For the following two pairs of groups, compute the list of conjugacy class representatives for each group in the pair. For each part, compare and contrast the results for the two groups in the pair, with thoughtful and insightful comments.
(a) The full symmetric group on 5 symbols, S5, and the alternating group on 5 symbols, A5.
(b) The dihedral groups that are symmetries of a 7-gon and an 8-gon, D7 and D8.