On the classification of distance-transitive orbital graphs of overgroups of the Jevons group

The Jevons group is the exponential group $S_2\uparrow S_n$. It is generated by the $(n\times n)$-matrices over $\operatorname{GF}(2)$ and the translation group on the $n$-dimensional vector space $V_n$ over $\operatorname{GF}(2)$. For a permutation group $G$ on $V_n$ being an overgraph of $S_2\uparrow S_n$, an orbital of $G$ is an orbit of $G$ in its natural action on $V_n\times V_n$. The orbital graph associated with an orbital $\Gamma$ is the graph with the vertex set $V_n$ and the edge set $\Gamma$. In this paper, we classify distance-transitive orbital graphs of overgroups of the Jevons group $S_2\uparrow S_n$ and show that some of them are isomorphic to the following graphs: the complete graph $K_{2^n}$, the complete bipartite graph $K_{2^{n-1},2^{n-1}}$, the halved $(n+1)$-cube, the folded $(n+1)$-cube, alternating forms graphs, the Taylor graph, the Hadamard graph.