SL(2,3) in GL(2,3)

Definition
The group $$G$$ is general linear group:GL(2,3): the general linear group of degree two over field:F3, i.e., the group of all invertible $$2 \times 2$$ matrices over the field, with matrix multiplication:

$$\! G := \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a,b,c,d \in \mathbb{F}_3, ad \ne bc \}$$

The subgroup $$H$$ is special linear group:SL(2,3): the special linear group of degree two over field:F3, i.e., the subgroup comprising matrices of determinant $$1$$.

$$\! H := \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a,b,c,d \in \mathbb{F}_3, ad - bc = 1 \}$$

Equivalently, $$H$$ is the kernel of the determinant homomorphism from $$G$$ to the multiplicative group of $$\mathbb{F}_3$$, i.e., the map:

$$\begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mapsto ad - bc$$

Direct construction
The group-subgroup pair can be constructed using GL and SL:

G := GL(2,3); H := SL(2,3);

Using a black-box group
If $$G$$ is given as a black-box group (i.e., not explicitly defined as $$GL(2,3)$$), $$H$$ can be constructed as its derived subgroup:

H = DerivedSubgroup(G);