# SmallGroup(32,35)

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## Definition

This group is a quaternion group-like variant of the generalized dihedral group of direct product of Z4 and Z4. It is given by the presentation: $\langle x,y,a \mid x^4 = y^4 = a^4 = e, xy = yx, axa^{-1} = x^{-1}, aya^{-1} = y^{-1}, a^2 = x^2 \rangle$.

The group can also be described (up to isomorphism) as a subgroup of the direct product of Q8 and Q8 that is not isomorphic to the direct product of Q8 and Z4.

## Group properties

Property Satisfied? Explanation
Cyclic group No
Abelian group No
Metacyclic group No
Group of nilpotency class two Yes
Frattini-in-center group Yes
Special group Yes
Extraspecial group No

## GAP implementation

### Group ID

This finite group has order 32 and has ID 35 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(32,35)

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

gap> G := SmallGroup(32,35);

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

IdGroup(G) = [32,35]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

### Other descriptions

gap> F := FreeGroup(3);
<free group on the generators [ f1, f2, f3 ]>
gap> G := F/[F.1^4,F.2^4,F.3^4,F.3^2 * F.1^(-2), F.1*F.2*F.1^(-1)*F.2^(-1),F.3*F.1*F.3^(-1)*F.1,F.3*F.2*F.3^(-1)*F.2];
<fp group on the generators [ f1, f2, f3 ]>