Split special orthogonal group of degree two

Definition
Suppose $$K$$ is a field. The split special orthogonal group of degree two over $$K$$ is defined as the subgroup of the general linear group of degree two over $$K$$ given as follows:

$$\{ A \in GL(2,K) \mid \operatorname{det}(A) = 1, A\begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix}A^T = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix}\} $$

For characteristic not equal to two, an alternative definition, which gives a conjugate subgroup and hence an isomorphic group, is:

$$\{ A \in GL(2,K) \mid \operatorname{det}(A) = 1, A\begin{pmatrix} 1 & 0 \\ 0 & -1 \\\end{pmatrix}A^T = \begin{pmatrix} 1 & 0 \\ 0 & -1 \\\end{pmatrix} \}$$

Over a finite field
For a finite field $$K$$, this group is denoted $$SO(+1,2,K)$$ and is termed the orthogonal group of "+" type. It is also denoted $$SO(+1,2,q)$$ where $$q$$ is the size of the field.

It turns out that:


 * If $$q$$ is a power of 2, i.e., if $$q$$ is even, then the group is a dihedral group of order $$2(q - 1)$$ and degree $$q - 1$$.
 * If $$q$$ is odd, then the group is a cyclic group of order $$q - 1$$.

Over a finite field
We consider here the group $$SO(+1,2,q) = SO(+1,2,K)$$ where $$K$$ is a field (unique up to isomorphism) of size $$q$$.