Element structure of special linear group:SL(2,7)

This article gives specific information, namely, element structure, about a particular group, namely: special linear group:SL(2,7).
View element structure of particular groups | View other specific information about special linear group:SL(2,7)

This article gives detailed information about the element structure of special linear group:SL(2,7), which is a group of order 336.

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

Conjugacy class structure

Compare with element structure of special linear group of degree two over a finite field#Conjugacy class structure.

In the table below, we consider the group as $SL(2,q),q=7$. The information is stated for generic odd $q$ and then computed numerically for $q=7$.

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 ($q=7$)	Number of such conjugacy classes (generic odd $q$)	Number of such conjugacy classes ($q=7$)	Total number of elements (generic odd $q$)	Total number of elements ($q=7$)	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 $x^2+2x+1$	$(q^2-1)/2$	24	4	4	$2(q^2-1)$	96	[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 ${\mathbb{F}}_{q^2}$, i.e., field:F49, not over ${\mathbb{F}}_q$, i.e., field:F7. Must necessarily have no repeated eigenvalues.	For $q=7$: $\{\sqrt{-1},-\sqrt{-1}\}$, $\{2+\sqrt{3},2-\sqrt{3}\}$, $\{-2+\sqrt{3},-2-\sqrt{3}\}$	For $q=7$: $x^2+1$, $x^2-4x+1$, $x^2-3x+1$	For $q=7$: $x^2+1$, $x^2-4x+1$, $x^2-3x+1$	$q(q-1)$	42	$(q-1)/2$	3	$q(q-1{)}^2/2$	126	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
Diagonalizable over ${\mathbb{F}}_q$, i.e., field:F7 with distinct diagonal entries	For $q=7$: $\{2,4\}$, $\{3,5\}$	For $q=7$: $x^2-x+1$, $x^2+x+1$	For $q=7$: $x^2-x+1$, $x^2+x+1$	$q(q+1)$	56	$(q-3)/2$	2	$q(q+1)(q-3)/2$	112	[SHOW MORE] $\left(\begin{array}{cc}2& 0\\ 0& 4\\ \end{array}\right)$, $\left(\begin{array}{cc}3& 0\\ 0& 5\\ \end{array}\right)$
Total	NA	NA	NA	NA	NA	$q+4$	11	$q^3-q$	336	NA