# Unitriangular matrix group:UT(4,3)

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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 three elements.
- It is a -Sylow subgroup of general linear group:GL(4,3), and hence also of special linear group:SL(4,3), projective general linear group:PGL(4,3), and projective special linear group:PSL(4,3).

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 729#Arithmetic functions

### Basic arithmetic functions

## GAP implementation

### Group ID

This finite group has order 729 and has ID 307 among the groups of order 729 in GAP's SmallGroup library. For context, there are groups of order 729. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(729,307)`

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

`gap> G := SmallGroup(729,307);`

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

`IdGroup(G) = [729,307]`

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

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

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

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