# Difference between revisions of "Element structure of special linear group:SL(2,5)"

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This article gives detailed information about the element structure of special linear group:SL(2,5), which is a group of order 120.

## Conjugacy class structure

### Conjugacy classes

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 ($q = 5$) Number of such conjugacy classes (generic odd $q$) Number of such conjugacy classes ($q = 5$) Total number of elements (generic odd $q$) Total number of elements ($q = 5$) 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 $x^2 + 2x + 1$ $(q^2 - 1)/2$ 12 4 4 $2(q^2 - 1)$ 48 [SHOW MORE]
Diagonalizable over field:F25, not over field:F5. Must necessarily have no repeated eigenvalues. $\{ 2 + \sqrt{3}, 2 - \sqrt{3} \}$ and $\{ -2 + \sqrt{3}, -2 - \sqrt{3} \}$, where $\sqrt{3}$ is interpreted a an element of field:F25 that squares to 3 $x^2 - x + 1$, $x^2 + x + 1$ $x^2 - x + 1$, $x^2 + x + 1$ $q(q - 1)$ 20 $(q - 1)/2$ 2 $q(q - 1)^2/2$ 40 $\begin{pmatrix}0 & -1\\ 1 & 1\\\end{pmatrix}$, $\begin{pmatrix}0 & -1\\ 1 & -1\\\end{pmatrix}$
Diagonalizable over field:F5 with distinct diagonal entries $\{ 2,3 \}$ $x^2 + 1$ $x^2 + 1$ $q(q+1)$ 30 $(q - 3)/2$ 1 $q(q+1)(q-3)/2$ 30 $\begin{pmatrix} 2 & 0 \\ 0 & 3 \\\end{pmatrix}$
Total NA NA NA NA NA $q + 4$ 9 $q^3 - q$ 120 NA