On the generic family of Cayley graphs of a finite group

Let $G$ be a finite group. For each $m>1$ we define the symmetric canonical subset $S=S(m)$ of the Cartesian power $G^m$ and we consider the family of Cayley graphs ${\mathcal{G}}_m(G)=Cay(G^m,S)$. We describe properties of these graphs and show that for a fixed $m>1$ and groups $G$ and $H$ the graphs ${\mathcal{G}}_m(G)$ and ${\mathcal{G}}_m(H)$ are isomorphic if and only if the groups $G$ and $H$ are isomorphic. We describe also the groups of automorphisms $\mathbf{A}\mathbf{u}\mathbf{t}({\mathcal{G}}_m(G))$. It is shown that if $G$ is a non-abelian group, then $$\mathbf{A}\mathbf{u}\mathbf{t}$(\mathscr{G}m(G))\simeq \big(G^m \rtimes $\mathbf{A}\mathbf{u}\mathbf{t}$(G)\big)\rtimes D{m+1}$,where$D_{m+1}$ is the dihedral group of order $2m+2$. If $G$ is an abelian group (with some exceptions for $m=3$), then $$\mathbf{A}\mathbf{u}\mathbf{t}$(\mathscr{G}m(G))\simeq G^m\rtimes \big($\mathbf{A}\mathbf{u}\mathbf{t}$(G)\times S{m+1}\big)$,where$S_{m+1}$isthesymmetricgroupofdegree$m+1$. As an example of application we discuss relations between Cayley graphs ${\mathcal{G}}_m(G)$ and Bergman-Isaacs Theorem on rings with fixed-point-free group actions.