# A family of $2$-groups and an associated family of semisymmetric, locally $2$-arc-transitive graphs

## TL;DR

This paper constructs a new family of semisymmetric, locally 2-arc-transitive graphs from specific 2-groups, demonstrating their properties and their role as normal covers of basic graphs, advancing understanding of symmetric graph structures.

## Contribution

It introduces a novel family of semisymmetric graphs derived from non-2-generated 2-groups, expanding the class of known such graphs and their structural properties.

## Key findings

- Graphs are semisymmetric with transitive edge action but intransitive vertex action.
- Constructed graphs are locally 2-arc-transitive and are normal covers of complete bipartite graphs.
- First known semisymmetric graphs from groups requiring more than two generators.

## 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, ${\rm Aut}(\Sigma)$ 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 ${\rm K}_{2^n,2^n}$. 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.