Unitriangular matrix group:UT(3,3)
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 group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]
This group is defined in the following equivalent ways:
- It is the unique (up to isomorphism) non-abelian group of order and exponent .
- It is the unitriangular matrix group of degree three over the field of three elements.
- It is the inner automorphism group of wreath product of groups of order p for .
- It is the Burnside group : the quotient of the free group of rank two by the subgroup generated by all cubes in the group.
|Generic name for family member||Definition||Parametrization of family||Parameter value(s) for this member||Other members||Comments|
|unitriangular matrix group of degree three over a field, or more generally, a unital ring||group of upper triangular matrices over the unital ring with 1s on the diagonal.||The family is parametrized by the field or the unital ring.||field:F3||click here for a complete list|
|unitriangular matrix group:UT(3,p)||unitriangular matrix group of degree three over the field of elements for a prime number . For an odd prime , this is the unique non-abelian group of order and exponent .||value of prime number||click here for a complete list|
|Burnside group||Quotient of free group on generators by subgroup generated by all powers.||Values of||click here for a complete list|
|group of prime power order||Yes|
|nilpotent group||Yes||prime power order implies nilpotent|
Other associated constructs
|Associated construct||Isomorphism class||Comment|
|Lazard Lie ring||upper-triangular nilpotent Lie ring:u(3,3)|
Linear representation theory
Further information: linear representation theory of unitriangular matrix group:UT(3,3)
|degrees of irreducible representations over a splitting field|| 1,1,1,1,1,1,1,1,1,3,3 (1 occurs 9 times, 3 occurs 2 times)|
maximum: 3, lcm: 3, number: 11, sum of squares: 27
|Schur index values of irreducible representations||1,1,1,1,1,1,1,1,1,1,1|
|smallest field of realization (characteristic zero)||or|
|condition for a field to be a splitting field|| characteristic not 3, contains a primitive cube root of unity, i.e., the polynomial splits.|
For a finite field of size , equivalent to 3 dividing
|smallest size splitting field||field:F4, i.e., the field with 4 elements|
|orbit structure of irreducible representations under automorphism group||?|
This finite group has order 27 and has ID 3 among the groups of order 27 in GAP's SmallGroup library. For context, there are groups of order 27. It can thus be defined using GAP's SmallGroup function as:
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(27,3);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [27,3]
or just do:
to have GAP output the group ID, that we can then compare to what we want.