# Unitriangular matrix group:UT(3,Z4)

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## Contents

## Definition

### As a matrix group

This group is defined as the unitriangular matrix group of degree three over ring:Z4. Explicitly, it is the group (under matrix multiplication) of upper-triangular unipotent matrices over the ring , i.e., matrices of the form:

### Definition by presentation

This group is given by the following presentation:

Here's how this presentation relates to the matrix description:

We can relate this with the matrix group definition by setting:

## 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(3,Z4)`

## Linear representation theory

`Further information: linear representation theory of unitriangular matrix group:UT(3,Z4)`

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

Degrees of irreducible representations over a splitting field (such as or ) | 1 (16 times), 2 (4 times), 4 (2 times) maximum: 4, lcm: 4, number: 22, sum of squares: 64 |

## GAP implementation

### Group ID

This finite group has order 64 and has ID 18 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(64,18)`

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

`gap> G := SmallGroup(64,18);`

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

`IdGroup(G) = [64,18]`

or just do:

`IdGroup(G)`

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

### Description as a matrix group

This description uses the functions SL, ZmodnZ, IsZero, and IsOne.

gap> L := SL(3,ZmodnZ(4));; gap> G := Group(Filtered(L,x -> ForAll([x[1][1],x[2][2],x[3][3]],IsOne) and ForAll([x[2][1],x[3][1],x[3][2]],IsZero)));; gap> IdGroup(G); [ 64, 18 ]

### Description by presentation

gap> F := FreeGroup(3); <free group on the generators [ f1, f2, f3 ]> gap> G := F/[F.1^4,F.2^4,F.3^4,F.1*F.3*F.1^(-1)*F.3^(-1),F.1*F.2*F.1^(-1)*F.2^(-1),F.2*F.3*F.2^(-1)*F.3^(-1)*F.1^(-1)]; <fp group on the generators [ f1, f2, f3 ]> gap> IdGroup(G); [ 64, 18 ]