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The Maximal Subgroups of the Low-Dimensional Finite Classical Groups by Colva M. Roney-Dougal, Derek F. Holt, John N. Bray

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Proposition 3.7.3 Let n = 8 and let H be a image-subgroup of Ω, preserving a decomposition into four subspaces. Then H is maximal amongst the geometric subgroups of Ω if and only if q > 2. If q = 2 then H does not extend to a novel maximal subgroup.

Proof The statement for q = 2 is immediate from Proposition 2.3.6, so we assume that q > 2. Assume, by way of contradiction, that image, where K is maximal amongst the geometric subgroups of Ω and is not of the same type as H. Note that if q > 3 then H ≅ SL2(q)4.

It is immediate from Lemma 2.3.7 (iv) that . Suppose ...

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