# String C-group representations of transitive Groups: a case study with degree $11$

## TL;DR

This paper provides a non-computer-assisted proof that certain transitive groups of degree 11 with specific string C-group representations are isomorphic to PSL_2(11), linking group theory with polytope automorphisms.

## Contribution

It offers a novel non-computer proof characterizing transitive groups of degree 11 with string C-group representations and introduces techniques for analyzing permutation representation graphs.

## Key findings

- Groups of degree 11 with rank 4 or 5 string C-group representations are isomorphic to PSL_2(11).
- The rank 4 string C-group corresponds to the automorphism group of the 11-cell polytope.
- Techniques developed can be applied to study other transitive groups.

## Abstract

In this paper we give a non-computer-assisted proof of the following result: if $G$ is an even transitive group of degree $11$ and has a string C-group representation with rank $r\in\{4,5\}$ then $G\cong\PSL_2(11)$. Moreover this string C-group is the group of automorphisms of the rank $4$ polytope known as the $11$-cell. The insights gained from this case study include techniques and observations concerning permutation representation graphs of string C-groups. The foundational lemmas yield a natural and intuitive understanding of these groups. These and similar approaches can be replicated and are applicable to the study of other transitive groups.