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I'd like to know the largest almost quasisimple group $G$ which acts faithfully and irreducibly on the spin module for an odd dimensional orthogonal group $\Omega(2k+1,q)$, or on each of the two half-spin modules for an even dimensional orthogonal group $\Omega^\epsilon(2k,q)$.

I've convinced myself using the weights of the representation in the corresponding algebraic group that $G/Z(G)$ contains no field or graph automorphisms.

I'm also half convinced that the correct answer is the Clifford group or the spin group $\mathrm{Spin}^\epsilon(n,q)$, but I'm not sure.

Any help or guidance would be much appreciated!

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    $\begingroup$ The examples $Spin_5(q)\cong Sp_4(q)$ and $Spin_6^+(q)\cong SL_4(q)$ might guide you to the right answer. $\endgroup$
    – Richard Lyons
    Commented Nov 19, 2017 at 15:01

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