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The inverse Galois problem conjectures that every finite group is (isomorphic to) the Galois group of some Galois extension of $\mathbb Q$, however it is not known.

My question is: what is the smallest finite group such that it is not known whether it is the Galois Group of such an extension?

Shaun
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Régis
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  • https://en.wikipedia.org/wiki/Inverse_Galois_problem#Partial_results seems to contain what you want (no idea whether it's state of the art) – user8268 Sep 05 '18 at 22:05
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    This is a research question, you'd better ask it on MO – Tsemo Aristide Sep 05 '18 at 22:11
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    For examples, see the first part of the Wiki. article on the subject. The "Bible" is of course the book of Malle-Matzat (1999), but a good not too technical and more recent (2011= bicentennial of Galois' birth) account can be found in www.galois.ihp.fr/wp-content/uploads/2012/03/P.-Debes.pdf – nguyen quang do Sep 06 '18 at 06:52
  • Thx nguyen quang and thx Tsemo Aristide for the tip. I did not know that MO is for reasearch questions. – Régis Sep 06 '18 at 11:52

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