# Central product of D8 and Q8

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## Definition

This group is defined in the following equivalent ways:

1. It is the central product of dihedral group:D8 and the quaternion group over a commonly identified cyclic central subgroup cyclic group:Z2.
2. It is the extraspecial group of order $2^5$ and type $-$.

## 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

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
extraspecial group Yes Hence also satisfies property special group, group of nilpotency class two, UL-equivalent group, metabelian group.

## GAP implementation

### Group ID

This finite group has order 32 and has ID 50 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(32,50)

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

gap> G := SmallGroup(32,50);

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

IdGroup(G) = [32,50]

or just do:

IdGroup(G)

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