Exterior square of a group

Definition
The exterior square of a group $$G$$ (sometimes also called the non-abelian exterior square), denoted $$G \wedge G$$ or $$\bigwedge^2G$$, can be defined in the following equivalent ways:


 * 1) It is the exterior product of groups of $$G$$ with itself, where both copies of $$G$$ are viewed as living inside the same group $$G$$ as the whole group.
 * 2) It is the defining ingredient::derived subgroup of any defining ingredient::Schur covering group of $$G$$. Note that the Schur covering groups need not be isomorphic groups, but they are isoclinic groups, so the definition is independent of the choice of Schur covering group.
 * 3) If $$G \cong F/R$$ where $$F$$ is a free group and $$R$$ is a normal subgroup of $$F$$, it is, up to isomorphism, the same as $$[F,F]/[F,R]$$ (for more on this perspective, see Hopf's formula for Schur multiplier).
 * 4) It can be defined as follows:
 * 5) * Denote by $$\mathcal{F}$$ the free group on the set $$G \times G$$.
 * 6) * For any central extension $$E$$ with quotient group $$G$$, the commutator map defines a set map $$\omega:G \times G \to [E,E]$$ and therefore a group homomorphism $$\Omega_E: \mathcal{F} \to [E,E]$$.
 * 7) * Let $$\mathcal{R}$$ the intersection of the kernels of all group homomorphisms $$\Omega_E$$ for all possible central extensions $$E$$. Note that although the collection of all possible homomorphisms is too big to be a set, the collection of all possible kernels is a set, so the intersection is defined.
 * 8) * The group $$G \wedge G$$ is defined as the quotient group $$\mathcal{F}/\mathcal{R}$$. The image of the freely generating element $$(x,y)$$ (with $$x,y \in G$$) is denoted $$x \wedge y$$.

Facts

 * Commutator map is homomorphism from exterior square to derived subgroup: For any group $$G$$, the commutator map gives a surjective homomorphism $$G \wedge G \to [G,G]$$.
 * Commutator map is homomorphism from exterior square to derived subgroup of central extension: Even better, if $$G$$ is the quotient group of a group $$E$$ by a central subgroup, the commutator map in $$E$$ descends to a homomorphism $$G \wedge G \to [E,E]$$.
 * Schur multiplier is kernel of commutator map homomorphism from exterior square to derived subgroup: The kernel of the homomorphism $$G \wedge G \to [G,G]$$ mentioned above is the Schur multiplier $$M(G)$$, which can also be written as $$H_2(G;\mathbb{Z})$$. In particular, this tells us that $$G \wedge G$$ is a group extension with normal subgroup $$M(G)$$ and quotient group $$[G,G]$$.
 * Exterior square of finite group is finite: This follows from the above and the fact that Schur multiplier of finite group is finite. The proof generalizes to showing that exterior product of finite groups is finite, which can also be used to show that tensor product of finite groups is finite.
 * For an abelian group, this coincides with the exterior square of abelian group.

Group families
For various families, the exterior square can be described in general terms based on the family.

Groups of a particular order
Prime numbers are not included below, because they are already covered under the group cohomology of finite cyclic groups.