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	${\begin{pmatrix}1&0\\0&1\\\end{pmatrix}}$ 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 $x^{2}+2x+1$	$(q^{2}-1)/2$	24	4	4	$2(q^{2}-1)$	96	[SHOW MORE] ${\begin{pmatrix}1&1\\0&1\\\end{pmatrix}}$, ${\begin{pmatrix}1&-1\\0&1\\\end{pmatrix}}$, ${\begin{pmatrix}-1&1\\0&-1\\\end{pmatrix}}$, ${\begin{pmatrix}-1&-1\\0&-1\\\end{pmatrix}}$
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] ${\begin{pmatrix}2&0\\0&4\\\end{pmatrix}}$, ${\begin{pmatrix}3&0\\0&5\\\end{pmatrix}}$
Total	NA	NA	NA	NA	NA	$q+4$	11	$q^{3}-q$	336	NA