Suppose $G$ is a finite perfect group, $N$ is an Abelian minimal normal subgroup of $G$ and $$G/N=SL_2(q),$$ where $q=2^f$ for some integer $f\geq5$.
What can we say about the order of $N$?
Thanks!
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2$\begingroup$ There are no perfect central extensions of this form. So you are asking for the dimensions of the nontrivial irreducible modules of ${\rm SL}_2(q)$ over prime fields. Those are certainly known. $\endgroup$– Derek HoltCommented Aug 14, 2017 at 8:26
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$\begingroup$ @Derek Holt, thanks for your comment. Could you please introduce a reference for irreducible modules of SL2(q) over prime fields? $\endgroup$– asadCommented Aug 21, 2017 at 6:42
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