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Let $G$ be a finite group and $G \wedge G$ denote the exterior square of $G$. It is well known that the second integral homology $H_2(G,\mathbb{Z})$ is the kernel of homomorphism $x \wedge y \mapsto [x,y]$. Graham Ellis and others have studied the hyper relative-homology group as the natural extension in this context. I want to know whether the higher integral homology is related to (some kind of) generalization of exterior square of $G$.

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