Define the outer-semi direct product in the standard way


I found out that $Q_8$ is the smallest finite group that is neither finite simple, nor can be expressed as an outer-semi-direct product of other groups.

What is the next smallest finite group (/set of smallest finite groups if they share the same order) that cannot be expressed an outer-semi-direct product?

(i'm not looking for a proof, just the names of these group(s))

  • 3
    $\begingroup$ What about a cyclic group of order $4$? $\endgroup$ – Eric Wofsey Feb 19 '17 at 6:15
  • $\begingroup$ ah i'm silly, and thanks! that works $\endgroup$ – frogeyedpeas Feb 19 '17 at 6:28

It's not true that $Q_8$ is the smallest such group: the smallest such group is a cyclic group of order $4$. The next smallest such group after $Q_8$ is a cyclic group of order $9$.


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