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.