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Why is $ q^2+1 $ the minimal degree of a faithful permutation representation of the Suzuki group $ Sz(q) $?

Context: I am trying to show that $ Sz(q) $ never embeds in the Weyl group of $ SU(d) $, where $ d $ is the minimal degree of an irreducible complex representation of $ Sz(q) $. This follows from the result about $ q^2+1 $ being the minimal permutation degree of $ Sz(q) $. But I would like to understand the $ q^2+1 $ result better.

I'm also curious in general if there is a good reference for the minimal permutation degrees of the infinite families of finite simple groups. Is there a unified approach to determining the minimal permutation degree?

a bit like this question Minimal degree faithful permutation representations of some finite simple groups, but I'm more interested in families of groups than the sporadic ones

Ian Gershon Teixeira
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    For a finite simple group, the minimum permutation degree is equal to minimum index of a maximal subgroup. Maximal subgroups of simple groups are extremely well studied and I think the largest ones are known for all finite simple groups. Most sources of information about simple groups will contain information about the maximal subgroups, so you can often recover what you want from these. – verret Oct 08 '24 at 19:11

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For the minimal degree permutation representations of the finite simple groups of Lie type, see Table 4 of

S. Guest, J. Morris, C.E. Praeger, and P. Spiga. On the maximum orders of elements of finite almost simple groups and primitive permutation groups. Trans. Amer. Math. Soc. 367 (2015), 7665-7694.

You can find the paper on arXiv here.

Most of the results in the table are cited from other sources, but many earlier such tables had contained mistakes, so the authors of this paper thought it a good idea to publish a citable correct table!

Derek Holt
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