Normal edge-transitive Cayley graphs and Frattini-like subgroups

For a finite group G and an inverse-closed generating set C of G, let Aut(G;C) consist those automorphisms which leave invariant. We define Aut(G;C)-invariant normal subgroup Φ(G;C) has the property that, for any generators if we remove from it all elements Φ(G;C), then remaining is still G. The contains Frattini Φ(G) but inclusion may be proper. Cayley graph Cay(G,C) edge-transitive acts transitively on pairs {c,c−1} C. show Cay(G,C), its quotient modulo unique largest isomorphic to subdirect product graphs characteristically simple groups. In particular, therefore view groups as building blocks whenever have trivial. explore several questions these results raise, some concerned with sets in given family. particular use this theory classify 4-valent dihedral groups; involves new construction infinite family examples, disproves conjecture Talebi.