Direct product of Z9 and E9

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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Definition

This group is defined as the direct product of the cyclic group of order 9 and elementary abelian group of order 9. Equivalently, it is the direct product of the cyclic group of order 9 and two copies of the cyclic group of order 3.

Arithmetic functions

Function Value Explanation
order 81
exponent 9
minimum size of generating set 3
subgroup rank 3

Group properties

Property Satisfied Explanation
cyclic group No
metacyclic group No
homocyclic group No
abelian group Yes
group of prime power order Yes
nilpotent group Yes

GAP implementation

Group ID

This finite group has order 81 and has ID 11 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(81,11)

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

gap> G := SmallGroup(81,11);

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

IdGroup(G) = [81,11]

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 described using GAP's DirectProduct, CyclicGroup and ElementaryAbelianGroup functions:

DirectProduct(CyclicGroup(9),ElementaryAbelianGroup(9))

Alternatively:

DirectProduct(CyclicGroup(9),CyclicGroup(3),CyclicGroup(3))