Automorphisms of a family of cubic graphs

A Cayley graph Cay(G,S) on a group G with respect to a Cayley subset S is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay(G,S). For a positive integer n, let Γn be a graph having vertex set {xi, yi | i ∈ Z2n} and edge set {{xi, xi+1}, {yi, yi+1}, {x2i, y2i+1}, {y2i, x2i+1} | i ∈ Z2n}. In this paper, it is shown that Γn is a Cayley graph and its full automorphism group is isomorphic to Z2 o S3 for n = 2, or to Z2 o D2n for n > 2. Furthermore, we determine all pairs of group G and Cayley subset S satisfying Γn = Cay(G,S) is non-normal for G. Using this, all connected cubic non-normal Cayley graphs of order 8p are constructed explicitly for each prime p.