3 Dimensional Representation of D3

[ChrisVer]

I was wondering how can I obtain the three dimensional representation of the Dihedral group of order 6, $D_3$.

If this group has the elements: $D_3 = \left \{ e,c,c^2,b,bc,bc^2 \right \}$

Where $c$ corresponds to rotation by $120^o$ on the xy plane (so about z-axis) and $b$ to reflections of the $x$ axis, I don't see how $z$ would change at all... so when I try to obtain it I'm getting the known 2-dimensional representation of $D_3$ together with an extra $1$ on the diagonal corresponding to the transformations $z \rightarrow z'=z$...
Any help?

 

[pasmith]

$D_3$ is isomorphic to $S_3$, which acts naturally (but reducibly) on $\mathbb{R}^3$ by permuting the standard basis vectors.

 

[ChrisVer]

So you are suggesting it will have the 3dim repr of S3?
I also thought that... but then the rotations and reflections as defined don't seem to help in this correspondence...:s