A Note on the Rank 5 Polytopes of M24

TL;DR

This paper investigates the maximal rank 5 regular polytopes associated with the Mathieu group M24, describing them via Curtis's MOG and providing bounds on their chamber graph diameters.

Contribution

It identifies and describes the four rank 5 polytopes of M24 using Curtis's MOG, and establishes an upper bound for their chamber graph diameters.

Findings

Four rank 5 polytopes of M24 identified and described.

Description of polytopes using Curtis's MOG.

Upper bound established for chamber graph diameters.

Abstract

The maximal rank of an abstract regular polytope for M24, the Mathieu group of degree 24, is 5. There are four such polytopes of rank 5 and in this note we describe them using Curtis's MOG. This description is then used to give an upper bound for the diameter of the chamber graphs of these polytopes.