Dihedral group:D12

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 member of family::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 $$\langle a^3 \rangle$$.