# 2-Sylow subgroup of special linear group:SL(2,5)

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$G$ is the special linear group:SL(2,5), i.e., the special linear group of degree two over field:F5. 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 0,1,2,3,4 with $4 = -1$.
$H$ is the subgroup:
$\{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}, \begin{pmatrix} 4 & 0 \\ 0 & 4 \\\end{pmatrix}, \begin{pmatrix} 0 & 4 \\ 1 & 0 \\\end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 4 & 0 \\\end{pmatrix}, \begin{pmatrix} 4 & 4 \\ 4 & 1 \\\end{pmatrix}, \begin{pmatrix} 1 & 1 \\ 1 & 4 \\\end{pmatrix}, \begin{pmatrix} 1 & 4 \\ 4 & 4 \\\end{pmatrix}, \begin{pmatrix} 4 & 1 \\ 1 & 1 \\\end{pmatrix} \}$