# Direct product of Q8 and Q8

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## Definition

This group can be defined as the external direct product of two copies of the quaternion group (which is a group of order eight).

## GAP implementation

### Group ID

This finite group has order 64 and has ID 239 among the groups of order 64 in GAP's SmallGroup library. For context, there are groups of order 64. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(64,239)

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

gap> G := SmallGroup(64,239);

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

IdGroup(G) = [64,239]

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 the SmallGroup function for the quaternion group and the DirectProduct function:

DirectProduct(SmallGroup(8,4),SmallGroup(8,4))