# Central product of D16 and Z4

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## Contents

## Definition

This group is defined in the following equivalent ways:

- It is the central product of dihedral group:D16 and cyclic group:Z4, with a common central cyclic group:Z2 identified.
- It is the central product of semidihedral group:SD16 and cyclic group:Z4, with a common central cyclic group:Z2 identified.
- It is the central product of generalized quaternion group:Q16 and cyclic group:Z4, with a common central cyclic group:Z2 identified.

## 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

## GAP implementation

### Group ID

This finite group has order 32 and has ID 42 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(32,42)`

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

`gap> G := SmallGroup(32,42);`

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

`IdGroup(G) = [32,42]`

or just do:

`IdGroup(G)`

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