  A permutation polytope is the convex hull of a group of permutation matrices.
In this paper we investigate the combinatorics of permutation polytopes and
their faces. As applications we completely classify permutation polytopes in
dimensions 2,3,4, and the corresponding permutation groups up to a suitable
notion of equivalence. We also provide a list of combinatorial types of
possibly occuring faces of permutation polytopes up to dimension four.
