Glasnik Matematicki, Vol. 58, No. 2 (2023), 259-287. \( \)

A FAMILY OF \(2\)-GROUPS AND AN ASSOCIATED FAMILY OF SEMISYMMETRIC, LOCALLY \(2\)-ARC-TRANSITIVE GRAPHS

Daniel R. Hawtin, Cheryl E. Praeger and Jin-Xin Zhou

Faculty of Mathematics, University of Rijeka, 51000 Rijeka, Croatia
e-mail:dan.hawtin@gmail.com

Department of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia
e-mail:cheryl.praeger@uwa.edu.au

School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, P.R. China
e-mail:jxzhou@bjtu.edu.cn

Abstract.   A mixed dihedral group is a group \(H\) with two disjoint subgroups \(X\) and \(Y\), each elementary abelian of order \(2^n\), such that \(H\) is generated by \(X\cup Y\), and \(H/H'\cong X\times Y\). In this paper, for each \(n\geq 2\), we construct a mixed dihedral \(2\)-group \(H\) of nilpotency class \(3\) and order \(2^a\) where \(a=(n^3+n^2+4n)/2\), and a corresponding graph \(\Sigma\), which is the clique graph of a Cayley graph of \(H\). We prove that \(\Sigma\) is semisymmetric, that is, \({\mathop{\rm Aut}}(\Sigma)\) acts transitively on the edges but intransitively on the vertices of \(\Sigma\). These graphs are the first known semisymmetric graphs constructed from groups that are not \(2\)-generated (indeed \(H\) requires \(2n\) generators). Additionally, we prove that \(\Sigma\) is locally \(2\)-arc-transitive, and is a normal cover of the `basic' locally \(2\)-arc-transitive graph \({\rm\bf K}_{2^n,2^n}\). As such, the construction of this family of graphs contributes to the investigation of normal covers of prime-power order of basic locally \(2\)-arc-transitive graphs – the `local' analogue of a question posed by C. H. Li.

2020 Mathematics Subject Classification.   05C38, 20B25

Key words and phrases.   Semisymmetric, \(2\)-arc-transitive, edge-transitive, normal cover, Cayley graph

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https://doi.org/10.3336/gm.58.2.08

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