Two names for one symmetry
The last guide sorted every crystal into one of exactly 32 crystallographic point groups — the 32 crystal classes — by asking which rotation axes, mirror planes, inversion center and rotoinversion axes it holds fixed about a single point. That gave us the objects. This guide gives us their names. And here is a wrinkle nobody warns you about: each of those 32 classes carries two different names at once, because two communities invented labels independently and both stuck. Learning to read both is not busywork — you will meet each notation in different books, and being fluent in the pair is what lets you cross between a chemistry paper and a crystallography table without getting lost.
The first name is the Schoenflies symbol, a compact letter-and-subscript code (C2v, D4h, Oh) that grew up in molecular chemistry and spectroscopy. It is wonderful for talking about a single molecule's symmetry out loud, and it dominates the vibration and orbital literature. Its weakness is that it says nothing about direction and cannot be extended cleanly once you add the translations that make a full crystal — for space groups it degenerates into ugly numbered superscripts like D4h with a little 14 tacked on.
The second name is the Hermann-Mauguin symbol — also called the international symbol, because it is the one the International Tables for Crystallography use as standard. It looks busier (1, m, 2/m, mm2, 4/mmm, m-3m), but every character is doing real work: it tells you exactly which symmetry element sits along which direction of the crystal, and it scales up gracefully to name all 230 space groups later in this rung. Think of Schoenflies as a nickname and Hermann-Mauguin as the full legal name printed on the address label.
Schoenflies: a skeleton letter plus its trimmings
Read a Schoenflies symbol in three bites: a capital letter for the skeleton, a subscript number for how many-fold the main axis is, and a subscript letter for the mirrors. The skeleton letter names the backbone. C (cyclic) means a single rotation axis and nothing crossing it. D (dihedral) means one main axis with n two-fold axes running perpendicular to it, like the spokes of a wheel. S means a lone rotoreflection axis (rotate-then-mirror). And the two cubic heavyweights, T (tetrahedral) and O (octahedral), stand for the high-symmetry classes of the cube and tetrahedron.
The subscript number is just the order of the main axis: C4 has a 4-fold axis, C6 a 6-fold. The subscript letter then bolts on mirrors and tells you where they sit relative to that axis. A 'v' (vertical) mirror contains the main axis, standing up alongside it. An 'h' (horizontal) mirror lies perpendicular to the main axis, capping it like a lid. A 'd' (diagonal) mirror is a vertical mirror that bisects the angle between the two-fold spokes. And a bare 'i' flags a center of inversion. Two short special names round it out: Ci is a crystal with only an inversion center (nothing else), and Cs is one with only a single mirror.
Put it together and the labels read almost like words. C2v is a 2-fold axis with two vertical mirrors (this is water's symmetry). C4v is a 4-fold axis with vertical mirrors. D4h is a 4-fold axis, four perpendicular 2-folds, and a horizontal mirror — a very common metal-and-mineral class. D3 is a 3-fold axis with three perpendicular 2-folds and no mirror at all (hold that one; it is quartz, and it returns in the last guide). Oh is the full symmetry of the cube. The scheme is compact and memorable — its only real blind spot is that a bare C2v never tells you which way those mirrors face in the crystal.
Hermann-Mauguin: every symbol points somewhere
The Hermann-Mauguin symbol fixes exactly that blind spot by encoding direction. Its vocabulary is small. A bare number 1, 2, 3, 4 or 6 is a rotation axis of that order — and note there is no 5 and no 7, exactly as the crystallographic restriction of guide 2 demanded. The letter m is a mirror plane. A number with a bar over it is a rotoinversion axis (rotate, then invert through a point); since the bar is awkward to type, we will write it here as a leading minus, so -4 means 'four-bar'. Finally, a slash reads as 'perpendicular to': 2/m means a 2-fold axis with a mirror plane perpendicular to it.
Now the part that makes Hermann-Mauguin powerful: the ORDER of the symbols is not decorative. Each slot in the symbol names the symmetry along a specific crystallographic direction, and which direction each slot means is fixed by the crystal system. So mm2 and 2mm use the same three characters but are read differently — the positions tell you that in mm2 the two mirrors stand along the first two directions and the 2-fold along the third. A symbol is therefore a tiny map: read left to right and you are walking down the crystal's principal, secondary and tertiary directions in a fixed order, told at each step what symmetry lives there.
