GTPack is a free Mathematica group theory package containing more than 200 additional group theory modules for the Mathematica language. GTPack builds a bridge between computational algebra, university education and modern research, with wide-ranging applications in condensed matter and solid-state physics, photonics, and quantum chemistry (Latest version: GTPack 1.2 (June 2020)).
Basic functionality: install point and space groups; calculate multiplication tables, classes, cosets, character tables, representation matrices, projection operators, Clebsch-Gordan coefficients, etc.
Structure: store, manipulate and generate structures of crystals and molecules; import/export to/from standard formats such as cif, POSCAR, etc.
Applications: construct Hamiltonians based on tight-binding, plane-wave expansion, crystal field theory; construct master equation for photonic crystals; calculate phonons based on harmonic approximation; analyze symmetry of electronic, photonic, phononic states; etc.
See examples at http://gtpack.org/tutorials