A locally $5$-arc transitive graph related to ${\rm PSU}_3(8)$

Abstract

We construct a locally 5-arc transitive $G$-graph $\Delta$, where $G \in \{ {\rm PSU}_3(8) \rtimes C_2, {\rm PSU}_3(8) \rtimes C_6\}$. The corresponding vertex stabilizer amalgams are of local characteristic $3$ but are not weak $(B,N)$-pairs. They are the first examples of this kind where there exists a vertex $z$ such that the extension of $G_z/O_p(G_z^{[1]})$ over $O_p(G_z^{[1]})$ is non-split.