Dihedral group:D12

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
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Definition

This group, usually denoted $D_{12}$ (though denoted $D_{6}$ in an alternate convention) is defined in the following equivalent ways:

• It is the dihedral group of order twelve. In other words, it is the dihedral group of degree six, i.e., the group of symmetries of a regular hexagon.
• It is the direct product of the symmetric group of degree three and the cyclic group of order two.
• It is the outer linear group of degree two over the field of two elements, i.e., the group $OL(2,2)$.
• It is Borel subgroup of general linear group for general linear group:GL(2,3), i.e., the general linear group of degree two over field:F3.

The usual presentation is:

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

With this presentation, the symmetric group of degree three is the direct factor $\langle a^{2},x\rangle$ and the complement of order two is the subgroup $\langle a^{3}\rangle$.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 12#Arithmetic functions

Function	Value	Explanation
order	12
exponent	6
nilpotency class	--	not a nilpotent group.
derived length	2
Frattini length	1
Fitting length	2
minimum size of generating set	2
subgroup rank	2
max-length	3

Group properties

Basic properties

Property	Satisfied	Explanation	Comment
abelian group	No
complete group	Yes
Group isomorphic to its automorphism group	Yes		Being a complete group is a stronger property

GAP implementation

Group ID

This finite group has order 12 and has ID 4 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(12,4)

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

gap> G := SmallGroup(12,4);

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

IdGroup(G) = [12,4]

or just do:

IdGroup(G)

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

Other definitions

Description	Functions used
DihedralGroup(12)	DihedralGroup
DirectProduct(SymmetricGroup(3),CyclicGroup(2))	DirectProduct, SymmetricGroup, CyclicGroup