# Element structure of unitriangular matrix group:UT(3,4)

From Groupprops

This article gives specific information, namely, element structure, about a particular group, namely: unitriangular matrix group:UT(3,4).

View element structure of particular groups | View other specific information about unitriangular matrix group:UT(3,4)

## Summary

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

number of conjugacy classes | 19 As : |

order | 64 As : |

exponent | 4 As is a power of 2: 4 |

conjugacy class size statistics | size 1 (4 classes), size 4 (15 classes) |

order statistics | order 1 (1 element), order 2 (27 elements), order 4 (36 elements) |

## Conjugacy class structure

### Interpretation as unitriangular matrix group of degree three

Compare with element structure of unitriangular matrix group of degree three over a finite field

Nature of conjugacy class | Jordan block size decomposition | Minimal polynomial | Size of conjugacy class (generic ) | Size of conjugacy class () | Number of such conjugacy classes (generic ) | Number of such conjugacy classes () | Total number of elements (generic ) | Total number of elements () | Order of elements in each such conjugacy class (generic ) | Order of elements in each such conjugacy class (, so ) | Type of matrix |
---|---|---|---|---|---|---|---|---|---|---|---|

identity element | 1 + 1 + 1 + 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||

non-identity element, but central (has Jordan blocks of size one and two respectively) | 2 + 1 | 1 | 1 | 3 | 3 | 2 | , | ||||

non-central, has Jordan blocks of size one and two respectively | 2 + 1 | 4 | 6 | 24 | 2 | , but not both and are zero | |||||

non-central, has Jordan block of size three | 3 | 4 | 9 | 36 | if odd 4 if |
4 | both and are nonzero | ||||

Total (--) | -- | -- | -- | -- | 19 | 64 | -- | -- | -- |