General semilinear group of degree one

Definition
Let $$K$$ be a field. The general semilinear group of degree one over $$K$$, denoted $$\Gamma L(1,K)$$, is defined as the general semilinear group of degree one over $$K$$. Explicitly, it is the external semidirect product:

$$\Gamma L (1,K) = GL(1,K) \rtimes \operatorname{Aut}(K) = K^\ast \rtimes \operatorname{Aut}(K) $$

where $$GL(1,K) = K^\ast$$ is the multiplicative group of $$K$$, and $$\operatorname{Aut}(K)$$ denotes the group of field automorphisms of $$K$$.

If $$k$$ is the prime subfield of $$K$$, and $$K$$ is a Galois extension of $$k$$ (note that this case always occurs for $$K$$ a finite field), then $$\operatorname{Aut}(K) = \operatorname{Gal}(K/k)$$ and we get:

$$\Gamma L (1,K) = GL(1,K) \rtimes \operatorname{Gal}(K/k) = K^\ast \rtimes \operatorname{Gal}(K/k) $$

If $$K$$ is a finite field of size $$q$$, this group is written as $$\Gamma L(1,q)$$.

For a finite field
Suppose $$K$$ is a finite field of size $$q$$, where $$q$$ is a prime power with underlying prime $$p$$, so that $$q = p^r$$ for a positive integer $$r$$. $$p$$ is the characteristic of $$K$$. In this case, $$K^\ast$$ is cyclic of order $$q - 1$$ (see multiplicative group of a finite field is cyclic) and $$\operatorname{Gal}(K/k)$$ is cyclic of order $$r$$ (generated by the Frobenius map $$a \mapsto a^p$$).

Thus, $$\Gamma L(1,K)$$ is a metacyclic group of order $$r(q - 1)$$ with presentation:

$$\langle a,x \mid a^q = a, x^r = e, xax^{-1} = a^p \rangle$$

(here $$e$$ denotes the identity element).