# The ranks of alternating string C-groups

## TL;DR

This paper classifies all possible ranks of string C-group representations of alternating groups $A_n$ for $n
geq 12$, providing a complete understanding of their structure within certain rank bounds.

## Contribution

It determines the exact values of $n$ for which $A_n$ can be represented as a string C-group of a given rank between 3 and $n/2$.

## Key findings

- Classifies string C-group ranks for all $A_n$, $n
geq 12$
- Provides explicit rank bounds for $A_n$ representations
- Completes the classification for $A_n$ with $n
eq 11$

## Abstract

In this paper, string C-groups of all ranks $3 \leq r \leq \frac{n}{2}$ are provided for each alternating group $A_n$, $n \geq 12$. As the string C-group representations of $A_n$ have also been classified for $n \leq 11$, and it is known that larger ranks are impossible, this paper provides the exact values of $n$ for which $A_n$ can be represented as a string C-group of a fixed rank.