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/ .
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