Affine Galois group of the splitting field of x to the p minus two
= Affine Galois group of the splitting field of x to the p minus two
For an odd prime $p$, the splitting field
$$
\mathbb Q(\zeta_p,2^{1/p})
$$
of $X^p-2$ has Galois group
$$
C_p\rtimes\mathbb F_p^\times
\cong\operatorname{AGL}_1(\mathbb F_p).
$$
The translation subgroup multiplies $2^{1/p}$ by powers of $\zeta_p$, while the multiplicative subgroup acts on $\zeta_p$ by cyclotomic automorphisms.