The dihedral group of order six as a matrix group

The dihedral group of order six is the group generated by a rotation of angle $2\pi/3$ and a reflection.

Let us verify that the group generated by the matrices $\begin{pmatrix} -1/2 & -\sqrt{3}/2\\ \sqrt{3}/2 & -1/2 \end{pmatrix}$ and $\begin{pmatrix} 1&0\\0&-1 \end{pmatrix}$ is isomorphic to the dihedral group of order six.

> Q<x> := PolynomialAlgebra(Rationals());
> f := x^2 - 3;
> K<z> := NumberField(f);
> a := Matrix(2, [-1/2,-z/2,z/2,-1/2]);
> b := Matrix(2, [1,0,0,-1]);
> G := MatrixGroup<2,K|a,b>;

Which group is this?

GroupName(G);` 
S3

And we all know that the dihedral group of order six is isomorphic to $\mathbb{S}_3$.