Wreath product of Z3 and Z3
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This group is defined in the following equivalent ways:
- It is the wreath product of the cyclic group of order three and the cyclic group of order three, acting regularly. In other words, if denotes the cyclic group of order three, this is .
- It is the -Sylow subgroup of the symmetric group of degree nine.
|minimum size of generating set||2|
|rank as p-group||3|
|normal rank as p-group||3|
|characteristic rank as p-group||3|
|group of prime power order||Yes|
|maximal class group||Yes|
|directly indecomposable group||Yes|
|centrally indecomposable group||Yes|
This finite group has order 81 and has ID 7 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:
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(81,7);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [81,7]
or just do:
to have GAP output the group ID, that we can then compare to what we want.