SmallGroup(32,24)

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Definition

This group is defined by the following presentation: $\langle a,b,c \mid a^4 = b^4 = c^2 = e, ab = ba, ac = ca, cbc^{-1} = a^2b \rangle$

It can alternatively be defined in the following equivalent ways:

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
metabelian group Yes
finite group that is 1-isomorphic to an abelian group Yes via cocycle halving generalization of Baer correspondence

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 51 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 ]>