2-Sylow subgroup of special linear group:SL(2,3)

Definition
$$G$$ is the special linear group:SL(2,3), i.e., the special linear group of degree two over field:F3. In other words, it is the group of invertible $$2 \times 2$$ matrices of determinant 1 over the field with three elements. The field has elements 0,1,2, with $$2 = -1$$.

$$H$$ is the subgroup:

$$\{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}, \begin{pmatrix} 2 & 0 \\ 0 & 2 \\\end{pmatrix}, \begin{pmatrix} 0 & 2 \\ 1 & 0 \\\end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 2 & 0 \\\end{pmatrix}, \begin{pmatrix} 2 & 2 \\ 2 & 1 \\\end{pmatrix}, \begin{pmatrix} 1 & 1 \\ 1 & 2 \\\end{pmatrix}, \begin{pmatrix} 1 & 2 \\ 2 & 2 \\\end{pmatrix}, \begin{pmatrix} 2 & 1 \\ 1 & 1 \\\end{pmatrix} \}$$

$$H$$ is a normal subgroup of $$G$$ and is isomorphic to the quaternion group of order 8.

Subgroup-defining functions
The subgroup is a characteristic subgroup of the whole group and arises as the result of many subgroup-defining functions. Some of these are given below.