  Quasi-vertex-transitive maps are the homogeneous maps on the plane with
finitely many vertex orbits under the action of their automorphism groups. We
show that there exist quasi-vertex-transitive maps of types $[p^3, 3]$ for $p
\equiv 1$ (mod $6$), but there doesn't exist vertex-transitive map of such
types. In particular, we determine the surface with the lowest possible genus
that admit a polyhedral map of type $[5^3, 3]$.
