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

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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]
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