Normal edge-transitive Cayley graphs on non-abelian groups of order 4p, where p is a prime number

Abstract

We determine all connected normal edge-transitive Cayley graphs on non-abelian groups with order 4p, where p is a prime number. As a consequence we prove if |G| = 2^δ p, δ = 0, 1,2 and p prime, then Λ = Cay(G, S) is a connected normal $\tfrac{1} {2}$ arc-transitive Cayley graph only if G = F _4p , where S is an inverse closed generating subset of G which does not contain the identity element of G and F _4p is a group with presentation $F_{4p} = \left\langle {\left. {a,b} \right|a^p = b^4 = 1,b^{ - 1} ab = a^\lambda } \right\rangle$, where λ ² ≠ −1 (mod p).