Finite groups as prescribed polytopal symmetries

We study properties of the realizations of groups as the combinatorial automorphism group of a convex polytope. We show that for any non-abelian group G with a central involution there is a centrally symmetric polytope with G as its combinatorial automorphisms. We show that for each integer n, there are groups that cannot be realized as the combinatorial automorphisms of convex polytopes of dimension at most n. We also give an optimal lower bound for the dimension of the realization of a group as the group of isometries that preserves a convex polytope.