Reading Hermann–Mauguin

The symbol p4g contains four pieces of information and looks like it contains none. It is not a name; it is a set of instructions for building the group, and the instructions can be read off left to right.

Half an hour spent on the rules converts the seventeen from a list to be memorised into something that can be reconstructed, and the same rules scale unchanged to the two hundred and thirty space groups.

The first character: the cell #

The symbol opens with a lower-case letter naming the lattice centring.

p means a primitive cell — one lattice point per cell, counting corners as quarters. Fifteen of the seventeen begin with p.

c means a centred cell — a rectangular cell with an extra lattice point at its centre, so two lattice points per cell. Two groups begin with c: cm and cmm.

That is the whole of the first character, and it is a statement about the description rather than about the pattern. Both cm and cmm could be described with a primitive rhombic cell instead, at the cost of axes that no longer lie along the mirror directions — the trade discussed in the essay on unit cells, which crystallography resolves in favour of clarity every time.

The first number: the rotation #

Next comes a digit giving the highest order of rotation present: 1, 2, 3, 4 or 6, and nothing else, by the crystallographic restriction.

A digit of 1 is usually omitted where it would be uninformative — the group written pm is strictly p1m1 — and this abbreviation is the single largest source of confusion in reading the symbols. The full symbol always has the same number of positions; the short symbol drops the ones that say nothing.

So p4 is a primitive cell with fourfold rotation and no reflection at all. p6 is a primitive cell with sixfold rotation and no reflection. p2 is a primitive cell with half turns and no reflection. Three groups, fully described by two characters each.

The remaining characters: reflections, positionally #

Here is where the notation earns its keep and where readers most often give up.

After the rotation come one or two characters describing reflections, and their position in the symbol says which direction of the lattice they refer to. An m means a mirror perpendicular to that direction; a g means a glide perpendicular to it; a 1 means neither.

The directions are fixed by convention for each lattice type. For a rectangular lattice the first position refers to the $x$ axis and the second to the $y$ axis. For a square lattice the first refers to the axes and the second to the diagonals. For a hexagonal lattice the first refers to the axes and the second to the directions bisecting them.

So pmm has mirrors perpendicular to both axes. pmg has a mirror perpendicular to one axis and a glide perpendicular to the other. pgg has glides perpendicular to both, and no mirror. p4m has mirrors along the axes and along the diagonals. p4g has a mirror in one family and a glide in the other.

The two pairs, and why the symbols differ #

The positional convention is what makes the classification’s two awkward pairs describable at all.

p3m1 and p31m have the same lattice, the same point group and the same number of operations. The symbols place the m in different positions: p3m1 has mirrors perpendicular to the axes and nothing in the second family, and p31m has nothing perpendicular to the axes and mirrors in the bisecting family. The digits are not decoration — they are the notation stating which of two genuinely different arrangements is meant.

p4m and p4g are the same story on a square lattice. Both have mirrors and glides available; the difference is which family carries the mirror. Their own essay works through the threefold pair in detail, and the fourfold pair follows the identical logic.

A reader who treats the symbols as arbitrary names finds these pairs inexplicable. A reader who knows the symbols are positional finds them obvious, which is the argument for learning the rules rather than the list.

The seventeen, decoded #

With the rules in hand, the whole list can be read rather than recalled. Taking them in order of rotation:

p1 — primitive cell, no rotation, no reflection. Translations only. p2 — primitive cell, half turns, no reflection. pm — mirror perpendicular to the first axis; strictly p1m1. pg — glide perpendicular to the first axis; strictly p1g1. cm — the same as pm on a centred cell, which forces glides between the mirrors. pmm — mirrors perpendicular to both axes. pmg — a mirror one way, a glide the other. pgg — glides both ways, no mirror. cmm — the pmm arrangement on a centred cell. p3 — threefold rotation, no reflection. p3m1 — mirrors perpendicular to the axes. p31m — mirrors in the bisecting directions instead. p4 — fourfold rotation, no reflection. p4m — mirrors along both the axes and the diagonals. p4g — a mirror in one family, a glide in the other. p6 — sixfold rotation, no reflection. p6m — sixfold rotation with mirrors in both families.

