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we have known

there are 3 simple groups arising from $SO(8)$, i.e.$SO(8,F_{q})$, $2D_4(q,q^2)$,$3D_4(q,q^3),$ so I want to know For the $3D_4(q,q^3),$, what is the corresponding Frobenius?

In fact, I know the non-split Frobenius map is the standard Frobenius followed by a graph automorphism

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# rationalpointsofnilpotentorbitson$D_4$non-splitFrobeniusmap

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# Frobeniusmorphismrationalpointsofnilpotentorbits on $D_4$

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