A family of \(2\)-groups and an associated family of semisymmetric, locally \(2\)-arc-transitive graphs

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.