Reading the list this way also makes the gaps informative. There is no p6g, because a glide compatible with sixfold rotation composes with it to force a mirror, so the arrangement collapses into p6m. There is no p3g1 for the same reason. Every absence in the list is an absence the composition rules produce, not a case somebody forgot.

Reading the four glide groups #

The four non-symmorphic groups — pg, pgg, pmg, p4g — are exactly the ones whose symbols contain a g, and the correspondence is not a coincidence.

A g in the symbol means the group contains a glide whose axis is not also a mirror. That is precisely the condition for non-symmorphism: no choice of origin removes the translation part from every operation at once. So the symbol announces, in one character, the structural property that makes a group awkward.

This is worth knowing because those four are the ones most often mislabelled by hand. A pattern in pg presents reversed motifs and the eye reads reversal as evidence of a mirror; the symbol pm is written; and the resulting figure is quietly of the wrong group. The g is the notation warning that the reflection has a slide attached and that checking is required.

Why positional and not descriptive #

It is worth asking why crystallography chose a notation this terse, since the alternative — spelling out the operations — would be easier to read cold.

The answer is that the symbol has to be short enough to appear in a table header, on a diagram, and in the running text of a structure paper, thousands of times. Two hundred and thirty space groups, each appearing constantly, in a literature where every paper reports one: economy is not a stylistic preference but a practical requirement.

The second answer is that the positional convention encodes precisely the information a diffraction experiment can measure. The directions in a crystal are what an experiment sees — a reflection is indexed by three integers naming a direction in reciprocal space — so a notation organised by direction maps directly onto the observable. A notation organised by abstract group type, as Schoenflies is, does not.

That is why the two notations survive side by side rather than one displacing the other. Chemists asking what a molecule’s symmetry is want the abstract type. Crystallographers asking which reflections will be absent want the directions. Both questions are legitimate and they want different symbols.

What the symbol does not tell #

Two things a group symbol is silent about, and both cause trouble.

It does not describe the pattern. Infinitely many patterns share the group p4m, and knowing the symbol tells a reader what symmetries the pattern has and nothing about what it looks like. The group determines the symmetry; the motif determines the appearance.

It does not fix the origin. The symbol says which operations exist and in which directions, not where the origin sits. Two descriptions of the same structure under different origin conventions differ by a fixed shift in every coordinate, and the International Tables list two origin choices for many space groups precisely because the symbol cannot settle it.

There is also a subtlety about settings. Because the symbol is positional, exchanging which axis is called first can change the symbol without changing the group. In the plane this is mild — pmg and pgm are the same group with the axes swapped, and only one is standard. In three dimensions it is not mild at all, and the same space group appears in the literature under more than one symbol.

What the terseness costs #

Economy is bought with two specific losses, and a reader who knows what they are stops being surprised by them.

The symbol cannot be read without knowing the lattice first. Every character after the first is positional — it says which operation is perpendicular to the first family of directions, then the second — and which families those are depends on the crystal system. So the symbol for a hexagonal group cannot be parsed by someone who does not already know that a hexagonal lattice has two inequivalent families of directions at $30°$ to one another. That is a genuine barrier: the notation is transparent to a reader who knows the classification and opaque to one learning it, which is the wrong way round for a teaching notation and the right way round for a working one. It is also the direct cause of the p3m1 and p31m confusion, since the whole distinction lives in which family the character occupies.

The symbol names generators, not contents. It lists enough operations to generate the group and stops, so it systematically under-reports what is present. The symbol p4m names a fourfold rotation and two mirror families; the group closes to eight operations per cell, and the detector on this site — which enumerates what the point set has rather than what the label claims — reports two fourfold rotations, four mirrors, and a twofold rotation that no character of the symbol mentions. The half turn is there because a quarter turn applied twice is one, and the notation leaves it to the reader to notice.

The under-reporting can go the other way as well. The symbol p4g names a fourfold rotation and a glide; the closure contains eight operations, and among them is one genuine mirror, which the letter g does not lead a reader to expect. Presence is a consequence of closure, not of naming, and only in the simplest groups do the two coincide.

