# Extraspecial commutator-in-center subgroup is central factor

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Let $G$ be a finite $p$-group, i.e. a group of prime power order. Suppose $H$ is a subgroup of $G$ satisfying the following two conditions:
1. $H$ is an extraspecial group
2. $[G,H] \le Z(H)$
Then $HC_G(H) = G$, i.e., $H$ is a central factor of $G$.