SmallGroup(32,2)
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Contents
Definition
This group is a semidirect product , and can be defined by the following presentation:
See the section #Description by presentation for how to construct the group using this presentation in GAP.
Arithmetic functions
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 32#Arithmetic functions
Group properties
Want to compare and contrast group properties with other groups of the same order? Check out groups of order 32#Group properties
Property | Satisfied? | Explanation | Comment |
---|---|---|---|
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 | |
abelian group | No | ||
T-group | No | ||
monolithic group | No | ||
one-headed group | No | ||
group of nilpotency class two | Yes | ||
metabelian group | Yes | ||
finite group that is 1-isomorphic to an abelian group | Yes | via cocycle skew reversal generalization of Baer correspondence |
GAP implementation
Group ID
This finite group has order 32 and has ID 2 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:
SmallGroup(32,2)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(32,2);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [32,2]
or just do:
IdGroup(G)
to have GAP output the group ID, that we can then compare to what we want.
Description by presentation
gap> F := FreeGroup(3); <free group on the generators [ f1, f2, f3 ]> gap> G := F/[F.1^4,F.2^4,F.3^2,Comm(F.1,F.2)*F.3^(-1),Comm(F.1,F.3),Comm(F.2,F.3)]; <fp group on the generators [ f1, f2, f3 ]> gap> IdGroup(G); [ 32, 2 ]
GAP verification of function values and group properties
Below is a GAP implementation verifying the various function values and group properties as stated in this page. Before beginning, set G := SmallGroup(32,2); or set it to this group in any other manner:
gap> Order(G); 32 gap> Exponent(G); 4 gap> NilpotencyClassOfGroup(G); 2 gap> DerivedLength(G); 2 gap> FrattiniLength(G); 2 gap> Rank(G); 2 gap> RankAsPGroup(G); 3 gap> NormalRank(G); 3 gap> CharacteristicRank(G); 3