a point group

A point group is the complete set of symmetry operations, rotations, reflections, inversions, rotoinversions, that an object has when at least one point stays fixed the whole time. It is a full inventory of 'all the ways this shape can be moved onto itself without sliding it anywhere'. A shoe box (rectangular, no square faces) has a modest point group; a perfect cube has a large one.

When you keep only the operations allowed in a periodic crystal (rotation axes limited to 1, 2, 3, 4, 6 by the crystallographic restriction), the possibilities are not endless, they close into exactly 32 crystallographic point groups, also called the 32 crystal classes. Every crystal on Earth belongs to one of these 32. They sort into the seven crystal systems by which rotation axes they contain: for example, any class with four 3-fold axes is cubic. Each class is named twice, once in Hermann-Mauguin (like 4/mmm) and once in Schoenflies (like D4h).

The point group is the crystal's symmetry fingerprint, and it dictates the physics. Whether a crystal can be piezoelectric, pyroelectric, or optically active is decided entirely by its point group, specifically by whether it contains an inversion centre or a unique polar axis. Point groups describe symmetry about a point only; add translations and each point group blossoms into several space groups (230 in all).