Pentavalent symmetric graphs of order 2p3

A graph is symmetric or 1-regular if its automorphism group is transitive or regular on the arc set of the graph, respectively. We classify the connected pentavalent symmetric graphs of order 2p3 for each prime p. All those symmetric graphs appear as normal Cayley graphs on some groups of order 2p3 and their automorphism groups are determined. For p = 3, no connected pentavalent symmetric graphs of order 2p3 exist. However, for p = 2 or 5, such symmetric graph exists uniquely in each case. For p ≥ 7, the connected pentavalent symmetric graphs of order 2p3 are all regular covers of the dipole Dip5 with covering transposition groups of order p3, and they consist of seven infinite families; six of them are 1-regular and exist if and only if 5 | (p-1), while the other one is 1-transitive but not 1-regular and exists if and only if 5 | (p± 1). In the seven infinite families, each graph is unique for a given order.