Inclusions of innately transitive groups into wreath products in product action with applications to $2$-arc-transitive graphs

TL;DR

This paper investigates the embedding of innately transitive groups into wreath products in product action, classifies certain 2-arc-transitive graphs, and establishes non-existence results for others.

Contribution

It introduces new classifications of 2-arc-transitive graphs with innately transitive automorphism groups embedded in wreath products, identifying specific examples and proving non-existence under certain conditions.

Findings

Identifies two specific 2-arc-transitive graphs with given automorphism groups.

Proves non-existence of additional such graphs under certain conditions.

Abstract

We study $(G,2)$-arc-transitive graphs for innately transitive permutation groups $G$ such that $G$ can be embedded into a wreath product $\sym\Gamma\wr\sy\ell$ acting in product action on $\Gamma^\ell$. We find two such connected graphs: the first is Sylvester's double six graph with 36 vertices, while the second is a graph with $120^2$ vertices whose automorphism group is $\aut\sp 44$. We prove that under certain conditions no more such graphs exist.