Conjugacy in permutation representations of the symmetric group

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 Sn 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 Sn x Sn. We also consider the permutation representations of Sn which arise from the action of Sn on ordered tuples and on unordered subsets, and classify which of them unite conjugacy classes and which do not.