Dicyclic group:Dic12

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This group, sometimes denoted $\operatorname {Dic} _{12}$ and sometimes denoted $\Gamma (3,2,2)$, 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:

$\langle a,b,c\mid a^{3}=b^{2}=c^{2}=abc\rangle$.

It also has the presentation:

$\langle a,x\mid a^{6}=x^{4}=e,a^{3}=x^{2},xax^{-1}=a^{-1}\rangle$

Group properties

Property	Satisfied?	Explanation	Comment
Abelian group	No
Nilpotent group	No
Leinster group	Yes	Sum of orders of proper normal subgroups equals order	Smallest non-abelian & non-nilpotent Leinster group

Subgroups

Subgroup-defining functions

Subgroup-defining function	What it means	Value as subgroup	Value as group	Order
center	elements that commute with every group element	$\{e,x^{2}\}$	cyclic group:Z2	2

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)];