  Each group G of nxn permutation matrices has a corresponding permutation
polytope, P(G):=conv(G) in R^{nxn}. We relate the structure of P(G) to the
transitivity of G. In particular, we show that if G has t nontrivial orbits,
then min{2t,floor(n/2)} is a sharp upper bound on the diameter of the graph of
P(G); so if G is transitive, the diameter is at most 2. We also show that P(G)
achieves its maximal dimension of (n-1)^2 precisely when G is 2-transitive. We
then extend results of I. Pak on mixing times for a random walk on P(G). Our
work depends on a new result for permutation groups involving writing
permutations as products of indecomposable permutations.
