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Contents 1 Definition 2 Arithmetic functions 3 GAP implentation 3.1 Group ID 3.2 Other definitions

Definition

This group, usually denoted D₁₂ (though denoted D₆ 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).

The usual presentation is:

.

With this presentation, the symmetric group of degree three is the direct factor and the complement of order two is the subgroup .

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

GAP implentation

Group ID

The group has ID 4 among the groups of order 12. It can be created using GAP's SmallGroup function:

SmallGroup(12,4)

Other definitions

The group can also be defined using GAP's DihedralGroup command:

DihedralGroup(12)