# Direct product of Q8 and Z3

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## Definition

This group is defined as the direct product of the quaternion group (of order eight) and the cyclic group of order three.

## Arithmetic functions

Function Value Explanation
order 24
exponent 12
nilpotency class 2
derived length 2
Fitting length 1
Frattini length 2

## GAP implementation

### Group ID

This finite group has order 24 and has ID 11 among the groups of order 24 in GAP's SmallGroup library. For context, there are 15 groups of order 24. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(24,11)

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

gap> G := SmallGroup(24,11);

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

IdGroup(G) = [24,11]

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 defined using GAP's DirectProduct function:

DirectProduct(SmallGroup(8,4),CyclicGroup(3))