Let's have a stiffness tensor $ a^{ijkl}: a^{ijkl} = a^{jikl} = a^{klij} = a^{ijlk}. $
It has a 21 independent components for anisotropic body.
How does body symmetry (cubic, hexagonal etc.) changes the number of independent components of the tensor? For example, for absolutely isotropic body tensor has 2 independent components, and for hexagonal symmetry $C_{6}$ (with an z-axis symmetry) it has five components. How to explain it?