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 upper-triangular unipotent matrix group: the group of $3\times 3$ matrices 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

• Prime-cube order group:U(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 three elements.
• Burnside groups: This group is $B(2,3)$. In general, the Burnside groups $B(m,3)$ are all finite.

Arithmetic functions

Function	Value	Explanation
order	27
exponent	3
Frattini length	2
Fitting length	1

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)

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.