# Unitriangular matrix group of degree three over quotient of polynomial ring over F2 by square of indeterminate

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Contents

## Definition

### As a matrix group

This group is defined as the unitriangular matrix group of degree three over the ring . 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

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## 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

## GAP implementation

### Group ID

This finite group has order 64 and has ID 215 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(64,215)`

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

`gap> G := SmallGroup(64,215);`

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

`IdGroup(G) = [64,215]`

or just do:

`IdGroup(G)`

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