Representation of the dihedral group

Question

I'm familiar with the multiplication table of finite groups (used extensively in physics, and may be in mathematics as well). For example, the dihedral group $\mathbb{D}_3$.

Given a representation of $\mathbb{D}_3$, it is trivial to verify whether it satisfies the multiplication table. My question is, whether there is a rule to construct matrix representations (of dimension $>1$) of $\mathbb{D}_3$, starting from its multiplication table. For example, a two-dimensional or three-dimensional matrix representation of $\mathbb{D}_3$.

Answer 1

A little insight in the geometrical meaning of the dihedral group will permit you to replace the symbols in the multiplication table by matrices. If you start with the two generators $r$ and $s$ (the 120* rotation and the flip that switches $X-$ and $Y-$ axis respectively) you can use the matrices $r = \begin{pmatrix} \cos(2\pi/3) & -\sin(2\pi/3) \\ \sin(2\pi/3) & \cos(2\pi/3) \end{pmatrix}$ and $s = \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}$. And verify that they satisfy the rules of the multiplication table.