  Although the conjugacy classes of the general linear group are known, it is
not obvious (from the canonic form of matrices) that two permutation matrices
are similar if and only if they are conjugate as permutations in the symmetric
group, i.e. that conjugacy classes of S_n do not unite under the natural
representation. We prove this fact, and give its application to the enumeration
of fixed points under a natural action of S_n x S_n. We also consider the
permutation representations of S_n which arise from the action of S_n on
k-tuples, and classify which of them unite conjugacy classes and which do not.
