Cubic symmetric graphs of order

A graph is symmetric if its automorphism group is transitive on the arc set of the graph. In this paper, we classify connected cubic symmetric graphs of order 8 p 3 for each prime p . All those symmetric graphs are explicitly constructed as normal Cayley graphs on some groups of order 8 p 3 , and their automorphism groups are determined. There is a unique connected cubic symmetric graph of order 64 . All connected cubic symmetric graphs of order 8 p 3 for p ≥ 3 are regular covers of the three dimensional hypercube Q 3 , and consist of four infinite families, of which two families exist if and only if 3 ∣ ( p − 1 ) and the other two families exist for each odd prime p . In each family, there is a unique graph for a given order.