# Structure of the outer automorphism group of $D_n(q)$

From the ATLAS, I know that the outer automorphism group of the Chevalley group $$D_n(q)$$, $$q=p^f$$ for some prime $$p$$ and some $$n$$ even and $$n>4$$, is a semidirect product of three groups, $$(C_d \times C_d) \rtimes (C_f \times C_g)$$, where $$d=(2,q-1)$$ (the "diagonal" automorphisms), $$f$$ is such that $$q=p^f$$ (the "field" automorphisms) and $$g=2$$ (the graph automorphisms), so $$\operatorname{Out}(D_n(q))= (C_2 \times C_2) \rtimes (C_f \times C_2)$$

What I want to know is: when $$f=3k$$ for some $$k \in \mathbb{N}$$, does $$C_f$$ act on $$C_2 \times C_2$$? Equivalently, do the field automorphisms and the diagonal automorphisms commute?

I am also interested in the $$n=4$$ case, when $$\operatorname{Out}(D_4(q))= (C_2 \times C_2) \rtimes (C_f \times S_3)$$ and I ask the same question for $$C_f$$, but also for $$C_3 \leq S_3$$.

The subgroup $$(C_d \times C_d) \rtimes C_g$$ is dihedral of order $$8$$, and since $$C_f$$ commutes with $$C_g$$, it follows that $$C_f$$ must commute with $$C_d \times C_d$$, because $${\rm Aut}(C_2 \times C_2) \cong S_3$$, not $$C_6$$.
On the other hand, when $$n=4$$, the $$S_3$$ subgroup acts faithfully on $$C_d \times C_d$$, and the subgroup $$(C_d \times C_d) \rtimes S_3$$ is isomorphic to $$S_4$$. This is the only situation in which the outer automorphism group of a finite simple group has derived length $$3$$.
• Any insight on why $(C_d \times C_d) \rtimes C_g$ is dihedral? I was having trouble making these three different types of automorphism "interact" – AnalysisStudent0414 Apr 2 at 21:19
• I don't have any magical insight about this. You can study the automorphisms of the orthogonal matrix groups $\Omega^+_{2n}(q)$, and work it out, but it's not particularly easy. IIRC, one of the diagonal automorphisms lies in ${\rm SO}^+_{2n}(q)$, the graph automorphism lies in ${\rm GO}^+_{2n}(q)$, and the another diagonal automorphism has the effect of multiplying the from by a non-square in the field. – Derek Holt Apr 2 at 21:29
• Will do! (I think that I need to look at these actions in detail anyway, since this copy of $S_4$ could mess up something I am working on). Thank you very much – AnalysisStudent0414 Apr 2 at 21:35