C-groups of high rank for the symmetric groups

TL;DR

This paper classifies high-rank C-groups for symmetric groups and links them to hypertopes, extending the understanding of their geometric and combinatorial structures beyond regular polytopes.

Contribution

It provides a classification of C-groups of ranks n-1 and n-2 for symmetric groups and establishes their correspondence with hypertopes, generalizing previous results on string C-groups.

Findings

Classified C-groups of ranks n-1 and n-2 for S_n

Established correspondence between these C-groups and hypertopes

Extended results from regular polytopes to more general geometries

Abstract

We classify C-groups of ranks $n-1$ and $n-2$ for the symmetric group $S_n$. We also show that all these C-groups correspond to hypertopes, that is, thin, residually connected flag-transitive geometries. Therefore we generalise some similar results obtained in the framework of string C-groups that are in one-to-one correspondence with abstract regular polytopes.