# Element structure of special linear group:SL(2,5)

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This article gives specific information, namely, element structure, about a particular group, namely: 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.

## Summary

Item | Value |
---|---|

order of the whole group (total number of elements) | 120 |

conjugacy class sizes | 1,1,12,12,12,12,20,20,30 in grouped form: 1 (2 times), 12 (4 times), 20 (2 times), 30 (1 time) maximum: 30, number of conjugacy classes: 9, lcm: 60 |

order statistics | 1 of order 1, 1 of order 2, 20 of order 3, 30 of order 4, 24 of order 5, 20 of order 6, 24 of order 10 maximum: 10, lcm (exponent of the whole group): 60 |

## Conjugacy class structure

### Conjugacy classes

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### Interpretation as special linear group of degree two

`Further information: element structure of special linear group of degree two over a finite field`

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 ) | Size of conjugacy class () | Number of such conjugacy classes (generic odd ) | Number of such conjugacy classes () | Total number of elements (generic odd ) | Total number of elements () | Representative matrices (one per conjugacy class) |
---|---|---|---|---|---|---|---|---|---|---|

Scalar | or | or | or | 1 | 1 | 2 | 2 | 2 | 2 | and |

Not diagonal, Jordan block of size two | or | or | or | 12 | 4 | 4 | 48 | [SHOW MORE] | ||

Diagonalizable over field:F25, not over field:F5. Must necessarily have no repeated eigenvalues. | and , where is interpreted a an element of field:F25 that squares to 3 | , | , | 20 | 2 | 40 | , | |||

Diagonalizable over field:F5 with distinct diagonal entries |
30 | 1 | 30 | |||||||

Total | NA | NA | NA | NA | NA | 9 | 120 | NA |

### Interpretation as double cover of alternating group

`Further information: element structure of double cover of alternating group`

is isomorphic to . Recall that we have the following rules to determine splitting and orders. The rules listed below are *only* for partitions that already correspond to even permutations, i.e., partitions that have an even number of even parts:

Hypothesis: does the partition have at least one even part? | Hypothesis: does the partition have a repeated part? (the repeated part may be even or odd) | Conclusion: does the conjugacy class split from to in 2? | Conclusion: does the fiber in over a conjugacy class in split in 2? | Total number of conjugacy classes in corresponding to this partition (4 if Yes to both preceding columns, 2 if Yes to one and No to other, 1 if No to both) | Number of these conjugacy classes where order of element = lcm of parts | Number of these conjugacy classes where order of element = twice the lcm of parts |
---|---|---|---|---|---|---|

No | No | Yes | Yes | 4 | 2 | 2 |

No | Yes | No | Yes | 2 | 1 | 1 |

Yes | No | No | Yes | 2 | 0 | 2 |

Yes | Yes | No | No | 1 | 0 | 1 |

Partition | Partition in grouped form | Does the partition have at least one even part? | Does the partition have a repeated part? | Conclusion: does the conjugacy class split from to in 2? | Conclusion: does the fiber in over a conjugacy class in split in 2? | Total number of conjugacy classes in corresponding to this partition (4 if Yes to both preceding columns, 2 if Yes to one and No to other, 1 if No to both) | Size of each conjugacy class | Size formula (we take the size formula in , multiply by 2, and divide by the number (1,2, or 4) two columns preceding | Total number of elements (= twice the size of the -conjugacy class) | Element orders | Formula for element orders |
---|---|---|---|---|---|---|---|---|---|---|---|

1 + 1 + 1 + 1 + 1 | 1 (5 times) | No | Yes | No | Yes | 2 | 1 | 2 | 1 (1 class), 2 (1 class) | (1 class) (1 class) | |

2 + 2 + 1 | 2 (2 times), 1 (1 time) | Yes | Yes | No | No | 1 | 30 | 30 | 4 | (1 class) | |

3 + 1 + 1 | 3 (1 time), 1 (2 times) | No | Yes | No | Yes | 2 | 20 | 40 | 3 (1 class) 6 (1 class) |
(1 class) (1 class) | |

5 | 5 (1 time) | No | No | Yes | Yes | 4 | 12 | 48 | 5 (2 classes), 10 (2 classes) | (2 classes) (2 classes) | |

Total | -- | -- | -- | -- | -- | 9 | -- | -- | 120 | -- | -- |