Allowable irreducible representations of the point groups with five-fold rotations – that represent the symmetry of the quasicrystals in two and three dimensions – are derived by employing the little group technique in conjunction with the solvability property. The point groups $D_{5h}(\bar{10}m2)$ and $I_{h}(\dfrac{2}{m} \bar{3} \bar{5})$ are taken to illustrate the method.
