Conjugacy classes of group products (D&F 4.3.3)

Find all conjugacy classes and their sizes in the following groups:
(a) \(Z_2 \times S_3\)
(b) \(S_3 \times S_3\)
(c) \(Z_3 \times A_4\)

First, let \(K_G\) be the set of all conjugacy classes of \(G\), and denote elements of cyclic groups as \(a^k\) where \(k \in \mathbb{Z}\). Second, observe that \((a^i)^{(a^j)} = a^j a^i a^{-j} = a^i\).

(a) Let \((a^i,x),(a^j,y) \in Z_2 \times S_3\). As noted above, \((a^i,x)^{(a^j,y)} = (a^i,x^y)\). Thus we can identify \((a^i,x)\) with \(x\) and \((a^j,y)\) with \(y\) and conclude that the set of all conjugacy classes of the set \(\{(a^i,b) \mid b \in S_3\}\) (denoted \(K_i\)) is precisely that of \(S_3\), but with each individual permutation \(b\) replaced by \((a^i,b)\). Thus \(K_{Z_2 \times S_3} = K_0 \cup K_1\). (It would be too troublesome to list it out explicitly.)

(b) Suppose \((a,b)\) and \((c,d)\) are conjugate in \(S_3 \times S_3\). Then \((c,d) = (a,b)^{(x,y)} = (a^x,b^y)\) for some \((x,y) \in S_3 \times S_3\), so \(a\) and \(c\) as well as \(b\) and \(d\) are conjugate, i.e. have the same cycle type. In other words, \(K_{S_3 \times S_3}\) consists of the Cartesian products of all ordered pairs containing sets in \(K_{S_3}\).

(c) As in (a), let \(K_i\) be the set of conjugacy classes of the set \(\{(a^i,b) \mid b \in A_4\}\). Then \(K_{Z_3 \times A_4} = K_0 \cup K_1 \cup K_2\).