Hexavalent symmetric graphs of order 9p

Discrete Mathematics 340 (2017) 2378–2387

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Hexavalent symmetric graphs of order 9p Song-Tao Guo *, Hailong Hou, Yong Xu School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, PR China

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Article history: Received 17 August 2016 Received in revised form 17 May 2017 Accepted 22 May 2017

a b s t r a c t A graph is symmetric if its automorphism group acts transitively on the set of arcs of the graph. In this paper, we classify hexavalent symmetric graphs of order 9p for each prime p. © 2017 Elsevier B.V. All rights reserved.

Keywords: Symmetric graph s-transitive graph Cayley graph Coset graph Orbital graph

1. Introduction Let G be a permutation group on a set Ω and α ∈ Ω . Denote by Gα the stabilizer of α in G, that is, the subgroup of G fixing the point α . We say that G is semiregular on Ω if Gα = 1 for every α ∈ Ω and regular if G is transitive and semiregular. Throughout this paper, we consider undirected finite connected graphs without loops or multiple edges. For a graph X we use V (X ), E(X ) and Aut(X ) to denote its vertex set, edge set, and automorphism group, respectively. For u , v ∈ V (X ), denote by {u , v} the edge incident to u and v in X . A graph X is said to be vertex-transitive if Aut(X ) acts transitively on V (X ). An s-arc in a graph is an ordered (s + 1)-tuple (v0 , v1 , . . . , vs−1 , vs ) of vertices of the graph X such that vi−1 is adjacent to vi for 1 ≤ i ≤ s, and vi−1 ̸ = vi+1 for 1 ≤ i ≤ s − 1. In particular, a 1-arc is called an arc for short and a 0-arc is a vertex. For a subgroup G ≤ Aut(X ), a graph X is said to be (G, s)-arc-transitive or (G, s)-regular if G is transitive or regular on the set of s-arcs in X , respectively. A (G, s)-arc-transitive graph is said to be (G, s)-transitive if it is not (G, s+1)-arc-transitive. In particular, a (G, 1)-arc-transitive graph is simply called G-symmetric. A graph X is simply called s-arc-transitive, s-regular or s-transitive if it is (Aut(X ), s)-arc-transitive, (Aut(X ), s)regular or (Aut(X ), s)-transitive, respectively. Arc-transitive or s-transitive graphs have received considerable attention in the literature. For example, s-transitive graphs of order np was classified in [5,6,27] depending on n = 1, 2 and 3, where p is a prime. Li [17] showed that there exists an s-transitive graph of odd order if and only if s ≤ 3. For the case of valency 4, Gardiner and Praeger [9,10] characterized tetravalent symmetric graphs, and Li et al. [18] classified vertex-primitive tetravalent s-transitive graphs. The classification of tetravalent s-transitive Cayley graphs on abelian groups was given by Xu and Xu [31]. We may deduce a classification of tetravalent 1-regular Cayley graphs on dihedral groups from [16,22,28,29]. Zhou [35] gave a classification of tetravalent 1-regular graphs of order 2pq for p, q primes. Recently, Zhou [34] classified tetravalent s-transitive graphs of order 4p, and Zhou and Feng [36] classified tetravalent s-transitive graphs of order 2p2 . A series of pentavalent symmetric graphs was classified in [19,23,24,32,33]. The classification of tetravalent symmetric graphs of order 9p with p a prime was given in [12].

*

Corresponding author. E-mail addresses: [email protected] (S.-T. Guo), [email protected] (H. Hou), [email protected] (Y. Xu).

http://dx.doi.org/10.1016/j.disc.2017.05.011 0012-365X/© 2017 Elsevier B.V. All rights reserved.

S.-T. Guo et al. / Discrete Mathematics 340 (2017) 2378–2387

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Table 1 Hexavalent s-transitive graphs of order 9p. X

s-transitive

Aut(X )

Comments

G18 i G27 3 G27 4 G27 G45 i G63

1-transitive 1-transitive 1-transitive 1-transitive 1-transitive 1-transitive 1-transitive 1-transitive 1-regular 1-regular

(S3 × Z3 ) ⋊ D12 (Z9 × Z3 ) ⋊ D12 Z33 ⋊ (S4 × Z2 ) (Z23 ⋊ Z3 ).(D8 × S3 ) Z3 .S6 G2 (2) S3 wrD6p S3 wrD2p Z9p ⋊ Z6 (Z23 × Zp ) ⋊ Z6

Example 3.1, p = 2 Example 3.2, p = 3 and i = 1, 2 Example 3.3, p = 3 Example 3.4, p = 3 Example 3.5, p = 5 Example 3.6, p = 7 and i = 1, 2 Arbitrary prime p Example 3.7,p ≥ 3 Example 3.8, p ≡ 1( mod 6) and i = 1, 2, 3 Example 3.9, p ≡ 1( mod 6)

C3p [3K1 ] C (3, p, 2) i C A9p C A(32 ,p)

Thus, as a natural continuation, we classify hexavalent symmetric graphs of order 9p for each prime p in this paper. Note that this can also be viewed as an application of the exact structure of hexavalent case, which was determined by Guo et al. [13]. Throughout the paper we denote by Cn and Kn the cycle and the complete graph of order n, respectively. Denote by Zn the cyclic group of order n, by Z∗n the multiplicative group of Zn consisting of numbers coprime to n, by D2n the dihedral group of order 2n, and by Fn the Frobenius group of order n. Theorem 1.1. Let p be a prime. Then any connected hexavalent symmetric graph of order 9p is isomorphic to one of the graphs in Table 1. Furthermore, all graphs in Table 1 are pairwise non-isomorphic.

2. Preliminary results In this section we collect some notation and preliminary results which will be used later in this paper. Let X be a graph and let G ≤ Aut(X ) be an arc-transitive subgroup on X . Assume that G is imprimitive on V (X ) and B = {B1 , B2 , . . . , Bn } is a complete block system of G. The block graph or quotient graph XB of X relative to B is defined as the graph with vertex set the complete block system B, and with the two blocks adjacent if and only if there is an edge in X between those two blocks. Clearly, if X is G-symmetric then XB is G/K -symmetric, where K is the kernel of K on B. For a normal subgroup N of G, the set of the orbits of N forms a complete block system of G. In this case, we denote by XN the quotient graph of X relative to the set of the orbits of N. Note that there is no connected regular graph of odd order and odd valency. Thus, in view of [20, Theorem 9], we have the following: Lemma 2.1. Let X be a connected hexavalent G-symmetric graph with odd order and G ≤ Aut(X ), and let N be a normal subgroup of G. Then one of the following holds: (1) N is transitive on V (X ); (2) N has r ≥ 3 orbits on V (X ), the quotient graph XN ∼ = Cr , and G induces the full dihedral group D2r on XN ; (3) N has r ≥ 7 orbits on V (X ), N is semiregular on V (X ) and the quotient graph XN is a connected hexavalent G/N-symmetric graph. For a finite group G and a subset S of G such that 1 ̸ ∈ S and S = S −1 , the Cayley graph Cay(G, S) on G with respect to S is defined to have vertex set V (Cay(G, S)) = G and edge set E(Cay(G, S)) = {{g , sg } | g ∈ G, s ∈ S }. Clearly, a Cayley graph Cay(G, S) is connected if and only if S generates G. Furthermore, Aut(G, S) = {α ∈ Aut(G) | S α = S } is a subgroup of the automorphism group Aut(Cay(G, S)). Given a g ∈ G, define the permutation R(g) on G by x ↦ → xg , x ∈ G. Then R(G) = {R(g) | g ∈ G}, called the right regular representation of G, is a permutation group isomorphic to G. The Cayley graph is vertex-transitive because it admits the right regular representation R(G) of G as a regular group of automorphisms of Cay(G, S). A Cayley graph Cay(G, S) is said to be normal if R(G) is normal in Aut(Cay(G, S)). A graph X is isomorphic to a Cayley graph on G if and only if Aut(X ) has a subgroup isomorphic to G, acting regularly on vertices (see [26]). For two subsets S and T of G, if there is an α ∈ Aut(G) such that S α = T then S and T are said to be equivalent, denoted by S ≡ T . We may easily show that if S ≡ T then Cay(G, S) ∼ = Cay(G, T ) and Cay(G, S) is normal if and only if Cay(G, T ) is normal. About the vertex stabilizer of normal Cayley graphs, we have the following proposition (see [30, Proposition 1.5]). Proposition 2.2. A Cayley graph Cay(G, S) is normal if and only if Aut(Cay(G, S))1 = Aut(G, S), where Aut(Cay(G, S))1 is the stabilizer of 1 in Aut(Cay(G, S)). Let X be a (G, s)-transitive graph with G ≤ Aut(X ) and s ≥ 2. For an intransitive normal subgroup of G, if the valency of X is equal to the valency of the quotient graph XN , then X is called a cover of XN , and since a cover induced by the normal subgroup N, X is called a normal cover of XN . From [17, Theorem 1.1], we have the following proposition about s-transitive graphs of odd order.

