On the Semisymmetric Graphs of Order 2p3: faithful and primitive Case

A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. A semisymmetric graph must be bipartite whose automorphism group has two orbits of same size on the vertices. One of our long term goals is to determine all the semisymmetric graphs of order 2p(3), for any prime p. All these graphs Gamma with the automorphism group Aut(Gamma) are divided into two subclasses: (I) Aut(Gamma) acts unfaithfully on at least one bipart; and (II) Aut(Gamma) acts faithfully on both biparts. In [9], [19] and [20], a complete classification was given for Subclass (I). In this paper, a partial classification is given for Subclass (II), when Aut(Gamma) acts primitively on one bipart.