Center of general linear group:GL(2,3)

Definition
$$G$$ is the general linear group of degree two over field:F3. In other words, it is the group of invertible $$2 \times 2$$ matrices with entries over the field of three elements. The field has elements $$0,1,2$$ with $$2 = -1$$.

$$H$$ is the subgroup:

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

$$H$$ is isomorphic to cyclic group:Z2. It is the center of $$G$$ (see center of general linear group is group of scalar matrices over center). The quotient group $$G/H$$ is $$PGL(2,3)$$ (the projective general linear group of degree two over field:F3) which is isomorphic to symmetric group:S4.