Polytopes derived from sporadic simple groups

In this article, certain of the sporadic simple groups are analysed, and the polytopes having these groups as automorphism groups are characterised. The sporadic groups considered include all with order less than 4030387201, that is, all up to and including the order of the Held group. Four of these simple groups yield no polytopes, and the highest ranked polytopes are four rank 5 polytopes each from the Higman-Sims group, and the Mathieu group $M_{24}$.