a space group

Think of a crystal as an infinite three-dimensional wallpaper: the same motif of atoms stamped over and over at every lattice point. A space group is the complete list of ALL the ways you can move that wallpaper — rotate it, reflect it, invert it, slide it, screw it, glide it — and have it fall back onto itself unchanged. It is the full symmetry of a periodic crystal, the point symmetry about a spot COMBINED with the translational symmetry that repeats it through space.

There are exactly 230 space groups in three dimensions — no more, no fewer. This is a proven, complete count (worked out independently by Fedorov, Schoenflies and Barlow in the 1890s), not a running tally that might grow. Each one is a particular marriage of the 14 Bravais lattices (how the repeat points are arranged), the 32 crystallographic point groups (the rotations, mirrors and inversions), and the two translation-carrying operations — screw axes and glide planes. Every conceivable periodic crystal, from table salt to a protein, belongs to exactly one of these 230.

The space group is a crystal's symmetry DNA. Once you know it you can immediately say which crystal system and point group the material has, which diffraction reflections must vanish (its systematic absences), how many atoms of each kind sit in a cell, and what the smallest building block — the asymmetric unit — looks like. Real examples: copper and rock salt are Fm-3m; hexagonal close-packed metals are P6_3/mmc; the mineral olivine and many others are Pnma; most organic molecular crystals are P2_1/c.