Semisymmetric graphs of order $2p^3$

A simple undirected graph is said to be {\em 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 [{\em J. Combin. Theory Ser. B} {\bf 3}(1967), 215-232] that there exist no semisymmetric graphs of order $2p$ and $2p^2$, where $p$ is a prime. The classification of semisymmetric graphs of order $2pq$ was given in [{\em Comm. in Algebra} {\bf 28}(2000), 2685-2715], for any distinct primes $p$ and $q$. Our long term goal is to determine all the semisymmetric graphs of order $2p^3$, for any prime $p$. All these graphs $\G$ are divided into two subclasses: (I) $\Aut(\G)$ acts unfaithfully on at least one bipart; and (II) $\Aut(\G)$ 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.