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.

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