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.

Conjugacy class structure

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

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	Number of such conjugacy classes	Total number of elements	Semisimple?	Diagonalizable over ${\mathbb{F}}_q$?	Splits in $S{L}_2$ relative to $G{L}_2$?	Representative matrices (one per conjugacy class)
Diagonalizable over field:F3 with distinct diagonal entries	--	--	--	NA	0	0	NA	NA	NA	--
Diagonalizable over field:F9, not over field:F3. Must necessarily have no repeated eigenvalues.	pair of square roots of $-1$ in field:F9	$x^2+1$	$x^2+1$	6	1	6	Yes	No	No	$\left(\begin{array}{cc}0& -1\\ 1& 0\\ \end{array}\right)$
Scalar	$\{1,1\}$ or $\{-1,-1\}$	$x^2-2x+1$ or $x^2-x+1$	$x-1$ or $x+1$	1	2	2	Yes	Yes	No	$\left(\begin{array}{cc}1& 0\\ 0& 1\\ \end{array}\right)$ 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 $x^2-x+1$	4	4	16	No	No	Yes	$\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}1& -1\\ 0& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& 1\\ 0& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& -1\\ 0& -1\\ \end{array}\right)$
Total	NA	NA	NA	NA	7	24	8	2	16	NA