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For any group $G$ of order $36$ and any subgroup $H$ of $G$ of order $4$, is $H$ contained in $Z(G)$?

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There is a nontrivial operation of $C_3$ on the Klein four group $C_2^2$. This gives us a group $((C_2^2)\rtimes C_3)\times C_3$ where $C_2^2$ is not in the center.

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