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Proposition 2.3. Let s be a positive integer. Then (1) There exists an s-transitive graph of odd order if and only if s ≤ 3; (2) Assume that X is a (G, s)-transitive graph of odd order for some G ≤ Aut(X ) and some s = 2 or 3, there exists a normal subgroup N ⊴ G such that G/N is an almost simple group, XN is a (G/N , s)-transitive graph, and X is a normal cover of XN . The next proposition characterizes the vertex stabilizers of connected hexavalent s-transitive graphs (see [13, Theorem 3.1] and [17, Theorem 1.1]). Proposition 2.4. Let X be a connected hexavalent (G, s)-transitive graph of odd order. Then s ≤ 3 and the stabilizer Gv of a vertex v ∈ V (X ) in G is as follows: (1) Gv is a {2, 3}-group for s = 1; (2) Gv ∼ = PSL(2, 5), PGL(2, 5), A6 or S6 for s = 2; (3) Gv ∼ = D10 × PSL(2, 5), F20 × PGL(2, 5), A5 × A6 , S5 × S6 , (D10 × PSL(2, 5)).Z2 or (A5 × A6 ) ⋊ Z2 for s = 3. The classification of symmetric graphs of order p is due to Chao [5]. Let Zp = ⟨a⟩ with 6 | (p − 1). Define the Cayley graph: C Ap = Cay(Zp , {a, aw , aw , aw , aw , aw }), 2

3

4

5

where w ∈ Z∗p has order 6. Clearly, C A7 ∼ = K7 . Proposition 2.5. Let p ≥ 7 be a prime and X a connected hexavalent symmetric graph of order p. Then X ∼ = K7 and Aut(X ) ∼ = S7 , ∼ ∼ or X = C Ap and Aut(X ) = Zp ⋊ Z6 . To introduce hexavalent symmetric graphs of order a prime 3p for a prime p, we define some graphs. Let p be a prime with 3 | (p − 1) and let Z3p = Z3 × Zp = ⟨a⟩ × ⟨b⟩. Define the Cayley graph as follows: C A3p = Cay(Z3p , {ab, a−1 b−1 , abw , a−1 bw , abw , a−1 bw }), 2

2

where w ∈ Z∗p has order 3. By the definition of G(3p, 3) given in [27, Example 3.4], it is easy to see that C A3p ∼ = G(3p, 3) and Aut(C A3p ) = Z3p ⋊ Z6 . The lexicographic product Cp [3K1 ] is defined as the graph with vertex set V (Cp ) × V (3K1 ) such that for any two vertices u = (x1 , y1 ) and v = (x2 , y2 ) in V (Cp [3K1 ]), u is adjacent to v in Cp [3K1 ] if and only if {x1 , x2 } ∈ E(Cp ). Then Cp [3K1 ] is a connected hexavalent 1-transitive Cayley graph on the group Z3 × Zp and Aut(Cp [3K1 ]) = S3 wrD2p . Let X be a symmetric graph, and G an arc-transitive subgroup of Aut(X ). Let {u, v} be an edge of X . Assume that H = Gu is the stabilizer of u ∈ V (X ) and that g ∈ G interchanges u and v . It is easy to see that the core HG of H in G (the largest normal subgroup of G contained in H) is trivial, and that HgH consists of all elements of G which maps u to one of its neighbors in X . By [20,26], the graph X is isomorphic to the coset graph Cos(G, H , HgH), which is defined as the graph with vertex set {Ha | a ∈ G}, the set of right cosets of H in A, and edge set {{Ha, Hda} | a ∈ G, d ∈ HgH }. The valency of Cos(G, H , HgH) is |HgH |/|H | = |H : H ∩ H g |, and Cos(G, H , HgH) is connected if and only if HgH generates G. By right multiplication, every element in G induces an automorphism of Cos(G, H , HgH). Since HG = 1, the induced action of G on V (Cos(G, H , HgH)) is faithful, and hence we may view G as a group of automorphisms of Cos(G, H , HgH). Let G = ⟨(1, 2), (1, 2, 3, 4, 5, 6)⟩ ∼ = S6 . Then G has two maximal subgroups H = ⟨(1, 2, 3, 4), (1, 2), (5, 6)⟩ and K = ⟨(1, 2), (3, 4), (5, 6), (1, 3, 5)(2, 4, 6), (1, 3)(2, 4)⟩, which are isomorphic to S4 × Z2 . Set P = ⟨(1, 2), (3, 4), (5, 6)⟩. It is easy to see that P ≤ H and P ⊴ K . Thus, K has an element g of order 4 such that H g ∩ H = P. Since g ̸ ∈ H and H is maximal in G, we have ⟨H , g ⟩ = G. It follows that the coset graph G15 = Cos(G, H , HgH)

is a connected hexavalent symmetric graph of order 15. By [27, Example 2.2], it is easy to see that G15 ∼ = T6c and Aut(G15 ) ∼ = S6 . ∼ ∼ ∼ Let G = PSL(2, 19). Then by Atlas [8], G has a maximal subgroup H = A5 , H has a subgroup K = D10 and NG (K ) ∼ = D20 . Clearly, the center Z (NG (K )) ∼ = Z2 . Set Z (NG (K )) = ⟨g ⟩. Then o(g) = 2, H g ∩ H = K and ⟨H , g ⟩ = G. Thus, the coset graph G57 = Cos(G, H , HgH)

is a connected hexavalent symmetric graph of order 57. By [27, Example 2.2], we have G57 ∼ = L2 (19)657 and Aut(G57 ) ∼ = PSL(2, 19). This graph also known as Perkel graph. The next proposition is about the classification of connected hexavalent symmetric graphs of order 3p (see [27, Theorem]). Proposition 2.6. Let X be a connected hexavalent symmetric graph of order 3p with p a prime. Then X is isomorphic to C A3p , Cp [3K1 ], G15 or G57 .

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3. Graph constructions and isomorphisms In this section we introduce some connected hexavalent symmetric graphs of order 9p for p a prime. Clearly, the lexicographic product C3p [3K1 ] is a connected hexavalent symmetric graph of order 9p and has full automorphism group S3 wrD6p . In the following example, we construct a hexavalent symmetric graph of order 18, which is a non-normal Cayley graph. 2 2 Example 3.1. Let G = ⟨a, b, c | a3 = b2 = c 3 = [a, c ] = [b, c ] = 1, ab = a−1 ⟩ ∼ = S3 × Z3 and S = {b, ba , bc , bc 2 , ba c , ba c 2 }. Then define the Cayley graph

G18 = Cay(G, S).

By Magma [2], G18 is a connected hexavalent symmetric graph and Aut(G18 ) ∼ = (S3 × Z3 ).D12 . Next we consider the connected hexavalent symmetric graphs of order 27 and construct four Cayley graphs on the groups of order 27. Example 3.2. Let G = ⟨a⟩ × ⟨b⟩ ∼ = Z9 × Z3 . Define two connected hexavalent Cayley graphs: 1 G27 = Cay(G, {a±1 , (ab)±1 , (a5 b)±1 }),

2 G27 = Cay(G, {a±1 , (ab)±1 , (a2 b)±1 }).

1 Then by Magma [2], these two graphs are non-isomorphic 1-transitive normal Cayley graphs. Furthermore, Aut(G27 ) ∼ = 2 ∼ (Z9 × Z3 ) ⋊ D12 and Aut(G27 ) = (Z9 × Z3 ) ⋊ D12 .

