Number of conjugacy classes in a direct product is the product of the number of conjugacy classes in each factor

For two groups
Suppose $$G_1$$ and $$G_2$$ are groups. Suppose the number of conjugacy classes in $$G_1$$ is $$a_1$$ and the number of conjugacy classes in $$G_2$$ is $$a_2$$.

Then, the number of conjugacy classes in the external direct product $$G_1 \times G_2$$ is the product $$a_1a_2$$.

Note that this says that when either $$G_1$$ or $$G_2$$ has infinitely many conjugacy classes, so does $$G_1 \times G_2$$, and the product statement is interpreted in the sense of infinite cardinals. If both $$G_1$$ and $$G_2$$ have finitely many conjugacy classes, then so does $$G_1 \times G_2$$.

For multiple groups
Suppose $$G_1,G_2,\dots,G_n$$ are groups. Suppose the number of conjugacy classes in $$G_1$$ is $$a_1$$, in $$G_2$$ is $$a_2$$, and so on.

Then, the number of conjugacy classes in the external direct product $$G_1 \times G_2 \times \dots \times G_n$$ is the product $$a_1a_2 \dots a_n$$.

If all the $$a_i$$ are finite, then so is the number of conjugacy classes in the direct product, and the multiplication is the usual multiplication of natural numbers. If one or more of the $$a_i$$ is infinite, then the number of conjugacy classes in the direct product is also infinite, and the product is interpreted in terms of infinite cardinals.

Note that the statement also holds for an internal direct product, because of equivalence of internal and external direct product.

Related facts

 * Number of conjugacy classes in a subgroup may be more than in the whole group
 * Number of conjugacy classes in a quotient is less than or equal to number of conjugacy classes of group