Pentavalent symmetric graphs of order 2p~3

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 2p~3 for each prime p. All those symmetric graphs appear as normal Cayley graphs on some groups of order 2p~3 and their automorphism groups are determined. For p = 3, no connected pentavalent symmetric graphs of order 2p~3 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 2p~3 are all regular covers of the dipole Dip5 with covering transposition groups of order p~3, 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.