# Element structure of unitriangular matrix group:UT(4,2)

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This article gives specific information, namely, element structure, about a particular group, namely: unitriangular matrix group:UT(4,2).

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This article describes the element structure of unitriangular matrix group:UT(4,2).

## Summary

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

number of conjugacy classes | 16 As : |

order | 64 As : |

exponent | 4 As , a power of : |

conjugacy class size statistics | size 1 (2 classes), size 2 (3 classes), size 4 (8 classes), size 8 (3 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 four

Compare with element structure of unitriangular matrix group of degree four over a finite field#Conjugacy class structure

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 , power of prime ) | Order of elements in each such conjugacy class (, so ) | Type of matrix (constraints on ) |
---|---|---|---|---|---|---|---|---|---|---|---|

identity element | 1 + 1 + 1 + 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | all the are zero | |

non-identity element, but central (has Jordan blocks of size 1,1,2 respectively) | 2 + 1 + 1 | 1 | 1 | 1 | 1 | 2 | , all the others are zero | ||||

non-central but in derived subgroup, has Jordan blocks of size 1,1,2 | 2 + 1 + 1 | 2 | 2 | 4 | 2 | Among and , exactly one of them is nonzero. may be zero or nonzero | |||||

non-central but in derived subgroup, Jordan blocks of size 2,2 | 2 + 2 | 2 | 1 | 2 | 2 | Both and are nonzero. may be zero or nonzero | |||||

outside derived subgroup, inside unique abelian subgroup of maximum order, with Jordan blocks of size 1,1,2 | 2 + 1 + 1 | 4 | 1 | 4 | 2 | is nonzero and are arbitrary | |||||

outside derived subgroup, inside unique abelian subgroup of maximum order, with Jordan blocks of size 2,2 | 2 + 2 | 4 | 1 | 4 | 2 | and are both nonzero and are arbitrary | |||||

outside abelian subgroup of maximum order, Jordan blocks of size 1,1,2 | 2 + 1 + 1 | 4 | 2 | 8 | 2 | Two subcases: Case 1: , nonzero, arbitrary Case 2: , nonzero, arbitrary | |||||

outside abelian subgroup of maximum order, Jordan blocks of size 2,2 | 2 + 2 | 4 | 1 | 4 | 2 | both nonzero arbitrary uniquely determined by other values | |||||

outside abelian subgroup of maximum order, Jordan blocks of size 1,3, with centralizer of order | 3 + 1 | 4 | 3 | 12 | if odd 4 if |
4 | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
| ||||

outside abelian subgroup of maximum order, Jordan blocks of size 1,3, with centralizer of order | 3 + 1 | 8 | 2 | 16 | if odd 4 if |
4 | Two subcases: Case 1: nonzero, , other entries arbitrary Case 2: nonzero, , other entries arbitrary | ||||

Jordan block of size 4 | 4 | 8 | 1 | 8 | if if |
4 | nonzero arbitrary | ||||

Total (--) | -- | -- | -- | -- | 16 | 64 | -- | -- | -- |