# Direct product of prime-cube order group:U(3,3) and Z3

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## Definition

This group is defined as follows: It is the direct product of prime-cube order group:U(3,3) (the unique non-abelian group of order $27$ and exponent $3$) and the cyclic group of order three.

## Group properties

Property Satisfied? Explanation Comment
Abelian group No
Nilpotent group Yes
Group of nilpotency class two Yes
Metabelian group Yes
Metacyclic group No
UL-equivalent group No has nilpotency class two but the derived subgroup is not the same as the center; also, it is the external direct product of groups with different nilpotency class values See also nilpotent not implies UL-equivalent

## GAP implementation

### Group ID

This finite group has order 81 and has ID 12 among the groups of order 81 in GAP's SmallGroup library. For context, there are 15 groups of order 81. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(81,12)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(81,12);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [81,12]

or just do:

IdGroup(G)

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

### Other descriptions

The group can be defined using the DirectProduct, SmallGroup, and CyclicGroup functions:

DirectProduct(SmallGroup(27,3),CyclicGroup(3))