao
2(ao-1)f (2f-1 - 1)
T = <{ 2(ao-1)(f-1)(2f-1 - 1)
2(ao-1)f (2f-1 - 1)
if r0 = 1 and 2ao+1 | o(b), if r0 = 1 and 2ao+1 f o(b), if r0 = -1,
if r0 = 1 + 2ao-1,
2(ao-1)(f-1)-1(2f-1 - 1) if r0 = 1 + 2ao-1.
Example 4.5. Let A be a Z2-extension of a cyclic group Z1260 = Z4 x Z32 x Z5 x Z7 = (a0) x (a1) x (a2) x (a3). By Theorem 2.5, the b-conjugation on Zn corresponds to a 4-tuple
(r0, r1, r2, r3), where ri = ±1 for i G {0,1, 2, 3}. Take (r0, r1, r2, r3) = (1, -1, -1,1) and P = 3 as an example. One has Isoc(G; A) = Isoc(G; B) Isoc(G; Z7), where B is a Z2-extension of Z180. ByLemmas4.1 and4.2, |0(B;3)| = (23 - 1)|Q1(B;3)| = 34836480 and |Aut(B)| = 4320. It follows that Isoc(G; B) = 8064. By Isoc(G; Z7) = 57, one gets Isoc(G; A) = 459648.
1
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5 In cases of Z2-extensions of an abelian group
Naturally, we are interested in extending the counting problem of the previous two sections to the case of a Z2-extension of an abelian group. To do this, we need to classify Z2-extensions of an arbitrary abelian group, but we can not give a complete answer so far, see Section 6. So we just count two special cases, generalized dihedral groups or generalized dicyclic groups.
5.1 With generalized dihedral groups
Let H be an abelian group. A generalized dihedral group Dih(H), as a Z2-extension of H, is defined with relations
b2 = 1, b-1ab = a-1, for all a e H.
It is a semidirect product of H and Z2, with Z2 acting on H by inverting elements. When H is cyclic, Dih(H) is just a dihedral group.
Lemma 5.1. H is a characteristic subgroup of Dih(H).
Proof. Take an automorphism a e Aut(Dih(H)). Note that the order of a quotient type element is 2. For any element a of odd order in Dih(H), a(a) should be normal type since a is order-preserving. For an element a0 of even order, suppose that a(a0) is a quotient type element. Since a0 commutes with a as an element of odd order, a(a0) commutes a(a). Then b commutes a, which is a contradiction. Then a(a) e H for any a e H. Hence H is a characteristic subgroup of Dih(H).	□
Now, |Aut(Dih(H))| = |H| • |Aut(H)|. By Lemmas 3.1 and 3.2, one can show that |fi(Dih(H);P)| = (2^ - l)|fi{i}(H;P - 1)| = (2^ - l)|H||fi(H; P - 1)|. Each abelian group can be decomposed into direct product of abelian p-group, namely, H = Hpi x • • • x Hps with pi prime. Then |fi(Dih(H); P)| = (2^-1)|H||fi(HPi; P-1)| • • • |fi(Hp.; P-1)|. Itjustneeds to determine Isoc(G; Dih(Hp)) for aprime integer p. Since |Q(Hp; P - 1)| is determined in [14], one gets
Theorem 5.2. For a generalized dihedral group Dih(Hp) and Hp = m1Zpsi x • • • x m^Zpse with m1,... ,mg and s1,...,s£ are positive integers satisfying s£ < • • • < s1, one can obtain
Isoc(G; Dih(Hp)) = (2^ - 1)pfiii=1 P-,
( ; ( p)) ( )p j n==1 p=j-h+1 -1,
where m = m1 + • • • + m^ and
f (P - 1, mi, Si) = (P - 1 - m) ^ mi(si - 1) + ^ mi I ^ mj(si - Sj - 1)
\i=1	J i=1 \j=i+1
5.2 With generalized dicyclic groups
A generalized dicyclic group Dic(H), as another Z2-extension of an abelian group H, is defined with relations
b2 = c,b-1ab = a-1,
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Ars Math. Contemp. 15 (2018) 147-160
where c is an involution of H and a is an arbitrary element of H. Similarly, one can have the coming lemma.
Lemma 5.3. H is a characteristic group of Dic(H). Hence
|Aut(Dic(H))| = |H| • |Aut(H)|.
Theorem 5.4. For a generalized dicyclic group Dic(Hp) and Hp = m1Zp»i x- • •xm^Zp»« with m1,... ,m£ and s1,... ,s£ are positive integers satisfying s£ < • • • < s1; one can obtain
Isoc(G; Dih(Hp)) = 2(2^ - 1)pf	lli=1 p-,
( ; ( p)) ( )r j n==1 r=j-h+1 -1'
where m = m1 + • • • + m£ and
£-1
f (3- 1, mi, Si) = (¡3 - 1 - m) Emi(si - 1) + ^mi I E mj(si - Sj - 1)
\i=1	J i=1 \j=i+1
6 Further remarks
In this paper, we enumerate the regular coverings of a graph whose covering transformation groups are Z2-extensions of a cyclic group. However, we could not give a complete answer of this problem if A is a Z2-extension of any abelian group H.
However, we cannot answer the same enumeration problem when the cyclic group is replaced by an abelian group, even by an elementary abelian p-group. In fact the difficulty for authors is how to determine all involutions of Aut(H). The counting problem has studied by many researchers, for example, in [21], it gave a generating function for the number of involutions of GL(n,p) which is isomorphic to automorphism group of Zp x • • • x Zp. For more results, see [2], [5], [3] and so on. But it is still hard for us to determine the specific form of each involution of GL(n,p).
For further possible problems unsolved in this paper, we list in the following.
(1)	Isoc(G; A) if A is a Z2-extension of any abelian group.
(2)	Isoc(G; A) if A is a Zp-extension of any cyclic group.
(3)	Isoc(G; A) if A is any metacyclic group.
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