How is knowing about the symmetry of a crystal useful I am studying the space groups and know that we use them to talk about crystal symmetry, I would like to know what material properties may be produced or predicted from knowing the symmetry or lack thereof in a crystal system of a material?
Properties that are directly dependant on crystal symmetry are


birefringence
piezoelectricity
(obviously) x-ray and electron diffraction pattern
pyroelectricity
nonlinear optical properties
...