Example 3.3. Let G = ⟨a⟩ × ⟨b⟩ × ⟨c ⟩ ∼ = Z33 . Define the connected hexavalent Cayley graphs: 3 G27 = Cay(G, {a±1 , b±1 , c ±1 }). 3 3 )∼ is a 1-transitive normal Cayley graph and Aut(G27 Then by Magma [2], G27 = Z33 ⋊ (S4 × Z2 ).

Example 3.4. Let G = ⟨a, b | a9 = b3 = 1, ab = a4 ⟩. Define a connected hexavalent Cayley graph: 4 G27 = Cay(G, {a±1 , (ab)±1 , (ab−1 )±1 }). 4 4 Then by Magma [2], G27 is a 1-transitive non-normal Cayley graph and Aut(G27 )∼ = (Z9 ⋊ Z3 ).(D8 × S3 ).

The following example is a coset graph of order 45 on the group Z3 .S6 . Example 3.5. Let G = Z3 .S6 . Then by Magma [2], there is a unique connected hexavalent symmetric graph of order 45 admitting G as an arc-transitive automorphism group, denoted by G45 . Furthermore, Aut(G45 ) ∼ = Z3 .S6 and Aut(G45 )v ∼ = S4 × Z2 with v ∈ V (G45 ). Conversely, any connected hexavalent symmetric graph of order 45 admitting G as an arc-transitive automorphism group is isomorphic to G45 . As we all know that any vertex-primitive symmetric graph is isomorphic to an orbital graph of a primitive permutation group (see [18, Section 2]). Thus, we can construct some vertex-primitive symmetric graphs by orbital graph. Example 3.6. Let G = PSU(3, 3). Then by Atlas [8], G has two subgroups of index 63, which are maximal and isomorphic to Z4 .S4 and Z24 ⋊ S3 . Thus, G has exactly two primitive permutation representations on 63 points. By Magma [2], both 1 2 representations have self-paired suborbit of length 6, and the corresponding orbital graphs, denoted by G63 and G63 , are non-isomorphic connected hexavalent symmetric graphs. Furthermore, their full automorphism groups are isomorphic to PSU(3, 3) ⋊ Z2 = G2 (2). Conversely, any connected hexavalent symmetric graph of order 63 admitting G as an arc-transitive 1 2 automorphism group is isomorphic to G63 or G63 . From now on, we construct the connected hexavalent symmetric graphs of order 9p with an arbitrary prime. The next graph is due to Praeger [25, Definition 2.1]. Example 3.7. Let p ≥ 3. Then define the graph C (3, p, 2) = (V , E) as follows: V (X ) = Zp × Z23 ,

E = {{(i, x, y), (i + 1, y, z)}}

where Zp and Z3 are additive groups of order p and 3, i ∈ Zp and x, y, z ∈ Z3 . Then by [25, Theorem 2.10 and Lemma 2.12], C (3, p, 2) is a connected hexavalent symmetric graphs of order 9p and Aut(C (3, p, 2)) = S3 wrD2p . Remark. With the notation in [25, Definition 2.1], it is easy to check that C (3, 3p, 1) ∼ = C3p [3K1 ]. Clearly, C (3, p, 2) is not isomorphic to C3p [3K1 ] because Aut(C (3, p, 2)) ̸ = Aut(C3p [3K1 ]). By [25, Theorem 1], Aut(C (3, p, 2)) has a minimal normal p subgroup isomorphic to Z3 , which is not semiregular on V (C (3, p, 2)). Next, we construct three infinite families of normal Cayley graphs on abelian groups of order 9p.

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Example 3.8. Let p be a prime greater than 3 with 6 | (p−1) and G = ⟨a⟩×⟨b⟩ ∼ = Z9 ×Zp . Then Aut(G) = ⟨α⟩×⟨γ ⟩ ∼ = Z6 ×Zp−1 (p−1)/6 and ⟨γ ⟩ has an element β = γ of order 6. Define two connected hexavalent Cayley graphs: 5 3 C A19p = Cay(G, {(ab)⟨αβ⟩ }), C A29p = Cay(G, {(ab)⟨α β⟩ }), C A39p = Cay(G, {(ab)⟨α β⟩ }).

Then Z9p ⋊ Z6 ∼ = G ⋊ ⟨αβ⟩ ≲ Aut(C A19p ), Z9p ⋊ Z6 ∼ = G ⋊ ⟨α 3 β⟩ ≲ Aut(C A29p ) and Z9p ⋊ Z6 ∼ = G ⋊ ⟨α 5 β⟩ ≲ Aut(C A39p ). 1 2 3 Furthermore, C A9p , C A9p and C A9p are symmetric. Example 3.9. Let p be a prime greater than 3 with 6 | (p − 1) and G = ⟨a⟩×⟨b⟩×⟨c ⟩ ∼ = Z23 × Zp . Then Aut(G) ∼ = GL(2, 3) × Zp−1 . Since 6 | (p − 1), we have that Zp−1 has an element δ of order 6, which is trivial on Z23 and non-trivial on Zp . Set

⎧ ⎨a → a−1 b−1 σ = b → b−1 ⎩ c → c. Then σ ∈ Aut(G) has order 6 and σ commutes with δ . Define the connected hexavalent Cayley graph: C A(32 ,p) = Cay(G, {ac ⟨δσ ⟩ }).

Then G ⋊ ⟨σ δ⟩ ∼ = (Z23 × Zp ) ⋊ Z6 ≲ Aut(C A(32 ,p) ). Hence C A(32 ,p) is symmetric. Following the two examples, we determined the full automorphism groups of the Cayley graphs defined above. Lemma 3.10. Let p be a prime greater than 3 and let G be an abelian group of order 9p. Then any connected hexavalent symmetric normal Cayley graph on G is isomorphic to C A19p , C A29p or C A(32 ,p) . Moreover, Aut(C A19p ) ∼ = Z9p ⋊ ⟨α 2 β⟩, Aut(C A29p ) ∼ = Z9p ⋊ ⟨α 3 β⟩, 3 5 ∼ ∼ Aut(C A9p ) = G ⋊ ⟨α β⟩ and Aut(C A(32 ,p) ) = G ⋊ ⟨σ δ⟩. Proof. Let X = Cay(G, S) be a connected hexavalent symmetric normal Cayley graph on G. Then S −1 = S, |S | = 6, ⟨S ⟩ = G, Aut(G, S) = Aut(X )1 and Aut(X ) ∼ = G ⋊ Aut(G, S). Note that G is abelian. Thus, G ∼ = Z9 ⋊ Zp or Z23 × Zp . ∼ ∼ ∼ Suppose that G = ⟨a⟩ × ⟨b⟩ = Z9 × Zp . Then Aut(G) = Z6 × Zp−1 = ⟨α, γ ⟩ is regular on the elements in G of order 9p. It follows that Aut(G, S) ∼ = Z6 and we may assume that ab ∈ S. Assume that 6 (p − 1). Then Zp−1 has a unique subgroup of order 2 or 3, say ⟨θ⟩ or ⟨ϑ⟩. Then Aut(G, S) = ⟨α⟩, ⟨αθ ⟩ or ⟨αϑ⟩. If S = (ab)⟨α⟩ or (ab)⟨αϑ⟩ , then it is easy to check⏐ that S −1 ̸= S, a contradiction. If S = (ab)⟨αθ⟩ , then X ∼ = C3p [3K1 ], which

is not a normal Cayley graph, a contradiction. Thus, 6 ⏐ (p − 1). Then Aut(G, S) = ⟨α i β j ⟩ with o(α i β j ) = 6 and 1 ≤ i, j ≤ 5. Note

