# Dihedral group:D12

(Redirected from Direct product of S3 and Z2)
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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:

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

## 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 $G$:

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