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This group, usually denoted (though denoted 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 .
- 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:
With this presentation, the symmetric group of degree three is the direct factor and the complement of order two is the subgroup .
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 12#Arithmetic functions
|nilpotency class||--||not a nilpotent group.|
|minimum size of generating set||2|
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 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(12,4);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [12,4]
or just do:
to have GAP output the group ID, that we can then compare to what we want.
|DirectProduct(SymmetricGroup(3),CyclicGroup(2))||DirectProduct, SymmetricGroup, CyclicGroup|