It is well-known that if $G$ is a nilpotent group, then every non-trivial normal subgroup of $G$ has non-trivial intersection with $Z(G)$. I would like to find examples of non-nilpotent finite groups $G$ so that every non-trivial normal subgroup of $G$ intersects $Z(G)$ non-trivially. In particular, I am interested in examples having small order. I have checked some of the well-known groups of small order to see whether they satisfy the condition in question, and those that I checked do not.

  • $\begingroup$ I'm still doing the computations, but maybe this: take first $H = C_9 \rtimes C_3$ where $C_3$ acts non-trivially over $C_9$. Then, take $K = C_2 \times C_2$ and make $K$ act on $H$ by first quotienting out one factor $C_2$ and making the other be the inversion on $C_9$. I believe that the center of the semidirect product $H \rtimes K$ is of order 6 and has the desired property. $\endgroup$ – Henrique Augusto Souza Jul 31 '18 at 6:24

The group $G={\rm SL}(2,3)$ of order $24$ has this property and is probably the smallest such example. It has structure $Q_8.3$ with $4$ Sylow $3$-subgroups, and its centre has order $2$ with $G/Z(G) \cong A_4$. Its only normal subgroups are $1$, $Z(G)$, $Q_8$ and $G$.

Non-simple quasisimple groups, such as ${\rm SL}(2,q)$ for odd prime powers $q$, are non-solvable examples. All of their proper normal subgroups are central.

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  • $\begingroup$ Great answer, thanks! $\endgroup$ – Zilliput Jul 31 '18 at 23:52

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