# Direct product of Q8 and Z5

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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## Definition

This group is defined as the external direct product of the following two groups: the quaternion group of order $8$ and cyclic group:Z5.

## GAP implementation

### Group ID

This finite group has order 40 and has ID 11 among the groups of order 40 in GAP's SmallGroup library. For context, there are groups of order 40. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(40,11)

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

gap> G := SmallGroup(40,11);

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

IdGroup(G) = [40,11]

or just do:

IdGroup(G)

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

### Other descriptions

Description Functions used
DirectProduct(SmallGroup(8,4),CyclicGroup(5)) DirectProduct, SmallGroup, CyclicGroup