Let $V$ be an $8$-dimensional vector space over $GF(4)$ equipped with a nondegenerate plus type quadratic form, $G$ be an almost simple group with socle $L=\Omega^+(V)$, and $H$ be a soluble subgroup of $G$. If $H$ is transitive on nondegenerate 1-subspaces of $V$, then does there always exist a maximal parabolic of $G$ containing $H$?
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$\begingroup$ I did some computer calculations in Magma, and I found that the only soluble subgroups of $\Gamma O^+_8(4)$ that were transitive on the 16320 nondegenerate 1-subspaces had nontrivial normal 2-subgroups, so the answer would appear to be yes. $\endgroup$– Derek HoltCommented Jul 15, 2013 at 8:52
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$\begingroup$ @DerekHolt:By THEOREM A of Liebeck, Praeger and Saxl, The maximal factorizatins of the finite simple groups and their automorphism groups, I would be able to show that if $H$ is not contained in any maximal parabolic subgroup then $H\cap L$ is contained in $(L_2(16)\times L_2(16)).2^2$ (as $C_2$ or $C_3$ subgroup of $L$). Does this help? $\endgroup$– Binzhou XiaCommented Jul 16, 2013 at 2:38
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$\begingroup$ @DerekHolt: What if $G$ contains a graph automorphism? $\endgroup$– Binzhou XiaCommented Jul 18, 2013 at 10:58
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$\begingroup$ @DerekHolt:$N_G((PSL_2(16)\times PSL_2(16)).2^2)$ is maximal in $G$ if and only if $G=L$ or $L.2$, so by your computation for $\Gamma O_8^+(4)$, my statement holds. Thank you so much! $\endgroup$– Binzhou XiaCommented Jul 18, 2013 at 14:30
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