# Unitriangular matrix group:UT(4,2)

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## Contents

## Definition

This group is defined in the following equivalent ways:

- It is the unitriangular matrix group of degree four over the field of two elements.
- It is a -Sylow subgroup of the general linear group , which turns out to be isomorphic to alternating group:A8.
- It is the -Sylow subgroup of the holomorph of the elementary abelian group of order eight, which is the general affine group.
- It is the -Sylow subgroup of the holomorph of the direct product of Z4 and Z2.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 64#Arithmetic functions

### Basic arithmetic functions

### Arithmetic functions of a counting nature

## Elements

`Further information: element structure of unitriangular matrix group:UT(4,2)`

### 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 | -- | -- | -- |

## Linear representation theory

`Further information: linear representation theory of unitriangular matrix group:UT(4,2)`

## GAP implementation

### Group ID

This finite group has order 64 and has ID 138 among the groups of order 64 in GAP's SmallGroup library. For context, there are groups of order 64. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(64,138)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(64,138);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [64,138]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Other descriptions

Description | Functions used |
---|---|

UnitriangularPcpGroup(4,2) |
UnitriangularPcpGroup |

SylowSubgroup(GL(4,2),2) |
SylowSubgroup, GL |

SylowSubgroup(Holomorph(ElementaryAbelianGroup(8)),2) |
SylowSubgroup, Holomorph, ElementaryAbelianGroup |

SylowSubgroup(Holomorph(DirectProduct(CyclicGroup(4),CyclicGroup(2))),2) |
SylowSubgroup, Holomorph, DirectProduct, CyclicGroup |