⏐

that S −1 = S. An easy calculation implies that (i, j) have 8 choices: (1, 1), (1, 3), (1, 5), (3, 1), (3, 5), (5, 1), (5, 3), (5, 5). Since ⟨αβ⟩ = ⟨α 5 β 5 ⟩, ⟨αβ 3 ⟩ = ⟨α 5 β 3 ⟩, ⟨αβ 5 ⟩ = ⟨α 5 β⟩ and ⟨α 3 β⟩ = ⟨α 3 β 5 ⟩, we have four corresponding Cayley graphs. It is easy 3 to see that the Cayley graph relative to the set (ab)⟨αβ ⟩ is isomorphic to C3p [3K1 ]. Thus, we have that X ∼ = C Ai9p for i = 1, 2 or 1 2 3 5 3. Moreover, the normality of X implies that G ⋊ ⟨αβ⟩ ∼ = Aut(C A9p ), G ⋊ ⟨α β⟩ ∼ = Aut(C A9p ) and G ⋊ ⟨α β⟩ ∼ = Aut(C A39p ). Since i these three full automorphism groups are non-isomorphic, we have that C A9p (i = 1, 2, 3) are pairwise non-isomorphic. Suppose that G ∼ = GL(2, 3) × Zp−1 . Since ⟨S ⟩ = G, we have the = ⟨ a⟩ × ⟨ b ⟩ × ⟨ c ⟩ ∼ = Z23 × Zp . Note that Aut(G) ∼ elements in S have orders 3p, and since Aut(G) is transitive on the elements in G of order 3p, we may assume that ac ∈ S. Set GL(2, 3) ∼ = H ≤ Aut(G) and Zp−1 ∼ = K ≤ Aut(G). Then Aut(G) = H × K . Since Aut(G, S) is transitive on S, we have Aut(G, S) has order 6. If forces that Aut(G, S) ∼ = S3 or Z6 . Assume that Aut(G, S) ∼ = S3 . Since Aut(Zp ) ∼ = Zp−1 is cyclic, the action of Aut(G, S) on ⟨a, b⟩ is isomorphic to S3 . Note that the subgroup S3 of GL(2, 3) has two conjugacy classes, the representatives are ⟨π, ϕ⟩ and ⟨τ , ϕ⟩, where π , τ and ϕ have the following forms:

π=

{

a → a−1 b → b,

τ=

{

a→a b → b−1 ,

ϕ=

{

a → ab b → b.

If the action of Aut(G, S) on ⟨a, b⟩ is isomorphic to ⟨π, ϕ⟩, then S ′ = aAut(G,S) = {a, ab, ab−1 , a−1 , a−1 b−1 , a−1 b} and this action is not transitive on S ′ , a contradiction. If this action is isomorphic to ⟨τ , ϕ⟩, then S ′ = aAut(G,S) = {a, ab, ab−1 , a−1 , a−1 b−1 , a−1 b} and this action is transitive on S ′ . Since S −1 = S, we have that S = {ac , abc , ab−1 c , a−1 c −1 , a−1 b−1 c −1 , a−1 bc −1 } = ⟨b⟩ac ∪ ⟨b⟩a−1 c −1 . However, an easy calculation implies that X ∼ = C3p [3K1 ], a contradiction. Assume that Aut(G, S) ∼ = Z6 . Since S −1 = S and ⟨S ⟩ = G, an easy calculation implies that the actions of Aut(G, S) on ⟨a, b⟩ and ⟨c ⟩ are isomorphic to Z6 , respectively. Note that the elements of order 6 in H are conjugate to each other. Thus, we may assume that Aut(G, S) = ⟨σ i δ j ⟩ with (i, j) = (1, 1), (1, 5), (5, 1) or (5, 5). Since ⟨σ δ⟩ = ⟨σ 5 δ 5 ⟩ and ⟨σ 5 δ⟩ = ⟨σ δ 5 ⟩, 5 we have two corresponding Cayley graphs relative to (ac)⟨σ δ⟩ and (ac)⟨σ δ⟩ . Clearly, σ and σ 5 are conjugate in H. Thus, these two Cayley graphs are isomorphic, that is, X ∼ = C A(32 ,p) . The normality of X implies that Aut(X ) ∼ = G ⋊ ⟨σ δ⟩. □

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Since the automorphism groups of the graphs defined in Example 3.1–3.9 are pairwise non-isomorphic, we have the following lemma. i i Lemma 3.11. The graphs G18 , G27 (i = 1, 2, 3, 4), G45 , G63 (i = 1, 2), C3p [3K1 ], C (3, p, 2), C Ai9p (i = 1, 2, 3) and C A(32 ,p) are connected pairwise non-isomorphic hexavalent symmetric graphs.

4. Proof of Theorem 1.1 This section is devoted to classifying hexavalent symmetric graphs of order 9p for p a prime. First we have the following lemma. Lemma 4.1. Let p be a prime greater than 3 and G a non-abelian group of order 9p. Then any connected hexavalent normal Cayley graph on G cannot be symmetric. Proof. Let X = Cay(G, S) be a connected hexavalent normal Cayley graph. Then ⟨S ⟩ = G, S −1 = S and |S | = 6. Since |G| = 9p, we may assume S = {x, x−1 , y, y−1 , z , z −1 }, and since X is normal, Aut(G, S) = Aut(X )1 by Proposition 2.2. Suppose to the contrary that X is symmetric. Then Aut(G, S) is transitive on S, forcing that o(x) = o(y) = o(z). Note that p > 3. By Sylow Theorem, G has a normal Sylow p-subgroup, which means that o(x) ̸ = p because ⟨S ⟩ = G. Denote by Z (G) the center of G. From the elementary group theory, up to isomorphism, there are three non-abelian groups of order 9p for a prime p > 3: G1 = ⟨a, b | ap = b9 = 1, b−1 ab = ar ⟩, where r ∈ Z∗p and o(r) = 3; G2 = ⟨a, b | ap = b9 = 1, b−1 ab = as ⟩, where s ∈ Z∗p and o(s) = 9; G3 = ⟨a, b, c | ap = b3 = c 3 = [b, c ] = [a, b] = 1, c −1 ac = at ⟩, where t ∈ Z∗p and o(t) = 3. Case 1: G = G1 In this case, Z (G) = ⟨b3 ⟩ and Z (G) is the unique subgroup of order 3 in G. Since ⟨S ⟩ = G, we have o(x) ̸ = 3 and hence o(x) = o(y) = o(z) = 3p or 9. Similarly, if o(x) = 3p then G = ⟨S ⟩ ⊆ Z (G) × ⟨a⟩, a contradiction. Thus, o(x) = 9 and x, y, z have the form ai b3j+1 or ai b3j−1 . Each automorphism α in Aut(G) can be written as follows:

α:

{

a ↦ → ai , b ↦ → aj b3k+1 ,

1 ≤ i ≤ p − 1; 0 ≤ j ≤ p − 1, 0 ≤ k ≤ 2.

Clearly, Aut(G) is transitive on the set {{g , g −1 } | g ∈ G, o(g) = 9}. We may assume that x = b and y = ai b3k+1 . Since a ↦ → ai , b ↦ → b induces an automorphism of G, S ≡ {b, b−1 , ab3k+1 , (ab3k+1 )−1 , aj b3l+1 , (aj b3l+1 )−1 }. Note that every automorphism of G cannot map b to ai b3k−1 . It follows that Aut(G, S) cannot be transitive on S, a contradiction. Case 2: G = G2 Since o(x) ̸ = p, each element in S has order 3 or 9, and since ⟨a, b3 ⟩ is a metacyclic normal subgroup of order 3p containing all elements of order 3, we have o(x) ̸ = 3. Thus, o(x) = o(y) = o(z) = 9 and x, y, z have the form ai b3j+1 or ai b3j−1 . Each automorphism α in Aut(G) can be written as follows:

α:

a ↦ → ai , b ↦ → aj b,

{

1 ≤ i ≤ p − 1; 0 ≤ j ≤ p − 1.

