Determine the symmetric group G of this infinite hexagonal tiling $\varGamma$. i.e. G={g$\in$ G｜g$\varGamma$=$\varGamma$}. What I know is that G is generated by certain rotations, reflexions and translations. And the rotations and reflexions part is the dihedral group of order 6 $D_{6}$. If I choose the natural basis $e_{1}=(1,0)$ and $e_{2}=(0,1)$ of $R^{2}$, then $D_{6}$ is generated by R=\begin{pmatrix} cos(π/3) & -sin(π/3) \\ sin(π/3) & cos(π/3) \end{pmatrix} and S=\begin{pmatrix} cos(π/3) & sin(π/3) \\ sin(π/3) & -cos(π/3) \end{pmatrix} And I am down to the translation part T which is generated by $T_{i}$ which is translation by ($x_{i}$,$y_{i}$) Where ($x_{i}$,$y_{i}$) could be ($3/2,\sqrt 3/2$) and ($-3/2,\sqrt 3/2$) and ($0,-\sqrt 3/2$).But I don't know how to represent $T_{i}$ in matrix form. Any help? Then I Can write G=〈$R$,$S$,$T_{1}$,$T_{2}$,$T_{3}$〉 Am I right? Or did I miss something? Many Thanks!