# Generalized dihedral group for direct product of Z4 and Z4

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## Definition

This group is defined as the generalized dihedral group corresponding to the abelian group given as the direct product of Z4 and Z4, i.e., it is the semidirect product of this group by cyclic group:Z2 where the non-identity element acts via the inverse map. It has the presentation:

$\langle x,y,a \mid x^4 = y^4 = a^2 = e, xy = yx,axa^{-1} = x^{-1}, aya^{-1} = y^{-1} \rangle$

## 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
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
Group of nilpotency class two Yes
Frattini-in-center group Yes
special group Yes
extraspecial group No
metabelian group Yes
metacyclic group No

## GAP implementation

### Group ID

This finite group has order 32 and has ID 34 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(32,34)

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

gap> G := SmallGroup(32,34);

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

IdGroup(G) = [32,34]

or just do:

IdGroup(G)

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

### Other descriptions

The group can be constructed using its 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,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 ]>