Semisymmetric graphs of order 2p3

A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Every semisymmetric graph is a bipartite graph with two parts of equal size. It was proved in Folkman (1967)  [14] that there exist no semisymmetric graphs of order 2p and 2p2, where p is a prime. The classification of semisymmetric graphs of order 2pq was given in Du and Xu (2000)  [12], for any distinct primes p and q. Our long term goal is to determine all the semisymmetric graphs of order 2p3, for any prime p. All these graphs Γ are divided into two subclasses: (I) Aut(Γ) acts unfaithfully on at least one bipart; and (II) Aut(Γ) acts faithfully on both biparts. This paper gives a group theoretical characterization for Subclass (I) and based on this characterization, we shall give a complete classification for this subclass in our further research.