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

• Unitriangular matrix group:UT(3,p): For an odd prime $p$, this is the unique non-abelian group of order $p^{3}$ and exponent $p$. It is the group of unipotent upper-triangular $3\times 3$ matrices over the field of $p$ elements. it can more generally be viewed as a unitriangular matrix group of degree three.
• Burnside groups: This group is $B(2,3)$. In general, the Burnside groups $B(m,3)$ are all finite.

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 $B(n,3),n=2$: $3^{n+{\binom {n}{2}}+{\binom {n}{3}}}=3^{2+{\binom {2}{2}}+{\binom {2}{3}}}=3^{2+1+0}=3^{3}=27$
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.

Related pages

UT(3,$\_$) , UT(4,$\_$) , UT(3, p) , UT(4, 2 ) , UT(4, 3 ) , UT(4, p ) .