Regular polytopes of rank $n/2$ for transitive groups of degree $n$

Abstract

Previous research established that the maximal rank of the abstract regular polytopes whose automorphism group is a transitive proper subgroup of $\mbox{S}_n$ is $n/2+1$. Up to isomorphism and duality, when $n\ge 12$, there are only two polytopes attaining this rank and they occur when $n/2$ is odd, and hence have even rank. In this paper, we investigate the case where the rank is equal to $n/2$ ($n\ge 14$). Our analysis suggests that reducing the rank by one results in a substantial increase in the number of regular polytopes.