Let $T_1$ and $T_2$ be disjoint finite sets and let $G_1$ and $G_2$ be, respectively, some groups of permutations of this sets. Direct sum $G_1 \bigotimes G_2$ acts on $T_1 \cup T_2$:
$$ \langle g_1,g_2 \rangle t = \begin{cases} g_1 t&\text{if }t \in T_1,\\ g_2 t&\text{if } t \in T_2. \end{cases} $$
Prove that cyclic index of this operation can be expressed by the formula: $$ I_{G_1 \bigotimes G_2}=I_{G_1}I_{G_2} $$
The cycle index of a permutation group is a sum of monomials associated to the elements of the group.
Let $m_g$ be the monomial associated to $g\in G$. Show that for $(g_1,g_2)\in G_1\times G_2$ we have the relation $m_{(g_1,g_2)}=m_{g_1}m_{g_2}$. Do you see how to leverage this for the cycle index relation?