# Unitriangular matrix group:UT(3,Z)

From Groupprops

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 reduced free group

Abstractly, this group is a free class two group on a generating set of size two. Hence, it is a reduced free group.

### As a matrix group

This group, denoted or , is defined as the unitriangular matrix group of degree three over the ring of integers. Explicitly, it is the group, under multiplication:

The group is also sometimes called the **integral Heisenberg group**.

### Definition by presentation

The group can be defined by means of the following presentation:

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

### Structures

The group has the structure of an arithmetic group.

## Arithmetic functions

Function | Value | Similar groups | Explanation |
---|---|---|---|

nilpotency class | 2 | The derived subgroup and center are both equal to the subgroup | |

derived length | 2 | Follows from nilpotency class being 2. | |

Frattini length | 2 | The Frattini subgroup also coincides with the derived subgroup and center, and it is isomorphic to the group of integers, which is a Frattini-free group. | |

Hirsch length | 3 | We can use a polycyclic series that starts with the center, then goes to the subgroup , and then goes to the whole group. Each of the quotient groups is isomorphic to . | |

polycyclic breadth | 3 | We can use a polycyclic series that starts with the center, then goes to the subgroup , and then goes to the whole group. Each of the quotient groups is isomorphic to . |

## Group properties

Property | Satisfied? | Explanation |
---|---|---|

abelian group | No | |

nilpotent group | Yes | |

group of nilpotency class two | Yes | |

metacyclic group | No | |

polycyclic group | Yes | |

metabelian group | Yes | |

supersolvable group | Yes |