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	$\left(\begin{array}{cc}1& 0\\ 0& 1\\ \end{array}\right)$ and $\left(\begin{array}{cc}-1& 0\\ 0& -1\\ \end{array}\right)$
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] $\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}1& 2\\ 0& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& 1\\ 0& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& 2\\ 0& -1\\ \end{array}\right)$
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	$\left(\begin{array}{cc}0& -1\\ 1& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}0& -1\\ 1& -1\\ \end{array}\right)$
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	$\left(\begin{array}{cc}2& 0\\ 0& 3\\ \end{array}\right)$
Total	NA	NA	NA	NA	NA	$q+4$	9	$q^3-q$	120	NA