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?

... and the reason it doesn't matter is because the two choices are conjugate in $S_3$.
– Noam D. ElkiesJun 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?
– wisonJun 17 '11 at 5:51