Note that a ↦ → ai , b ↦ → b and a ↦ → a, bj ↦ → ak bj induce automorphisms of G. Then S ≡ {b3k1 +1 , (b3k1 +1 )−1 , ab3k2 +1 , (ab3k2 +1 )−1 , al b3k3 +1 , (al b3k3 +1 )−1 }. Since every automorphism of G cannot map bi to aj b−i , we have Aut(G, S) cannot be transitive on S, a contradiction. Case 3: G = G3 Since o(x) ̸ = p, each element in S has order 3p or 3. Since ⟨a, b⟩ contains all elements of order 3p in G, one has o(x) = 3 because ⟨S ⟩ = G. Note that Z (G) = ⟨b⟩. Thus, b, b2 ̸ ∈ S, and x, y, z have the form ai bj c or ai bj c −1 with 1 ≤ i ≤ p and 1 ≤ j ≤ 3. Each automorphism α in Aut(G) can be written as follows:

⎧ ⎨a ↦→ ai α : b ↦ → bj ⎩ c ↦ → ak b l c

1 ≤ i ≤ p − 1; 1 ≤ j ≤ 2; 0 ≤ k ≤ p − 1, 0 ≤ l ≤ 2.

Thus, we may assume that x = c, and since the map a ↦ → ai , b ↦ → bj , c ↦ → c induces an automorphism of G, S ≡ {c , c −1 , abc , (abc)−1 , ai bj c , (ai bj c)−1 }. Since every automorphism of G cannot map ai bj c to (ai bj c)−1 , we have Aut(G, S) cannot be transitive on S, a contradiction. □ To state the following lemmas, we introduce the so called Schur multiplier. Let G be a simple group and Z an abelian group. We call an extension E = Z .G of Z by G a central extension of G if Z ≤ Z (E). If E is perfect, that is, the derived group E ′ = E, we call E a covering group of G. Schur [15] proved that for every simple group G there is a unique maximal covering

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group M such that every covering group of G is a factor group of M. This group M is called the full covering group of G, and the center of M is called the Schur multiplier of G, denoted by Mult(G). Let X be a connected hexavalent s-transitive graph of order 9p with s ≥ 1. Set A = Aut(X ) and Av be the stabilizer of the i vertex v ∈ V (X ) in A. If p = 2 or 3, then by McKay [21] and [7], X ∼ (i = 1, 2, 3, 4). Thus, in what follows, = C6 [3K1 ], G18 , or G27 we suppose that p ≥ 5. The following lemma is about the case s ≥ 2. Lemma 4.2. The case s ≥ 2 never occurs.

⏐ ⏐

⏐ ⏐

Proof. Since s ≥ 2, then by Proposition 2.4, we have |Av | ⏐ 27 · 33 · 52 and |A| ⏐ 27 · 35 · 52 · p. Note that |V (X )| = 9p is an

odd number. By Proposition 2.3(2), there exists a normal subgroup N in A such that A/N is an almost simple group, the block graph ⏐ XN is a (A/N , s)-transitive graph, and X is a normal cover of XN . It follows that N is semiregular on V (X ) and hence

⏐ |N | ⏐ 9p.

Suppose that |N | = p. Then XN is a connected hexavalent (A/N , s)-transitive graph of order 9 with s ≥ 2. However, by McKay [21], there is no such graph of order 9, a contradiction. Suppose that |N | = 9. Then XN is a connected hexavalent (A/N , s)-transitive graph of order p. By Proposition 2.5, X ∼ = K7 with p = 7 or C Ap with p > 7. For the latter case, A/N ∼ = Aut(C Ap ) ∼ = Zp ⋊Z6 and Av ∼ = Z6 . By Proposition 2.4, X⏐ is 1-transitive,

⏐ contrary to our assumption that s ≥ 2. Thus, XN ∼ = K7 and A/N ≲ S7 . Note that s ≥ 2. By Proposition 2.4, 60 ⏐ |Av | and hence ⏐

⏐ ∼ A7 . Then M /N is a normal arc-transitive subgroup in A/N and 60 · 7 ⏐ |A/N |. By Atlas [8], A7 ≲ A/N ≲ S7 . Let M /N =

hence M ⊴ A. Note that |N | = 9. Thus, it is easy to see that N ∼ = Z9 or Z23 . By ⏐‘‘N/C-Theorem’’ (see [15, Chapter I, Theorem

⏐ 4.5]), M /CM (N) ≲ Aut(N) ∼ = Z6 or GL(2, 3). Since 7 |Aut(N)|, we have that 7 ⏐ |CM (N)| and CM (N)/N ̸= 1. The normality of

CM (N)/N in M /N ∼ = A7 implies that CM (N)/N = M /N. It follows that CM (N) = M and N ≤ Z (M). Let Z3 ∼ = K ≤ N. Then K ≤ Z (M), and hence K ⊴ M. The block graph XK is a hexavalent 2-arc-transitive graph of order 21. However, by [7], there is no hexavalent 2-arc-transitive graph of order 21, a contradiction. Suppose that |N | = 3. Then N ∼ = Z3 and XN is a connected hexavalent (A/N , s)-transitive graph of order 3p with s ≥ 2. By Proposition 2.6, XN ∼ = C A3p , Cp [3K1 ], G15 or G57 . For the former three cases, they are 1-transitive by their constructions and Proposition 2.4. Thus, XN ∼ = G57 and A/N ∼ = PSL(2, 19). It follows from the ‘‘N/C-Theorem’’ that N ≤ Z (A), hence A is a central extension of PSL(2, 19). By [1, 33.3], A = NA′ , where A′ is the derived subgroup of A. Since the Schur multiplier Mult(PSL(2, 19)) ∼ = Z2 (see the Atlas [8]), it follows that A ̸= A′ and hence N ∩ A′ = 1. We conclude that A ∼ = N × A′ ∼ = Z3 × PSL(2, 19). Set K ∼ = PSL(2, 19). Then K ⊴ A and Kv ⊴ Av ∼ = A5 . Clearly, Kv ̸= 1. If K is transitive on V (X ), then |Kv | = |K |/(9 · 19) = 20. By Atlas [8], Kv ∼ = D20 . This is impossible because Av ∼ = A5 has no normal subgroup isomorphic to D20 . Thus, K is intransitive on V (X ). Since Kv ̸ = 1, the block graph XK has valency 2 by Lemma 2.1. Clearly, A/K ∼ = Z3 cannot be arc-transitive on XK , a contradiction. Suppose that |N | = 1. Then A is an almost simple group. Let T be the socle of A. Then T is a non-abelian simple group and A ≲ Aut(T ). Note that Aut(T )/T is solvable. Thus, Av /Tv is solvable. By Proposition 2.4, ⏐Av is non-solvable ⏐ because s ≥ 2. This forces that Tv is non-solvable, Tv is transitive on the neighborhood N(v ) of v and 60 ⏐ |Tv |. Since |T | ⏐ |A|, we have that

⏐ ⏐ ⏐ ⏐ |T | ⏐ 27 · 35 · 52 · p. Since Tv ̸= 1, we have that T is transitive or has r ≥ 3 orbits on V (X ) and XT ∼ = Cr by Lemma 2.1. If XT ∼ = Cr , then X is 1-transitive, contrary to our assumption that s ≥ 2. Thus, T is transitive on V (X ). It follows that T is arc-transitive ⏐ ⏐ and 22 · 33 · 5 · p ⏐ |T |. Thus, the order |T | has 3-prime factor or 4-prime factor. Assume that |T | has 3-prime factor. Then by [11, pp. 12–14] or [14], T ∼ , 17), PSL(3, 3), = A5 , A6 , PSL(2, 7), PSL(2, 8), PSL(2 ⏐ ⏐ 2 3 PSU(3, 3) or PSU(4, 2). By checking the orders of these groups, none of them satisfies that 2 · 3 · 5 · p ⏐ |T |. ⏐ ⏐ Assume that |T | has 4-prime factor. Then |T | ⏐ 27 · 35 · 52 · p. By calculating the orders of the simple groups listed in ⏐ ⏐ [3, Theorem 1], T is isomorphic to one of the groups in Table 2. Note that 22 · 33 · 5 · p ⏐ |T |. Thus, we have T = A9 , A10 , PSL(2, 81), M12 or J2 . Let T = A9 , A10 , M12 or J2 . Then |Tv | = |T |/9p. By Magma [2], T has no subgroup of order |T |/9p, a contradiction. Let T = PSL(2, 81). Then |Tv | = |T |/9 · 41 = 720 and p = 41. By Magma [2], Tv ∼ = PGL(2, 9). By Proposition 2.4, Av has no subgroup isomorphic to PGL(2, 9), a contradiction. □ i Lemma 4.3. Let s = 1. Then X is isomorphic to G45 , G63 (i = 1, 2), C3p [3K1 ], C (3, p, 2), C Ai9p (i = 1, 2, 3) or C A(32 ,p) .

