Cayley digraphs of 2-genetic groups of odd prime-power order

A group is called 2-genetic if each normal subgroup of the group can be generated by two elements. Let G   be a non-abelian 2-genetic group of order pnpn for an odd prime p and a positive integer n  . In this paper, we investigate connected Cayley digraphs Cay(G,S)Cay(G,S) for non-abelian 2-genetic groups G   of odd order pnpn, and determine their full automorphism groups A=Aut(Cay(G,S))A=Aut(Cay(G,S)) in the case when Aut(G,S)={α∈Aut(G)|Sα=S} is a p′p′-group. It is shown that either Cay(G,S)Cay(G,S) is normal, that is, the right regular representation of G is normal in A  , or p=3,5,7,11p=3,5,7,11 and the largest normal p  -subgroup Op(A)Op(A) of A   has order pn−1pn−1 with ASL(2,p)≤A/Φ(Op(A))≤AGL(2,p)ASL(2,p)≤A/Φ(Op(A))≤AGL(2,p). Furthermore, a non-normal Cayley digraph with smallest order and smallest valency is constructed for each p=3,5,7,11p=3,5,7,11, respectively. In particular, the underlying graphs of the above non-normal Cayley digraphs for p=3,7,11p=3,7,11 are half-arc-transitive, and they are the first constructions of half-arc-transitive non-normal Cayley graphs of order a prime-power.