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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?

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So you must take the graph automorphism of order three. It does not matter which of the two you take. –  Wilberd van der Kallen Jun 15 '11 at 14:44
thank you very much –  wison Jun 17 '11 at 3:50
... and the reason it doesn't matter is because the two choices are conjugate in $S_3$. –  Noam D. Elkies Jun 17 '11 at 5:30
I knew this is classical result using Galois semi-automorphisms and 1-cohomology, furthermore, how to write down the Frobenius? –  wison Jun 17 '11 at 5:51

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