# Dicyclic group:Dic12

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

- It is the dicyclic group (i.e., the binary dihedral group) of order , and hence of degree .
- It is the binary von Dyck group with parameters .

A presentation for the group is given by:

.

## GAP implementation

### Group ID

This finite group has order 12 and has ID 1 among the groups of order 12 in GAP's SmallGroup library. For context, there are 5 groups of order 12. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(12,1)`

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

`gap> G := SmallGroup(12,1);`

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

`IdGroup(G) = [12,1]`

or just do:

`IdGroup(G)`

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

### Other definitions

The group can also be defined using its presentation:

F := FreeGroup(3); G := F/[F.1^3 * F.2^(-2), F.2^2 * F.3^(-2), F.1 * F.2 * (F.3)^(-1)];