Dihedral group:D12
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Contents
Definition
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
.
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 |
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 5 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 :
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:
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 |