Hexavalent (G, s)-transitive graphs

Let X be a finite simple undirected graph with a subgroup G of the full automorphism group Aut(X). Then X is said to be (G, s)-transitive for a positive integer s, if G is transitive on s-arcs but not on (s + 1)-arcs, and s-transitive if it is (Aut(X), s)-transitive. Let Gv be a stabilizer of a vertex v ∈ V (X) in G. Up to now, the structures of vertex stabilizers Gv of cubic, tetravalent or pentavalent (G, s)-transitive graphs are known. Thus, in this paper, we give the structure of the vertex stabilizers Gv of connected hexavalent (G, s)-transitive graphs.