A graph theoretic characterization of the classical generalized hexagon on $364$ vertices

Abstract

A tetravalent $2$-arc-transitive graph of order $728$ is either the known $7$-arc-transitive incidence graph of the classical generalized hexagon $GH(3,3)$ or a normal cover of a $2$-transitive graph of order $182$ denoted $A[182,1]$ or $A[182,2]$ in the $2009$ list of Poto\v{c}nik.