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This group is defined by the following presentation:
It can alternatively be defined in the following equivalent ways:
- It is the central product of SmallGroup(16,3) and cyclic group:Z4 over a common cyclic central subgroup of order two.
- It is the central product of nontrivial semidirect product of Z4 and Z4 (ID: (16,4)) and cyclic group:Z4 over a common cyclic central subgroup of order two.
|group of prime power order||Yes|
|nilpotent group||Yes||prime power order implies nilpotent|
|supersolvable group||Yes||via nilpotent: finite nilpotent implies supersolvable|
|solvable group||Yes||via nilpotent: nilpotent implies solvable|
|finite group that is 1-isomorphic to an abelian group||Yes||via cocycle halving generalization of Baer correspondence|
This finite group has order 32 and has ID 24 among the groups of order 32 in GAP's SmallGroup library. For context, there are 51 groups of order 32. 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(32,24);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [32,24]
or just do:
to have GAP output the group ID, that we can then compare to what we want.
Description by presentation
Here is the GAP code to define this group using a presentation:
gap> F := FreeGroup(3); <free group on the generators [ f1, f2, f3 ]> gap> G := F/[F.1^4,F.2^4, F.1*F.2*F.1^(-1)*F.2^(-1), F.3^2, F.1*F.3*F.1^(-1)*F.3^(-1),F.3*F.2*F.3^(-1)*F.2^(-1)*F.1^2 ]; <fp group on the generators [ f1, f2, f3 ]>