Proof. Since A is 1-transitive and by Proposition 2.4, Av is a {2, 3}-group. Let |Av | = 2s · 3t . Then |A| = 2s · 3t +2 · p. Since p ≥ 5, every Sylow 2-subgroup of A is also a Sylow 2-subgroup of a stabilizer of some vertex in A, implying that A has no non-trivial normal 2-subgroups. Suppose that A has an intransitive minimal normal subgroup, say N. Since |V (X )| = 9p and |A| | 2s · 3t +2 · p, N is either a non-abelian simple group, or an elementary abelian 3- or p-group. Let B = {B1 , B2 , . . . , Bn } be the set of orbits of N and K the kernel of A acting on B. Then N ≤ K . Let m = |B1 |. Then mn = 9p with 1 < m, n < 9p. The quotient graph XN has vertex

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Table 2 Non-abelian simple {2, 3, 5, p}-groups. T

Order

Out

A7 A8 A9 A10 PSL(2, 11) PSL(2, 16) PSL(2, 19) PSL(2, 25) PSL(2, 31) PSL(2, 81) PSL(3, 4) PSU(3, 4) M11 M12 J2

23 26 26 27 22 24 22 23 25 24 26 26 24 26 27

2 2 2 2 2 4 2 22 2 23 D12 4 1 2 2

· 32 · 5 · 7 · 32 · 5 · 7 · 34 · 5 · 7 · 34 · 52 · 7 · 3 · 5 · 11 · 3 · 5 · 17 · 32 · 5 · 19 · 3 · 52 · 13 · 3 · 5 · 31 · 34 · 5 · 41 · 32 · 5 · 7 · 3 · 52 · 13 · 32 · 5 · 11 · 33 · 5 · 11 · 33 · 52 · 7

set B and A/K ≤ Aut(XN ). Moreover, assume that B1 is adjacent to B2 in XN with v ∈ B1 and u ∈ B2 being adjacent in X . Since the order |V (X )| = 9p is odd, XN has valency 2 or 6. Case 1: XN has valency 2. In this case, XN is a cycle and A/K ∼ = D2n . Since X is symmetric, the induced subgraph ⟨B1 ∪ B2 ⟩ of B1 ∪ B2 in X is a bipartite graph of valency 3. Suppose that XN ∼ = C3p . Then |B1 | = 3 and the induced subgraph ⟨B1 ∪ B2 ⟩ ∼ = K3,3 . It follows that X ∼ = C3p [3K1 ]. Suppose that XN ∼ = C⏐ p . Then |B1 | = 9. Since ⟨B1 ∪B2 ⟩ has valency 3, 5 is not a divisor of |K |. It follows that N is an elementary

⏐ abelian 3-group and 9 ⏐ |N |. Let N ∼ = Zk3 . If k ≥ 3 then Nv ̸= 1. By [25, Theorem 1], X ∼ = C (3, p, 2) or C (3, 3p, 1) ∼ = C3p [3K1 ].

If k = 2, then N ∼ = Z23 is semiregular on V (X ). Set C = CA (N) and P be a Sylow p-subgroup of A. Then P ∼ = Zp , and by ‘‘N/C-Theorem’’ (see [15, Chapter I, Theorem 4.5]), A/C ≲ Aut(N) ∼ = GL(2, 3). Note that p ≥ 5 and p |GL(2, 3)|. Thus, P ≤ C and PN = P × N is regular on V (X ). It follows that X is a Cayley graph on the group P × N ∼ = Zp × Z23 . Since ⟨B1 , B2 ⟩ is ∼ an arc-transitive bipartite graph of order 18, ⟨B1 , B2 ⟩ = 3K3,3 or Pappus graph by [21]. Clearly, N induces an automorphism group of ⟨B1 , B2 ⟩. Assume that ⟨B1 , B2 ⟩ ∼ = 3K3,3 . Note that X is a Cayley graph on P × N. Thus, X = Cay(P × N , S) and B can be viewed as the right cosets |P × N : N |. It forces that S is a union of two right cosets of subgroup of order 3. An easy calculation implies that X ∼ = C3p [3K1 ]. Assume that ⟨B1 , B2 ⟩ is isomorphic to Pappus graph. Take a non-identity element x in K . If x fixes each vertex in ⟨B1 , B2 ⟩, then the connectivity of XN implies that there exists some Bi ∈ B such that x acts on Bi non-trivially. Without loss of generality, we may assume that i = 3 and ⟨B2 , B3 ⟩ is isomorphic to Pappus graph. Let {u, w} ∈ E(X ) with u ∈ B2 and w ∈ B3 . Note x acts trivially on B2 and non-trivially on B3 . Thus, ux = u and we may assume that w x = w ′ ̸ = w . Take u′ ∈ (N(w ) ∩ B2 ) \ {u}. Then {u, w}x = {u, w′ } and {u′ , w}x = {u′ , w′ }. It forces that the set {u, u′ } ⊆ N(w) ∩ N(w′ ). This is impossible in Pappus graph. Thus, x cannot fix each vertex in ⟨B1 , B2 ⟩. It follows that K acting on ⟨B1 , B2 ⟩ is faithful and induces an automorphism group of Pappus graph and K ≲ Z23 ⋊ (S3 × Z2 ). Since A/K ∼ = D2p , the symmetry of A implies that Z3 ≲ Kv ≲ S3 × Z2 . Let Q be a Sylow 3-subgroup of K . Then Q ∼ = Z23 ⋊ Z3 is normal in K and hence characteristic in K . The normality of K in A implies that Q ⊴ A and QP ≤ A. Since KP /Q has order p, 2p or 4p and any group of order kp with k = 1, 2 or 4 has a normal subgroup of order p with p ≥ 5, QP /Q ⊴ KP /Q . It follows that QP ⊴ KP. Since N ⊴ QP, we have QP /N is a group of order 3p. Note that any group of order 3p has a normal subgroup isomorphic to Zp . Thus, P ⊴ QP and hence P is a characteristic subgroup of QP. Thus, P is normal in KP. It implies that P × N is normal in A and hence X is a normal Cayley graph on Z23 × Zp . By Lemma 3.10, X ∼ = C A(32 ,p) and A ∼ = (Z23 × Zp ) ⋊ Z6 . Suppose that XN ∼ = C9 . Then |B1 | = p. It follows that N ∼ = Zp or a {2, 3, p}-simple group. Since A is arc-transitive, ⟨B1 , B2 ⟩ is a cubic symmetric graph of order 2p. By [6, Table 1], Aut(⟨B1 , B2 ⟩) ∼ = PGL(2, 7) or D2p ⋊ Z3 . A similar argument as above, we can deduce that K is faithful on ⟨B1 , B2 ⟩. It follows that K ≲ PGL(2, 7) or K ≲ D2p ⋊ Z3 . Clearly, K fixes B1 and B2 setwise. Thus, K ∼ = PSL(2, 7) or Zp ⋊ Z3 . Let K ∼ = PSL(2, 7). Then ⟨B1 , B2 ⟩ is isomorphic to Heawood graph and p = 7. Since A/K ∼ = D18 , we have that A⏐/K has a

⏐ normal subgroup M /K ∼ = Z9 . By ‘‘N/C-Theorem’’, M /CM (K ) ≲ Aut(K ) ∼ = PGL(2, 7). Note that 9 |PGL(2, 7)|. Thus, 3 ⏐ |CM (K )|.

