A List for Vertex-Primitive Symmetric Graphs of Order $6p$

Abstract

The aim of this paper is to classify the vertex-primitive symmetric graphs of order $6p$. These works were essentially done in $[1]$. But in $[1]$ there is no such situation: $G=\mathrm{P}\mathrm{S}\mathrm{L}(2,13)$ acting on the set of cosets of subgroup $H\cong D_{14}$. Then $m=|\mathrm{\Omega }|=78=6p$, $G$ has rank $9$, and the sub-orbits of $G$ have one of length $1$, five of length $7$, and three of length $14$. In this paper, we give a complete list of symmetric graphs of order $6p$.