Abstract
Abstract polytopes generalize the classical notion of convex polytopes to more general combinatorial structures. The most studied ones are regular and chiral polytopes, as it is well-known, they can be constructed as coset geometries from their automorphism groups. This is also known to be true for 2- and 3-orbit 3-polytopes. In this paper we show that every abstract n-polytope can be constructed as a coset geometry. This construction is done by giving a characterization, in terms of generators, relations and intersection conditions, of the automorphism group of a k-orbit polytope with given symmetry type graph. Furthermore, we use these results to show that for all k≠2, there exist k-orbit n-polytopes with Boolean automorphism groups, for all n≥3.
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