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

## Contents

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

## 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]
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]
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]
Total NA NA NA NA NA $q + 4$ 11 $q^3 - q$ 336 NA