Groups of Ree type in characteristic 3 acting on polytopes

Abstract

Every Ree group $R(q)$, with $q\ne 3$ an odd power of 3, is the automorphism group of an abstract regular polytope, and any such polytope is necessarily a regular polyhedron (a map on a surface). However, an almost simple group $G$ with $R(q)<G\le \mathsf{Aut}(R(q))$ is not a C-group and therefore not the automorphism group of an abstract regular polytope of any rank.