Element structure of special linear group:SL(2,5)

This article gives specific information, namely, element structure, about a particular group, namely: special linear group:SL(2,5).
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This article gives detailed information about the element structure of special linear group:SL(2,5), which is a group of order 120.

See also element structure of special linear group of degree two.

Conjugacy class structure

Conjugacy classes

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Relationship with conjugacy class structure for an arbitrary special linear group of degree two

Further information: element structure of special linear group of degree two over a finite field

Nature of conjugacy class	Eigenvalue pairs of all conjugacy classes	Characteristic polynomials of all conjugacy classes	Minimal polynomials of all conjugacy classes	Size of conjugacy class (generic odd $q$)	Size of conjugacy class ($q = 5$)	Number of such conjugacy classes (generic odd $q$)	Number of such conjugacy classes ($q = 5$)	Total number of elements (generic odd $q$)	Total number of elements ($q = 5$)	Representative matrices (one per conjugacy class)
Scalar	$\{ 1, 1 \}$ or $\{ -1,-1\}$	$x^2 - 2x + 1$ or $x^2 + 2x + 1$	$x - 1$ or $x + 1$	1	1	2	2	2	2	$\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}$ and $\begin{pmatrix} -1 & 0 \\ 0 & -1\\\end{pmatrix}$
Not diagonal, Jordan block of size two	$\{ 1, 1 \}$ or $\{ -1,-1\}$	$x^2 - 2x + 1$ or $x^2 + 2x + 1$	$x^2 - 2x + 1$ or $x^2 + 2x + 1$	$(q^2 - 1)/2$	12	4	4	$2(q^2 - 1)$	48	[SHOW MORE] $\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}$, $\begin{pmatrix} 1 & 2 \\ 0 & 1 \\\end{pmatrix}$, $\begin{pmatrix} -1 & 1 \\ 0 & -1 \\\end{pmatrix}$, $\begin{pmatrix} -1 & 2 \\ 0 & -1 \\\end{pmatrix}$
Diagonalizable over field:F25, not over field:F5. Must necessarily have no repeated eigenvalues.	$\{ 2 + \sqrt{3}, 2 - \sqrt{3} \}$ and $\{ -2 + \sqrt{3}, -2 - \sqrt{3} \}$, where $\sqrt{3}$ is interpreted a an element of field:F25 that squares to 3	$x^2 - x + 1$, $x^2 + x + 1$	$x^2 - x + 1$, $x^2 + x + 1$	$q(q - 1)$	20	$(q - 1)/2$	2	$q(q - 1)^2/2$	40	$\begin{pmatrix}0 & -1\\ 1 & 1\\\end{pmatrix}$, $\begin{pmatrix}0 & -1\\ 1 & -1\\\end{pmatrix}$
Diagonalizable over field:F5 with distinct diagonal entries	$\{ 2,3 \}$	$x^2 + 1$	$x^2 + 1$	$q(q+1)$	30	$(q - 3)/2$	1	$q(q+1)(q-3)/2$	30	$\begin{pmatrix} 2 & 0 \\ 0 & 3 \\\end{pmatrix}$
Total	NA	NA	NA	NA	NA	$q + 4$	9	$q^3 - q$	120	NA