Hermann-Mauguin notation
/ HAIR-mahn moh-GAN /
Hermann-Mauguin notation is the international shorthand for writing down a crystal's symmetry, adopted by the International Tables for Crystallography and used on essentially every modern crystal-structure database entry. It is a compact code: read the symbol and you read off the symmetry elements, in a fixed order tied to the crystal's main directions.
The rules are systematic. A bare number (2, 3, 4, 6) means a rotation axis of that order; a bar over the number (4-bar, 3-bar) means a rotoinversion axis; the letter m means a mirror plane; and X/m (a number over m, written with a slash, like 4/m) means an axis with a mirror perpendicular to it. Each position in the symbol refers to a specific direction in the crystal. So 4/mmm reads: along the main axis a 4-fold with a perpendicular mirror (4/m), then mirrors perpendicular to two further sets of directions (m, m). The full symbol for table salt is Fm-3m; its point-group part is m-3m.
Hermann-Mauguin is the language of space groups too (the leading letter P, I, F, C gives the lattice centring, then the symmetry follows). Because each character maps to a direction, the notation carries orientation information that the older Schoenflies labels throw away, which is why crystallographers reach for Hermann-Mauguin and spectroscopists often prefer Schoenflies.
Quartz (low, alpha) is point group 32: a 3-fold axis with three 2-fold axes perpendicular to it, and, note, no m and no bar, so no mirror and no inversion. That absence is why quartz is chiral.
Hermann-Mauguin encodes each symmetry element by direction: numbers for axes, bars for rotoinversion, m for mirrors.
The bar is written over the number (1-bar, 3-bar, 4-bar, 6-bar). In plain-text transcription people write it as '-3' or '3bar'; either way it means rotoinversion, not a minus sign.