Direct product of A4 and Z2
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This group is defined in the following equivalent ways:
- it is the external direct product of the alternating group of degree four and the cyclic group of order two.
- It is the wreath product of the cyclic group of order two and the cyclic group of order three, acting regularly. In other words, it is the group , or equivalently, the group , where the latter acts on the former by cylic permutations of coordinates.
|nilpotency class||--||not a nilpotent group.|
|minimum size of generating set||2|
|Subgroup-defining function||Subgroup type in list||Isomorphism class||Comment|
|commutator subgroup||Klein four-group|
|Frattini subgroup||trivial group|
|Fitting subgroup||elementary abelian group:E8|
This finite group has order 24 and has ID 13 among the groups of order 24 in GAP's SmallGroup library. For context, there are 15 groups of order 24. 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(24,13);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [24,13]
or just do:
to have GAP output the group ID, that we can then compare to what we want.