Superperfectness is finite direct product-closed

Statement for two groups
Suppose $$G_1$$ and $$G_2$$ are superperfect groups (possibly isomorphic, possibly non-isomorphic). Then, the external direct product $$G_1 \times G_2$$ is also a superperfect group.

Statement for finitely many groups
Suppose $$n > 1$$ with $$n$$ a positive integer. Suppose $$G_1,G_2,\dots,G_n$$ are superperfect groups (possibly isomorphic, possibly non-isomorphic). Then, the external direct product $$G_1 \times G_2 \times \dots \times G_n$$ is also a superperfect group.

Examples
Since the smallest nontrivial superperfect group is SL(2,5) which has order 120, the smallest nontrivial example of a superperfect group obtained as a direct product of smaller superperfect groups is the direct product of SL(2,5) and SL(2,5), which is a group of order 14400.

Facts used

 * 1) uses::Kunneth formula for group homology

Proof for two groups
The proof basically follows from Fact (1), which allows us to compute the homology groups of a direct product in terms of the homology groups of the direct factors.

Proof for finitely many groups
This follows from the proof for two groups and using mathematical induction.