Cubic semisymmetric graphs of order 6p36p3

A regular edge-transitive graph is said to be semisymmetric   if it is not vertex-transitive. By Folkman [J. Folkman, Regular line-symmetric graphs, J. Combin Theory 3 (1967) 215–232], there is no semisymmetric graph of order 2p2p or 2p22p2 for a prime pp and by Malnič, et al. [A. Malnič, D. Marušič, C.Q. Wang, Cubic edge-transitive graphs of order 2p32p3, Discrete Math. 274 (2004) 187–198], there exists a unique cubic semisymmetric graph of order 2p32p3, the so-called Gray graph of order 54. In this paper it is shown that a connected cubic semisymmetric graph of order 6p36p3 exists if and only if p−1p−1 is divisible by 3. There are exactly two such graphs for a given order, which are constructed explicitly.