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