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