Cubic symmetric graphs of order 8p3

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 8p3 for each prime p. All those symmetric graphs are explicitly constructed as normal Cayley graphs on some groups of order 8p3, and their automorphism groups are determined. There is a unique connected cubic symmetric graph of order 64. All connected cubic symmetric graphs of order 8p3 for p≥3 are regular covers of the three dimensional hypercube Q3, 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.