# Direct product of Q8 and Z2

From Groupprops

## Definition

This group is the external direct product of the quaternion group of order eight and the cyclic group of order two.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions

## Group properties

Property | Satisfied? | Explanation | Comment |
---|---|---|---|

Abelian group | No | ||

Nilpotent group | Yes | ||

Group of nilpotency class two | Yes | ||

Metabelian group | Yes | ||

Metacyclic group | No | ||

UL-equivalent group | No | Has nilpotency class two but the derived subgroup (order two) is not equal to the center (order four); also, the group is an external direct product of groups of different nilpotency class values | See also nilpotent not implies UL-equivalent |

## GAP implementation

### Group ID

This finite group has order 16 and has ID 12 among the groups of order 16 in GAP's SmallGroup library. For context, there are 14 groups of order 16. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(16,12)`

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

`gap> G := SmallGroup(16,12);`

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

`IdGroup(G) = [16,12]`

or just do:

`IdGroup(G)`

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