The simplicity of K forces that K ∩ CM (K ) = 1 and hence KCM (K ) = K × CM (K ). Since KCM (K )/K ≲ D18 , we have that CM (K ) has a subgroup H ∼ = Z3 and H ⊴ A. It follows that the block graph XH has order 21 and valency 2 or 6. However, by [7], there is no symmetric graph of order 21 admitting A/H as an automorphism group, a contradiction. Let K ∼ = Zp ⋊ Z3 . Then N ∼ = Zp . Since A/K ∼ = D18 , we have that A/K has a normal subgroup M /K ∼ = Z9 . Set Q be a Sylow ∼ 3-subgroup of M such that Q ∩ K = Kv = Z3 with v ∈ B1 . Then M = N ⋊ Q , |Q | = 33 , Q ∩ K ∼ = Z3 and Q /(Q ∩ K ) ∼ = Z9 . By the elementary group theory, Q ∼ = Z27 , Z9 × Z3 or Z9 ⋊ Z3 . If Q ∼ = Z27 , then Q ≤ CA (Q ). Since Kv ≤ Q , we have that

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⏐ ⏐ N ̸ ≤ NA (Q ). Clearly, 2 ⏐ |NA (Q )|. If NA (Q ) = CA (Q ), then by ‘‘Burnside Normal p-Complement Theorem’’ (see [4, pp. 327]), Q has a normal 3-complement subgroup H, that is, A = H ⋊ Q . It follows that XH ∼ = C9 . However, A/H ∼ = Z27 cannot be arc-transitive on XH , contrary to Lemma 2.1(2). Thus, CA (Q ) < NA (Q ). This implies that A/N ∼ = D54 . Since N = CA (N), we have that A/N ≲ Aut(N) ∼ = Zp−1 by ‘‘N/C-Theorem’’. This is impossible because A/N ∼ = D54 is non-abelian. Thus, Q ∼ = Z9 × Z3 or Z9 ⋊ Z3 . Since M = N ⋊ Q ∼ = Zp ⋊ (Z9 × Z3 ) or Zp ⋊ (Z9 ⋊ Z3 ), we have that M has a subgroup G ∼ = Zp ⋊ Z9 , which is normal in A and regular on V (X ). It follows that X is a normal Cayley graph on Zp ⋊ Z9 . By Lemmas 3.10 and 4.1, X ∼ = C Ai9p (i = 1, 2, 3) ∼ and A = (Z9 × Zp ) ⋊ Z6 . Suppose that XN ∼ = C3 . Then |B1 | = 3p and N is a {2, 3, p}-simple group. By [11, pp.12–14] or [14], N is isomorphic to A5 , A6 , PSL(2, 7), PSL(2, 8), PSL(2, 17), PSL(3, 3), PSU(3, 3), PSU(4, 2) and p = 5, 7, 13 or 17. Set C = CA (N). Then C ∩ N = 1 because N is a non-abelian simple group. If C = 1 then A ∼ = A/C ≲ Aut(N). Note that A/K ∼ = S3 . It follows that 6 | Out(N), which contradicts the information in [8]. Thus, C ̸= 1. Clearly, p |C |. Consider the block graph XC . Then XC is a symmetric graph of order p or 3p and valency 2 or 6 admitting A/C as an arc-transitive subgroup. Since N ≲ A/C is non-solvable, by Propositions 2.5 and 2.6, we have XC ∼ = K7 or G15 and A/C ≲ S7 or S6 . Assume that XC ∼ = K7 . Then p = 7 and |C | = 9. Since N ≲ A/C ≲ S7 and N is a {2, 3, 7}-simple group, we have N ∼ = PSL(2, 7) by Atlas [8]. Since N ≲ A/C ≲ Aut(N) and A/K ∼ = D6 , we have that A ∼ = (N × C ) ⋊ Z2 . Thus, A has a normal subgroup M ∼ = Z3 . Clearly, the block graph XM is a A/M-symmetric graph of order 21 and valency 6. Note that PSL(2, 7) ≲ A/M is non-solvable. This is impossible by Proposition 2.6. Assume that XC ∼ = G15 . Then C ∼ = Z3 and A/C ≲ S6 . By Atlas [8], N ∼ = A5 or A6 . Note that S5 acting on G15 is not arc∼ transitive. Thus, N = A6 . A similar argument as the above, we have that A ∼ = (N × C ) ⋊ Z2 with N ⋊ Z2 ∼ = S6 and C ⋊ Z2 ∼ = S3 . It follows that Av ∼ = S4 × Z2 . However, by Magma [2], (A6 × Z3 ) ⋊ Z2 has no connected hexavalent symmetric coset graph, a contradiction. Case 2: XN has valency 6. In this case, Kv fixes the neighborhood of v in X pointwise. Thus, K = N is semiregular on V (X ) and A/N ≲ Aut(XN ). Since |V (X )| = 9p, we have N = Zp , Z23 or Z3 . Suppose that N ∼ = Zp . Then the quotient graph XN has order 9. By [21], XN ∼ = C3 [3K1 ] and A/N ≲ S3 wrS3 . Set C = CA (N). Then N ≤ C and A/C ≲ Aut(N) ∼ = Zp−1 . It forces that C is transitive on V (X ). Consider a minimal normal subgroup M /N of A/N. Then M /N ∼ = C3 . Since M is a minimal normal subgroup and C ∩ M ⊴ A, we have C ∩ M = M and M ≤ C . = Zk3 and XM ∼ It follows that M ∼ = Zp × Zk3 . Thus, M has a normal Sylow 3-subgroup P ∼ = Zk3 and hence P is characteristic in M. This forces ∼ ∼ ∼ that P ⊴ A. In this case XP = C3p and X = C3p [3K1 ]. It follows that A = S3 wrD6p . However S3 wrD6p has no normal subgroup isomorphic to Zp , a contradiction. Suppose that N ∼ = K7 = Z23 . Then XN is a hexavalent (A/N , 1)-transitive graph of order p. By Proposition 2.5, either XN ∼ ∼ and A/N = PSL(2, 7), or XN is a normal Cayley graph. Assume that XN is a normal Cayley graph. Then A/N has a normal regular Sylow p-subgroup and hence A has normal regular subgroup isomorphic to Z23 × Zp . By Lemma 3.10, X ∼ = C A(32 ,p) . Assume that XN ∼ = K7 and A/N ∼ = PSL(2, 7). Then p = 7. Set C = CA (N). Then A/C ≲ Aut(N) ∼ = GL(2, 3). Since PSL(2, 7) is non-solvable and A/C ∼ = (A/N)/(C /N), we have C = A. By Atlas [8], Mult(PSL(2, 7)) = 2. Thus, A = N × M ∼ = Z23 × PSL(2, 7). This is impossible because A/M cannot be arc-transitive on the block graph XM . Suppose that N ∼ = Z3 . Then XN is a connected hexavalent A/N-1-transitive graph of order 3p. By Proposition 2.6, XN ∼ = C A3p , Cp [3K1 ] or G15 . Assume that XN ∼ = C A3p . Then A/N has a normal regular subgroup on V (XN ) because Aut(C A3p ) ∼ = Z3p ⋊ Z6 . It implies that X is a normal Cayley graph. By Lemma 3.10, X ∼ = C Ai9p (i = 1, 2, 3) or C A(32 ,p) . Assume that XN ∼ = Cp [3K1 ]. Then A/N ≲ S3 wrD2p . Let P /N be a Sylow 3-subgroup of A/N. Then P /N is normal in A/N and P /N ∼ = Zk3 . It follows that P ⊴ A and P ∼ = Z9 × Zk3−1 or Z3k+1 . If k > 1, then P is not semiregular on V (X ). By [25, Theorem 1], 2 X ∼ = C (3, p, 2) or C (3, 3p, 1) ∼ = C3p [3K1 ]. If k = 1, then P ∼ = Z9 or ⏐ Z3 . Since P is a Sylow 3-subgroup of A and |A| = |P | · |A/P |, we have that Av is a 2-group. This is contrary to the fact that 6 ⏐ |Av |.

