SmallGroup(32,24)

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

This group is defined by the following presentation:

$\langle a,b,c\mid a^{4}=b^{4}=c^{2}=e,ab=ba,ac=ca,cbc^{-1}=a^{2}b\rangle$

Group properties

Property	Satisfied?	Explanation
cyclic group	No
abelian group	No
metacyclic group No
group of nilpotency class two	Yes
metabelian group	Yes

Arithmetic functions

Function	Value	Explanation
order	32
exponent	4
nilpotency class	2
derived length	2
Frattini length	2
minimum size of generating set	3
subgroup rank	3
rank as p-group	3
normal rank as p-group	3
characteristic rank as p-group	3

GAP implementation

Group ID

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 groups of order 32. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(32,24)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(32,24);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [32,24]

or just do:

IdGroup(G)

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 ]>