Exterior product of finite groups is finite

Statement
Suppose $$G,H$$ are both finite groups that arise as normal subgroups inside a group $$Q$$. (We can assume without loss of generality that $$Q$$ is finite, because we can replace $$Q$$ by the subgroup $$GH$$ if necessary, whose size is given by the product formula).

Then, the exterior product of groups $$G \wedge H$$ is a finite group.

Related facts

 * Exterior square of finite group is finite
 * Exterior product of p-groups is p-group
 * Tensor product of finite groups is finite
 * Tensor product of p-groups is p-group