String C-group representations of transitive Groups: a case study with degree \(11\)

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.