Direct product of Z9 and E27
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The group is defined in the following equivalent ways:
- It is the external direct product of the cyclic group of order 9 and the elementary abelian group of order 27, i.e., .
- it is the external direct product of one copy of the cyclic group of order 9 and three copies of the cyclic group of order 3.
This finite group has order 243 and has ID 61 among the groups of order 243 in GAP's SmallGroup library. For context, there are groups of order 243. It can thus be defined using GAP's SmallGroup function as:
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(243,61);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [243,61]
or just do:
to have GAP output the group ID, that we can then compare to what we want.
|Description||GAP functions used||Mathematical translation of description|
|DirectProduct(CyclicGroup(9),ElementaryAbelianGroup(27))||CyclicGroup, DirectProduct, ElementaryAbelianGroup||external direct product of cyclic group of order 9 and elementary abelian group of order 27|
|DirectProduct(CyclicGroup(9),CyclicGroup(3),CyclicGroup(3),CyclicGroup(3))||CyclicGroup, DirectProduct||external direct product of one copy of cyclic group of order 9 and three copies of cyclic group of order 3|