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Although every non-simple group can be described as a group extension, it is well-known that not every non-simple group can be expressed as a semidirect product of smaller groups.

For example, a cyclic group of order $p^2$ is an extension of $\mathbb{Z}_p$ by $\mathbb{Z}_p$, but cannot be written as a semidirect product. The quaternion group $Q_8$ is also not a semidirect product. Indeed, this seems to be very common property for $p$-groups.

So my question is:

What's the smallest non-simple group $G$ such that:

  • $G$ isn't a $p$-group, and

  • $G$ can't be expressed as a semidirect product of smaller groups?

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What about $\mathtt{SmallGroup}(48,28)$, which is a group that is similar to ${\rm GL}(2,3)$ but has no non-central elements of order $2$? It is one of the two Schur covering groups $2 \cdot S_4$ of $S_4$ (the other one being ${\rm GL}(2,3)$).

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  • $\begingroup$ I'm a little confused. Is this an answer or a question? $\endgroup$ – zibadawa timmy Jun 22 '15 at 8:27
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    $\begingroup$ According to my magma computation, Derek is right and, SmallGroup(48,28) is the correct answer. (SL(2,5) is the second smallest and (144,31) the third smallest.) $\endgroup$ – verret Jun 22 '15 at 8:33
  • $\begingroup$ @zibadawatimmy Yes, it is an answer to the question. I was politely disputing the claim that ${\rm SL}(2,5)$ is the correct answer! Sorry to be confusing! $\endgroup$ – Derek Holt Jun 22 '15 at 9:21
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    $\begingroup$ You are correct. There was a bug in my GAP code! $\endgroup$ – Jim Belk Jun 22 '15 at 15:44

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