Affine group over a finite field

The one-dimensional affine group over ${\mathbb{F}}_q$ consists of maps $x\mapsto ax+b$ with $a\ne 0$.

For an odd prime $p$, the splitting fieldof $X^p-2$ has Galois groupThe translation subgroup multiplies $2^{1/p}$ by powers of ${\zeta }_p$, while the multiplicative subgroup acts on ${\zeta }_p$ by cyclotomic automorphisms.

Let $Q$ be the subgroup of nonzero squares in ${\mathbb{F}}_{11}^{\times }$. The affine mapsform the nonabelian semidirect product $C_{11}⋊C_5$, with multiplication

This group has seven conjugacy classes: the identity; two classes of five nonidentity translations, distinguished by square class of the translation parameter; and, for each of the four $r\in Q\setminus \{1\}$, one class $\{(r,b):b\in {\mathbb{F}}_{11}\}$ of size eleven.