Two atlases of abstract chiral polytopes for small groups

Abstract

We construct chiral abstract polytopes in two different ways. Firstly we seek them as quotients of regular polytopes arising from the Atlas of Small Regular Polytopes (http://www.abstract-polytopes.com/atlas/); the resulting atlas of chiral polytopes atlas is available on the website http://www.abstract-polytopes.com/chiral/. Secondly, for each almost simple group Γ such that S ≤ Γ ≤ Aut(S) where S is a simple group and Γ is a group of order less than 900,000 listed in the Atlas of Finite Groups, we give, up to isomorphism, the number of abstract chiral polytopes on which Γ acts regularly. The results have been obtained using a series of Magma programs. All these polytopes are made available on the third author's website, at http://math.auckland.ac.nz/~dleemans/CHIRAL/.