# Unitriangular matrix group:UT(3,3)

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## Definition

This group is defined in the following equivalent ways:

1. It is the unique (up to isomorphism) non-abelian group of order $27$ and exponent $3$.
2. It is the unitriangular matrix group of degree three over the field of three elements.
3. It is the inner automorphism group of wreath product of groups of order p for $p = 3$.
4. It is the Burnside group $B(2,3)$: 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 $3 \times 3$ 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 $p$ elements for a prime number $p$. For an odd prime $p$, this is the unique non-abelian group of order $p^3$ and exponent $p$. value of prime number $p$ $p = 3$ click here for a complete list
Burnside group $B(d,n)$ Quotient of free group on $d$ generators by subgroup generated by all $n^{th}$ powers. Values of $d,n$ $d = 2, n = 3$ click here for a complete list

## Arithmetic functions

Function Value Similar groups Explanation for function value
underlying prime of p-group 3
order (number of elements, equivalently, cardinality or size of underlying set) 27 groups with same order As $UT(n,q), n = 3, q = 3$: $q^{n(n-1)/2} = 3^{3(3-1)/2} = 3^3 = 27$
As $B(d,3), d = 2$: $3^{d + \binom{d}{2} + \binom{d}{3}} = 3^{2 + \binom{2}{2} + \binom{2}{3}} = 3^{2 + 1 + 0} = 3^3 = 27$
prime-base logarithm of order 3
exponent of a group 3 groups with same order and exponent of a group | groups with same exponent of a group
prime-base logarithm of exponent 1 groups with same order and prime-base logarithm of exponent | groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
Frattini length 2 groups with same order and Frattini length | groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length
derived length 2 groups with same order and derived length | groups with same prime-base logarithm of order and derived length | groups with same derived length
nilpotency class 2 groups with same order and nilpotency class | groups with same prime-base logarithm of order and nilpotency class | groups with same nilpotency class
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same prime-base logarithm of order and minimum size of generating set | groups with same minimum size of generating set
subgroup rank of a group 2 groups with same order and subgroup rank of a group | groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group
rank of a p-group 2 groups with same order and rank of a p-group | groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group
normal rank of a p-group 2 groups with same order and normal rank of a p-group | groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group
characteristic rank of a p-group 1 groups with same order and characteristic rank of a p-group | groups with same prime-base logarithm of order and characteristic rank of a p-group | groups with same characteristic rank of a p-group

## 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) $\mathbb{Q}(e^{2\pi i/3})$ or $\mathbb{Q}[x]/(x^2 + x + 1)$
condition for a field to be a splitting field characteristic not 3, contains a primitive cube root of unity, i.e., the polynomial $x^2 + x + 1$ splits.
For a finite field of size $q$, equivalent to 3 dividing $q - 1$
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 $G$:

gap> G := SmallGroup(27,3);

Conversely, to check whether a given group $G$ 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.