# Central product of UT(3,Z) and Z identifying center with 2Z

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]

## Definition

This group can be defined in the following equivalent ways:

- It is the central product of unitriangular matrix group:UT(3,Z) and the group of integers where the subgroup of is identified with the center of .
- It is the following group of matrices under multiplication:

This group is almost like unitriangular matrix group:UT(3,Z). In fact, occurs as a subgroup of index two inside it. However, unlike , it is a CS-Baer Lie group, and hence can participate in the CS-Baer correspondence.

## Arithmetic functions

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

nilpotency class | 2 | ||

derived length | 2 | ||

Frattini length | 2 | ||

Hirsch length | 3 | ||

polycyclic breadth | 3 |

## 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 |