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I'm interested in examples of "big" finite subgroups of $G(\mathbb C)$ for $G=\mathrm{Sp}_{2n}, \mathrm{SO}_{2n+1}$. A subgroup $H$ of $G(\mathbb C)$ is said to be big if the associated representation of $H$ is irreducible. For example, consider the Weyl group $W\cong N_G(T)/T$, we can take $H$ to be any finite subgroup of $N_G(T)$ for which $H/(H\cap T) \cong W$. Is there any other examples of big finite subgroups of $G(\mathbb C)$?

The work of Griess-Ryba http://www.math.lsa.umich.edu/~rlg/researchandpublications/pdffiles1/qfseag.pdf gives a classification of quasi-simple groups which embed into exceptional algebraic groups. But I don't know of any similar work for classical groups.

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    $\begingroup$ For $GL_n$ you're just asking about groups with an $n$-dimensional irrep, for $O_n$ and $SP_n$ you're asking for an $n$-dimensional irrep with FS-indicator $\pm 1$ (respectively). That's clearly too broad a question to hope to say much. You might be interested in Jordan's Theorem or this paper. $\endgroup$
    – Noah Snyder
    Commented Nov 22, 2017 at 0:39
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    $\begingroup$ Noah means faithful irrep, although this condition is automatic if $G$ is simple. $\endgroup$
    – Qiaochu Yuan
    Commented Nov 22, 2017 at 1:12
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    $\begingroup$ If you want to see how difficult this problem can be check out Geoff Robinsons DPhil Thesis. There he classifies the irreducible subgroups of $\mathrm{GL}_{11}(\mathbb{C})$. It's quite the feat considering it's pre-classification of finite simple groups. homepages.abdn.ac.uk/d.j.benson/papers/r/robinson/thesis.dvi $\endgroup$
    – Jay Taylor
    Commented Nov 22, 2017 at 15:38

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Take a look to the following list made by Tim Dokchitser.

Faithful irreducible representation
https://people.maths.bris.ac.uk/~matyd/GroupNames/R.html

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