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$\DeclareMathOperator\SO{SO}$The smallest degree of a nontrivial complex representation of the monster group $ M $ is $ 196883 $. This irrep has Schur indicator $ 1 $, so the image must lie in the orthogonal group. In fact, since $ M $ is perfect the image must lie in the special orthogonal group. Thus $ M $ is a subgroup of the Lie group $ \SO(196883) $.

Is $ M $ maximal among the closed subgroups of $ \SO(196883) $?

To give a very small example of why I think this might be true consider the simple group $ A_5 $. The smallest degree of a nontrivial complex representation of $ A_5 $ is $ 3 $. And the degree 3 irrep of $ A_5 $ has Schur indicator $ 1 $. So, for the same reasons described above, $ A_5 $ must be a subgroup of the Lie group $ \SO(3) $.

And it turns out that $ A_5 $ is maximal among the closed subgroups of $ \SO(3) $.

This seems interesting to me since $ \SO(3)/A_5 $ is the famous Poincare homology sphere. Perhaps $ \SO(196883)/M $ also has some interesting properties. For example maybe the isometry group of $ \SO(196883)/M $ acts primitively? Golubitsky - Primitive actions and maximal subgroups of Lie groups.

This question is cross posted from MSE where it has been up for over a week with 7 upvotes but no answers.

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    $\begingroup$ There are some simple cases to test before extrapolating from a single case to the most complicated. $\endgroup$
    – YCor
    Commented Feb 15 at 21:42
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    $\begingroup$ Maybe it would be better to ask whether this is true generally for finite simple groups in their minimal-degree representation (with SO or SU or Sp as appropriate depending on the Schur indicator), and if not then for which simple groups is it true. $\endgroup$
    – Sean Eberhard
    Commented Feb 16 at 10:02
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    $\begingroup$ See for example "M. W. Liebeck, G. M. Seitz, On the subgroup structure of classical groups. Invent. Math. 134 (1998), no. 2, 427-453.". By Theorem 1 there, to answer your question you need to look at all quasi-simple finite groups; see which ones have an irreducible representation of dimension 196883; check whether those contain $M$. $\endgroup$
    – testaccount
    Commented Feb 17 at 9:23
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    $\begingroup$ Yes, it should be. Although there are no tables of maximal subgroups that high, the only other option is that is contained in another simple group. But it isn't contained in a sporadic group, the smallest alternating group is way too big, and it cannot be in any classical group of smaller dimension. $\endgroup$
    – David A. Craven
    Commented Feb 17 at 23:39
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    $\begingroup$ @DavidA.Craven that's great! Would you be willing to write out your thoughts a little more and post it as an answer? $\endgroup$
    – Ian Gershon Teixeira
    Commented Feb 18 at 18:47

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