# Unitriangular matrix group:UT(3,4)

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## Contents

## Definition

### As a group of matrices

This group is the unitriangular matrix group of degree three (i.e., the group of upper-triangular matrices with s on the diagonal) over the field of four elements. It is isomorphic to the 2-Sylow subgroup of general linear group:GL(3,4), special linear group:SL(3,4), projective general linear group:PGL(3,4), projective special linear group:PSL(3,4).

The group can be described explicitly as:

The multiplication of matrices and gives the matrix where:

The identity element is the identity matrix.

The inverse of a matrix is the matrix where:

Note that all addition and multiplication in these definitions is happening over the field .

## 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

## Elements

`Further information: element structure of unitriangular matrix group:UT(3,4)`

### 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 ) | 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 | -- | -- | -- |

## GAP implementation

### Group ID

This finite group has order 64 and has ID 242 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(64,242)`

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

`gap> G := SmallGroup(64,242);`

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

`IdGroup(G) = [64,242]`

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 |
---|---|

SylowSubgroup(GL(3,4),2) |
SylowSubgroup, GL |

SylowSubgroup(SL(3,4),2) |
SylowSubgroup, SL |

SylowSubgroup(PGL(3,4),2) |
SylowSubgroup, PGL |

SylowSubgroup(PSL(3,4),2) |
SylowSubgroup, PSL |