Order of group $A\Gamma L_d(F)$ of Affine semi-linear transformation.

Question

Question What is the order of a group $A\Gamma L_d(q)$ of Affine semilinear transformation ? (here $q$ is a prime power of some prime $p$).

Answer 1

Write $q=p^f$. The group is a semidirect product of three groups:

• a $d$-dimensional $\mathbb{F}_q$-vector space of size $q^d$
• $\mathrm{GL}(d,q)$ of size $(q^d-1)(q^d-q)\ldots(q^d-q^{d-1})$.
• a cyclic group of order $f$ generated by the Frobenius automorphism of $\mathbb{F}_q$.

Now multiply the three factors.