# Cyclic group:Z40

From Groupprops

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## Definition

This group, denoted , , or , is defined in the following equivalent ways:

- It is the unique (up to isomorphism) cyclic group of order 40, i.e., it is a group of integers modulo n where .
- It is the external direct product of cyclic group:Z8 and cyclic group:Z5, i.e., is it .

## GAP implementation

### Group ID

This finite group has order 40 and has ID 2 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(40,2)`

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

`gap> G := SmallGroup(40,2);`

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

`IdGroup(G) = [40,2]`

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 |
---|---|

CyclicGroup(40) |
CyclicGroup |

DirectProduct(CyclicGroup(8),CyclicGroup(5)) |
CyclicGroup, DirectProduct |