Element structure of special linear group:SL(2,3)

This article gives specific information, namely, element structure, about a particular group, namely: special linear group:SL(2,3).
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This article gives detailed information about the element structure of special linear group:SL(2,3).

See also element structure of special linear group of degree two over a finite field.

Conjugacy class structure

Conjugacy classes

Note that since we are over field:F3, $-1=2$, so all the $-1$s below can be rewritten as $2$s.

Conjugacy class representative	Conjugacy class size	List of all elements of conjugacy class
$\left(\begin{array}{cc}1& 0\\ 0& 1\\ \end{array}\right)$	1	$\left(\begin{array}{cc}1& 0\\ 0& 1\\ \end{array}\right)$
$\left(\begin{array}{cc}-1& 0\\ 0& -1\\ \end{array}\right)$	1	$\left(\begin{array}{cc}-1& 0\\ 0& -1\\ \end{array}\right)$
$\left(\begin{array}{cc}0& -1\\ 1& 0\\ \end{array}\right)$	6	[SHOW MORE] $\left(\begin{array}{cc}0& -1\\ 1& 0\\ \end{array}\right)$, $\left(\begin{array}{cc}0& 1\\ -1& 0\\ \end{array}\right)$, $\left(\begin{array}{cc}1& 1\\ 1& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& 1\\ 1& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}1& -1\\ -1& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& -1\\ -1& 1\\ \end{array}\right)$
$\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right)$	4	[SHOW MORE] $\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& 1\\ -1& 0\\ \end{array}\right)$, $\left(\begin{array}{cc}1& 0\\ -1& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}0& 1\\ -1& -1\\ \end{array}\right)$
$\left(\begin{array}{cc}1& -1\\ 0& 1\\ \end{array}\right)$	4	[SHOW MORE] $\left(\begin{array}{cc}1& -1\\ 0& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& -1\\ 1& 0\\ \end{array}\right)$, $\left(\begin{array}{cc}1& 0\\ 1& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}0& -1\\ 1& -1\\ \end{array}\right)$
$\left(\begin{array}{cc}-1& 1\\ 0& -1\\ \end{array}\right)$	4	[SHOW MORE] $\left(\begin{array}{cc}-1& 1\\ 0& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}1& 1\\ -1& 0\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& 0\\ -1& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}0& 1\\ -1& 1\\ \end{array}\right)$
$\left(\begin{array}{cc}-1& -1\\ 0& -1\\ \end{array}\right)$	4	[SHOW MORE] $\left(\begin{array}{cc}-1& -1\\ 0& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}1& -1\\ 1& 0\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& 0\\ 1& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}0& -1\\ 1& 1\\ \end{array}\right)$

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 $q$)	Size of conjugacy class ($q=3$)	Number of such conjugacy classes (generic $q$)	Number of such conjugacy classes ($q=3$)	Total number of elements (generic $q$)	Total number of elements ($q=3$)	Representative matrices (one per conjugacy class)
Scalar	$\{1,1\}$ or $\{-1,-1\}$	$x^2-2x+1$ or $x^2-x+1$	$x-1$ 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 $x^2-x+1$	$x^2-2x+1$ or $x^2-x+1$	$(q^2-1)/2$	4	4	4	$2(q^2-1)$	16	[SHOW MORE] $\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}1& -1\\ 0& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& 1\\ 0& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& -1\\ 0& -1\\ \end{array}\right)$
Diagonalizable over field:F9, not over field:F3. Must necessarily have no repeated eigenvalues.	pair of square roots of $-1$ in field:F9	$x^2+1$	$x^2+1$	$q(q-1)$	6	$(q-1)/2$	1	$q(q-1{)}^2/2$	6	$\left(\begin{array}{cc}0& -1\\ 1& 0\\ \end{array}\right)$
Diagonalizable over field:F3 with distinct diagonal entries	--	--	--	$q(q+1)$	12	$(q-3)/2$	0	$q(q+1)(q-3)/2$	0	--
Total	NA	NA	NA	NA	NA	$q+4$	7	$q^3-q$	24	NA