⏐

Assume that XN ∼ = G15 . Then A/N ∼ = S6 . By Atlas [8], S6 has a minimal arc-transitive subgroup A6 . Thus, A/N has a subgroup ∼ M /N = A6 . Since Mult(A6 ) = Z6 , we have M ∼ = A6 × Z3 or Z3 .A6 . If M ∼ = A6 × Z3 , then M has a normal subgroup H ∼ = A6 . Clearly, M /H ∼ = Z3 cannot be arc-transitive on XM , a contradiction. If M ∼ = Z3 .A6 , then by Example 3.5, X ∼ = G45 . Now we may assume that A has no intransitive minimal normal subgroup. Thus, every non-trivial normal subgroup of A is transitive on V (X ). Again let N be a minimal normal subgroup of A. Then N is transitive on V (X ). Since |V (X )| = 9p and A is 1-transitive, N has 3 prime divisors and is a non-abelian simple group. By [11, pp.12–14] or [14], N is isomorphic to A5 , A6 , PSL(2, 7), PSL(2, 8), PSL(2, 17), PSL(3, 3), PSU(3, 3), PSU(4, 2).

⏐

⏐ Recall that N is transitive on V (X ). Thus, 9p ⏐ |N |. By checking the orders of the simple group above, we have N ∼ = A6 ,

PSL(2, 8), PSL(2, 17), PSL(3, 3), PSU(3, 3), PSU(4, 2). Set C = CA (N), the centralizer of N in A. Then C ∩ N = 1 and C is a {2, 3}-group. If C ̸ = 1 then C is an intransitive normal subgroup of A because |V (X )| = 9p, which is contrary to our assumption. Thus, C = 1 and A = A/C ≲ Aut(N). By Atlas [8],

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5 |Out(N)|. Since A is 1-transitive, 5 |Nv | by Proposition 2.4. Note that |Nv | = |N |/9p. Thus, an easy calculation implies that N ∼ = A6 , PSL(2, 8), PSL(2, 17), PSL(3, 3), PSU(3, 3), PSU(4, 2). By Magma [2], there is no symmetric coset graph of order 9p on the group A6 , PSL(2, 8), PSL(2, 17), PSL(3, 3), PSU(4, 2). i Thus, N ∼ (i = 1, 2). □ = PSU(3, 3) and |Nv | = 25 · 3. By Example 3.5, X ∼ = G63 Combining Lemmas 4.1–4.3, we can complete the proof of Theorem 1.1. Acknowledgments This work was supported by the National Natural Science Foundation of China (11301154, 11301159, 11601125), the Key Project of Education Department of Henan Province Scientific and Technological Research (13A110249) and the Scientific Research Foundation for Doctoral Scholars of HAUST (09001707), the Innovation Team Funding of Henan University of Science and Technology (2015XTD010). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]

M. Aschbacher, Finite Group Theory, second ed., Cambridge University Press, Cambridge, 2000. W. Bosma, C. Cannon, C. Playoust, The MAGMA algebra system I: The user language, J. Symbolic Comput. 24 (1997) 235–265. Y. Bugeaud, Z. Cao, M. Mignotte, On simple k4 -groups, J. Algebra 241 (2001) 658–668. W. Burnside, Theory of Groups of Finite Order, second ed., Cambridge, 1991; Dover Publications, New York, 1955. C.Y. Chao, On the classification of symmetric graphs with a prime number of vertices, Trans. Amer. Math. Soc. 158 (1971) 247–256. Y. Cheng, J. Oxley, On weakly symmetric graphs of order twice a prime, J. Combin. Theory Ser. B 42 (1987) 196–211. M.D.E. Conder, A complete list of all connected symmetric graphs of order 2 to 30. https://www.math.auckland.ac.nz/~conder/symmetricgraphsorderupto30.txt. H.J. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A Wilson, Atlas of Finite Group, Clarendon Press, Oxford, 1985. http://brauer.maths.qmul.ac.uk/ Atlas/v3/. A. Gardiner, C.E. Praeger, A characterization of certain families of 4 -valent symmetric graphs, European J. Combin. 15 (1994) 383–397. A. Gardiner, C.E. Praeger, On 4-valent symmetric graphs, European J. Combin. 15 (1994) 375–381. D. Gorenstein, Finite Simple Groups, Plenum Press, New York, 1982. S.T. Guo, Y.Q. Feng, Tetravalent symmetric graphs of order 9p, J. Korean Math. Soc. 49 (2012) 1111–1121. S.T. Guo, X.H. Hua, Y.T. Li, Hexavalent (g , s)-transitive graphs, Czechoslovak Math. J. 63 (2013) 923–931. M. Herzog, On finite simple groups of order divisible by three primes only, J. Algebra 10 (1968) 383–388. B. Huppert, Eudiche Gruppen I, Springer-Verlag, Berlin, 1967. J.H. Kwak, J.M. Oh, One-regular normal Cayley graphs on dihedral groups of valency 4 or 6 with cyclic vertex stabilizer, Acta Math. Sin. (Engl. Ser.) 22 (2006) 1305–1320. C.H. Li, On finite s-transitive graphs of odd order, J. Combin. Theory Ser. B 81 (2001) 307–317. C.H. Li, Z.P. Lu, D. Marušič, On primitive permutation groups with small suborbits and their orbital graphs, J. Algebra 279 (2004) 749–770. B. Ling, C.X. Wu, B. Lou, Pentavalent symmetric graphs of order 30p, Bull. Aust. Math. Soc. 90 (2014) 353–362. P. Lorimer, Vertex-transitive graphs: Symmetric graphs of prime valency, J. Graph Theory 8 (1984) 55–68. B.D. McKay, Transitive graphs with fewer than 20 vertices, Math. Comp. 33 (1979) 1101–1121. J.M. Oh, K.W. Hwang, Construction of one-regular graphs of valency 4 and 6, Discrete Math. 278 (2004) 195–207. J. Pan, B. Lou, C. Liu, Arc-transitive pentavalent graphs of order 4pq, Electron. J. Combin. 20 (2013) 1215–1230. J.M. Pan, X. Yu, Pentavalent symmetric graphs of order twice a prime square, Algebra Colloq. 22 (2015) 383–394. C.E. Praeger, M.Y. Xu, A characterization of a class of symmetric graphs of twice prime valency, European J. Combin. 10 (1989) 91–102. B.O. Sabidussi, Vertex-transitive graphs, Monatsh. Math. 68 (1964) 426–438. R.J. Wang, M.Y. Xu, A classification of symmetric graphs of order 3p, J. Combin. Theory Ser. B 58 (1993) 197–216. C.Q. Wang, M.Y. Xu, Non-normal one-regular and 4-valent Cayley graphs of dihedral groups D2n , European J. Combin. 27 (2006) 750–766. C.Q. Wang, Z.Y. Zhou, 4-valent one-regular normal Cayley graphs of dihedral groups, Acta Math. Sinica (Chin. Ser.) 49 (2006) 669–678. M.Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math. 182 (1998) 309–319. J. Xu, M.Y. Xu, Arc-transitive Cayley graphs of valency at most four on abelian groups, Southeast Asian Bull. Math. 25 (2001) 355–363. D.W. Yang, Y.Q. Feng, Pentavalent symmetric graphs of order 2p3 , J. Korean Math. Soc. 18 (2016) 1–18. D.W. Yang, Y.Q. Feng, J.L. Du, Pentavalent symmetric graphs of order 2pqr, Discrete Math. 339 (2016) 522–532. J.X. Zhou, Tetravalent s-transitive graphs of order 4p, Discrete Math. 309 (2009) 6081–6086. J.X. Zhou, Y.Q. Feng, Tetravalent one-regular graphs of order 2pq, J. Algebraic Combin. 29 (2009) 457–471. J.X. Zhou, Y.Q. Feng, Tetravalent s-transitive graphs of order twice a prime power, J. Aust. Math. Soc. 88 (2010) 277–288.