Dicyclic group:Dic12

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
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

This group, sometimes denoted ${\mathrm{D}}_{\mathrm{i}}$ and sometimes denoted $\mathrm{\Gamma }(3,2,2)$, is defined in the following equivalent ways:

• It is the dicyclic group (i.e., the binary dihedral group) of order $12$, and hence of degree $3$.
• It is the binary von Dyck group with parameters $(3,2,2)$.

A presentation for the group is given by:

$⟨a,b,c\mid a^3=b^2=c^2=abc⟩$.

It also has the presentation:

$⟨a,x\mid a^6=x^4=e,a^3=x^2,xa{x}^{-1}=a^{-1}⟩$

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 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 $G$:

gap> G := SmallGroup(12,1);

Conversely, to check whether a given group $G$ 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)];