This is the sense in which a group symbol is a compressed instruction rather than a description. It says: take these, close under composition, and see what results. What comes out is larger than what went in, always, and the diagrams in the International Tables exist because the closure is not something a reader should be asked to perform mentally.

Reading a symbol backwards #

The notation is most useful when run in reverse: given a pattern, produce the symbol. The procedure is short and it is exactly the decision tree the classification supplies.

Find the highest rotation order. One, two, three, four or six — nothing else is available. This fixes the digit and narrows the candidates to at most five.

Decide whether any reflection is present. Not a reversal of handedness, which glides also produce, but a pure reflection with no slide attached. If there is none, the symbol ends at the digit unless a glide is present.

For each family of directions, decide mirror, glide or neither. The families are fixed by the lattice type. This fills the remaining positions.

Decide whether the cell is centred. A centred cell shows itself as mirrors alternating with glides at half-cell spacing, and it is the last thing to check because it is the easiest to mistake for something else.

Each step is a question about the pattern rather than about the picture, and each can be answered by applying an operation and comparing rather than by judging. That distinction is the difference between a classification a reader can trust and one they merely believe, and it is why every figure here is generated and checked rather than drawn and labelled.

The same rules in three dimensions #

The notation scales without modification, which is its real achievement.

A space-group symbol opens with a centring letter — P, C, I, F, R for primitive, base-centred, body-centred, face-centred and rhombohedral — and continues with up to three positions describing symmetry along the three principal directions of the crystal system. The characters are richer: as well as m and g there are $2_1$, $3_1$, $4_1$ and their relatives, denoting screw axes, which are rotations combined with a slide along the axis.

So P2₁/c, one of the commonest space groups in organic crystallography, reads as: primitive cell, a twofold screw axis along $b$, and a glide plane perpendicular to $b$ with its slide along $c$. Four pieces of information in six characters, and every one of them checkable against a diffraction pattern.

Where the symbols come from #

The notation is named for Carl Hermann and Charles Mauguin, who introduced it independently in the late 1920s — Hermann in 1928, Mauguin in 1931 — and it was adopted as standard by the International Tables for X-ray Crystallography in 1935.

What it replaced was Schoenflies notation, which describes abstract group types rather than generator positions. Schoenflies symbols are more compact and remain standard in chemistry and spectroscopy, and they have a specific weakness for this subject: because they name the abstract group, they cannot distinguish the pairs where position is the entire difference. p4m and p4g have the same Schoenflies designation in the point-group sense, and the crystallographic distinction is invisible in it.

That is the reason crystallography went its own way. A notation for crystals has to record where the elements sit, because where they sit is what a diffraction experiment measures.

What this site does with it #

Every pattern figure here carries its Hermann–Mauguin symbol, and every symmetry element is drawn in the International Tables marks — lens for twofold, triangle for threefold, square for fourfold, hexagon for sixfold, solid line for a mirror and dashed for a glide.

That is a deliberate choice, and the alternative would have been worse. Inventing a friendlier notation would make this site’s figures unreadable everywhere else, and a reader who learns the marks here meets the same marks in every other source, from a textbook to a structure paper to the Tables themselves.

The marks are placed by computation rather than by hand: the detector finds an operation, reads its determinant to decide handedness, reads its trace to get the rotation order, and locates the element from the operation’s own matrix and translation. So the mark on the picture and the character in the symbol are two readings of the same object, and they cannot disagree.

Where the ladder goes next #

The obvious next step is the pair the notation exists to distinguish: p3m1 and p31m, where the position of a mirror is the whole difference between two groups.

The obvious hazard is that a figure can display the wrong symbol without looking wrong, which is the comma problem.

And the confirmation route is diffraction, where a g in a symbol becomes a testable prediction about which reflections vanish.

What the pictures here cannot show. A symbol is a claim about which operations exist and where; a figure displays a pattern. The correspondence between them is checked by comparing the detected operations with the symbol’s, which is arithmetic rather than looking, and no drawing on this page establishes it.