  We investigate the combinatorics and geometry of permutation polytopes
associated to cyclic permutation groups, i.e., the convex hulls of cyclic
groups of permutation matrices. We give formulas for their dimension and vertex
degree. In the situation that the generator of the group consists of at most
two orbits, we can give a complete combinatorial description of the associated
permutation polytope. In the case of three orbits the facet structure is
already quite complex. For a large class of examples we show that there exist
exponentially many facets.
