# Unitriangular matrix group:UT(3,3)

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## Contents

## Definition

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.

## Families

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 |

## Arithmetic functions

## Group properties

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

abelian group | No | |

group of prime power order | Yes | |

nilpotent group | Yes | prime power order implies nilpotent |

solvable group | Yes | |

extraspecial group | Yes | |

Frattini-in-center group | Yes |

## 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)`

### Summary

Item | Value |
---|---|

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 | ? |

## GAP implementation

### Group ID

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:

`SmallGroup(27,3)`

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:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

## Related pages

UT(3,) , UT(4,) , UT(3, p) , UT(4, 2 ) , UT(4, 3 ) , UT(4, p ) .