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A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$, or equivalently, if any $1$-dimensional complex representation is trivial.

A finite group $G$ is linearly primitive if it has a faithful complex irreducible representation.

Question: Are the perfect finite groups linearly primitive?

Remark: the finite simple groups are perfect and linearly primitive.

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No. Any linearly primitive group must have cyclic center by Schur's lemma, but there's no reason a perfect group should have this property. I think that, for example, the universal central extension of $A_5 \times A_5$ is perfect but has center $C_2 \times C_2$.

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    $\begingroup$ More precisely $G = {\rm SL}(2,5) \times {\rm SL}(2,5)$ is a counterexample. $\endgroup$ – Derek Holt Sep 16 '15 at 16:57
  • $\begingroup$ @DerekHolt: Is there a counterexample which is not a (semi-)direct product? $\endgroup$ – Sebastien Palcoux Sep 16 '15 at 18:48
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    $\begingroup$ The perfect group number 4 of order 3840 (which is called A5 2^4 E ( 2^1 x N 2^1 in the book of Holt/Plesken) has a centre of type $C_2\times C_2$ but cannot be written as a semidirect product of any proper subgroups. (This is the smallest example.) $\endgroup$ – ahulpke Sep 16 '15 at 19:30

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