Decoding a symbol, step by step
Let us decode a few real symbols. Take 4/mmm. The first slot, 4/m, is a 4-fold axis with a mirror perpendicular to it (the principal direction, [001]). The second m is a mirror along the <100> directions; the third m is a mirror along the <110> directions. So 4/mmm is a rich, fully-mirrored tetragonal class — and because it contains that perpendicular mirror on a 4-fold, it also contains a center of inversion. Its Schoenflies nickname is D4h. Now m-3m: the tell-tale sign is the 3 in the SECOND slot, which always means cubic — those are the four body-diagonal 3-fold axes of a cube. m-3m is the full symmetry of the cube (Schoenflies Oh), the class of copper, salt and diamond.
- Spot the crystal system first. A 3 or -3 in the SECOND slot means cubic (four body-diagonal 3-fold axes). Otherwise the first slot holds the single principal axis of the crystal.
- Translate each character. A bare n (1,2,3,4,6) is an n-fold rotation axis; a barred -n is a rotoinversion axis, with -1 a pure center and -2 always rewritten as m; the letter m is a mirror plane.
- Read every slash as 'perpendicular to'. n/m means the axis of that slot carries a mirror plane at right angles to it (2/m, 4/m, 6/m).
- Attach each element to its direction. Slot 1 is the principal axis; slots 2 and 3 are the secondary and tertiary directions fixed by the crystal system — that is what makes mm2 and 2mm different.
- Scan for a center. If the group contains -1 anywhere (often visible as m sitting perpendicular to every axis, as in mmm, 4/mmm, m-3m), the crystal is centrosymmetric.
SYSTEM SCHOENFLIES HERMANN-MAUGUIN H-M SLOTS READ BY DIRECTION ----------- ----------- --------------- ---------------------------------- triclinic C1 / Ci 1 / -1 one slot: the whole cell monoclinic C2h 2/m one slot: the unique axis (b) orthorhombic D2h mmm a , b , c tetragonal D4h 4/mmm [001] , <100> , <110> trigonal D3 32 [001] , <100> (quartz) hexagonal D6h 6/mmm [001] , <100> , <120> cubic Oh m-3m <100> , <111> , <110> (3 = cubic)
One more, going the other way. The Schoenflies label C2v has a 2-fold axis and two vertical mirrors; translated to Hermann-Mauguin it becomes mm2 — two mirrors (the vertical planes) along the first two directions and the 2-fold along the third. And the humble triclinic pair: Schoenflies C1 (no symmetry but the identity) is Hermann-Mauguin 1, while Ci (a lone inversion center) is -1. Practise a dozen of these and the two dialects start to feel like the same language spoken with different accents.
What the symbol tells you at a glance
Once you can read the symbol, it becomes a compact fact sheet. The first question worth asking is whether the crystal has a center of inversion. Exactly 11 of the 32 classes are centrosymmetric (-1, 2/m, mmm, 4/m, 4/mmm, -3, -3m, 6/m, 6/mmm, m-3, m-3m) — each one either shows -1 outright or is built entirely from m's sitting perpendicular to axes. The other 21 classes are non-centrosymmetric, and among them the 10 'polar' classes (1, 2, m, mm2, 3, 3m, 4, 4mm, 6, 6mm) keep one unique direction that is not flipped by any symmetry — a built-in arrow.
There is one more thing the symbol quietly warns you about, and it is honest to state now. Diffraction cannot see a crystal's handedness: because scattering imposes an apparent center of inversion (Friedel's law), the 32 point groups collapse into just 11 Laue classes that an ordinary diffraction pattern cannot tell apart — those 11 are exactly the centrosymmetric point groups. So the symbol distinguishes classes that your diffractometer, by itself, cannot. The next and final guide of this rung takes these threads — chirality, centrosymmetry, and the Laue classes — and draws them on the stereographic projection, the flat map crystallographers use to picture all of this symmetry at once.