Exterior square preserves powering for nilpotent groups

Definition
Suppose $$\pi$$ is a set of primes and $$G$$ is a $$\pi$$-powered nilpotent group, i.e., $$G$$ is both a nilpotent group and a $$\pi$$-powered group. Then, the exterior square $$G \wedge G$$ is also a $$\pi$$-powered nilpotent group.

Facts used

 * 1) uses::Schur multiplier preserves powering for nilpotent groups
 * 2) uses::Derived subgroup is divisibility-closed in nilpotent group
 * 3) uses::Powering is central extension-closed

Proof of powering
Recall that we have a short exact sequence, where $$M(G)$$ denotes the Schur multiplier of $$G$$:

$$0 \to M(G) \to G \wedge G \to [G,G] \to 1$$

where the image of $$M(G)$$ in $$G \wedge G$$ is a central subgroup of $$G \wedge G$$.

Fact (1) says that $$M(G)$$ is $$\pi$$-powered. Fact (2) says that $$[G,G]$$ is $$\pi$$-powered. Fact (3) then gives us that $$G \wedge G$$ is $$\pi$$-powered.

Proof of nilpotency
Since nilpotency is subgroup-closed, $$[G,G]$$ is nilpotent. Since $$G \wedge G$$ is a central extension, it is also nilpotent. In fact, the nilpotency class of $$G \wedge G$$ is at most $$\lceil c/2 \rceil$$. This follows from the fact that lower central series is strongly central applied to any Schur covering group of $$G$$.