# Finite subgroups of classical groups over $\mathbb C$

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.

• 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. – Noah Snyder Nov 22 '17 at 0:39
• Noah means faithful irrep, although this condition is automatic if $G$ is simple. – Qiaochu Yuan Nov 22 '17 at 1:12
• 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 – Jay Taylor Nov 22 '17 at 15:38