One group, three symbols

Search the structural literature for P2₁/c and it returns a third of all published small-molecule structures. Search for P2₁/n and it returns a great many more. Search for P2₁/a and there are thousands, mostly older.

They are the same group. Not related groups, not settings of a family — one group, number 14, written on three different choices of axes, and a crystallographer who has not met the fact can read three papers and believe they concern three different symmetries.

The reason is completely specific and it is visible in the symbol. A glide letter names which lattice vector the glide’s translation is half of, and a change of basis changes which vector that is. Replace c by a + c and a c-glide becomes an n-glide, without a single atom moving.

Deriving a symbol instead of looking one up #

The site has derived a point group’s symbol from its own operations since the point-groups phase: walk the system’s symmetry directions, write down the highest-ranking axis on each and the mirror across it. A space group’s symbol is that walk with two additions — the centring letter in front, and the translation parts turning rotations into screws and mirrors into glides.

Both additions are decided by arithmetic that this site already had. A screw’s subscript is how many n-ths of a cell the operation advances, which is its intrinsic translation divided by the lattice vector. A glide’s letter is decided by which vector its intrinsic translation is half of: a, b or c for an axial glide, n for half a face diagonal, d for a quarter of one, and m for none at all.

Every group this site defines derives its own name #

A derivation is only worth having if it is checked, and the check is the obvious one: run it over every group the site defines and compare against the symbol each was entered under.

The two entries that took the most care are worth naming, because both are mistakes this site has made before in other places.

A rotoinversion’s order is not its matrix order. 3̅ has matrix order six, so a classifier reading the order off the matrix writes P3̅m1 as P6̅2/m — a symbol for something that does not exist. The pair (determinant, trace) decides it in one step, which is the same repair the point-groups phase made for the same reason.

A screw’s subscript depends on the sense of the turn. P4₃ contains an operation that is a 4₁ about the reversed axis, so a derivation taking whichever operation it meets first calls half of the eleven enantiomorphic pairs by their partner’s name. The convention is the right-handed sense, and the test is the sign of the vector part of M − Mᵀ against the axis.

What the derivation found here #

The check above says forty-five of forty-five, and it says so because one entry was corrected.

Since the space-groups phase this site has carried a group under the symbol Cmcm. Deriving its symbol from its own operations gives Ccmm: the c-glide is perpendicular to a in these operations and perpendicular to b in the standard setting of Cmcm. The two are settings of one group — number 63 either way — and the entry was in the second.

Nothing drawn from it was wrong. Every figure showed the operations the entry actually contains, every caption described the picture in front of it, and the extinction conditions computed from it were the conditions those operations produce. What was wrong was the name, and no gate on this site could have caught it: a symbol is a string in a table, and nothing compared it against the operations it labels until a derivation existed to do the comparing.

The entry is now named Ccmm, with a comment saying what it was called and why it changed. That is the honest repair — renaming the group to what it is, rather than reorienting its generators and invalidating four essays’ worth of captions.

The same group, drawn twice #

The clearest demonstration that nothing has changed is to draw the group in two of its settings and compare the pictures.

There is a check on that claim built into the settings figure, and it is worth stating because it is what separates a setting from a different group. Three things are required of every row: the basis change has determinant ±1, so it maps the lattice onto itself; conjugating the operations back by the inverse reproduces the original set exactly; and the census of operation kinds — how many rotations, screws, mirrors, glides — is unchanged, since conjugation cannot turn one kind into another. A row failing any of the three would be a different group under a different name, and the figure would say so.

Why monoclinic groups have the most names #

The six cell choices in the first figure are a monoclinic phenomenon, and the reason is that a monoclinic lattice has one constrained direction and two free ones.

b is the unique axis, fixed by the symmetry. a and c span the plane perpendicular to it, and any pair of vectors spanning the same lattice plane is as good a basis as any other. The conventional choices — six of them — differ by which vector is called a, which is called c, and whether their sum is used instead. Each keeps the cell monoclinic; each renames the glide.

In a group with no glide, nothing changes. P2₁ has one symbol under all six choices, because a screw along b is a screw along b whatever is done to the other two axes — the operation’s translation is along the one axis the cell choices leave alone, so there is no other name available for it.

What a name is for, and what it is not for #

There is a general point underneath, and it is one this site keeps arriving at from different directions.

A Hermann–Mauguin symbol is a description of a group in a basis. It is not a name of the group in the way a number is: number 14 is a fact about the group, and P2₁/c is a fact about the group together with a choice of axes. The two hundred and thirty numbers are canonical and the five hundred and thirty symbols are not.

That is exactly the distinction the cell is a choice makes about lattices, and the origin is a choice makes about coordinates, one level up. Every level of this subject has a layer of convention on top of a layer of fact, and reading the convention as fact is the standing hazard: three symbols get taken for three groups, six cell choices for six lattices, two origins for two structures.

The symptom is always the same — a count that comes out too large — and the repair is always the same: derive the thing from the operations and see how many distinct answers there really are.

How many names a group has, and what fixes the number #

The six choices in the first figure are a monoclinic accident. The general question — how many symbols one group can be written under — has a clean answer, and it is the answer this whole site keeps arriving at: count the descriptions, then quotient by the changes of basis that do nothing.

A change of basis carries a group’s operation set to another operation set. Some of those changes carry the group onto itself: they permute its operations without producing anything new, and the set of them is the group’s affine normaliser. Every other change gives a genuinely different description, and the number of distinct symbols is the number of orbits the remaining changes have.

That is why symmetrical groups have fewer names. P1 has one symbol under every basis whatever, because there is nothing in the group for a basis change to relabel — and it has infinitely many cells, which is the same freedom seen from the other side. At the other end, a group whose symmetry directions are all inequivalent has a symbol for each way of ordering them.

Orthorhombic is where that bites hardest, and it is a more common nuisance than the monoclinic case because it is not confined to glides. The three axes of an orthorhombic group are three distinct symmetry directions, so the six permutations of a, b and c can give six symbols for one group. Pnma is the standard setting of a group the perovskite literature very often writes as Pbnm, and Pmcn, Pnam, Pmnb and Pcmn are the other four. Nothing distinguishes them but which axis was called which, and a reader who has met only one of the six has no reason to suspect the other five exist.

The number is the invariant, and the symbol is not #

There is a canonical name and it is not a symbol at all: the group’s type number, one to two hundred and thirty. It is what survives every change of basis, and it is the thing to compare when the question is whether two structures are in the same group.

This is why a structure file carries the number alongside the symbol rather than instead of it. A symbol has to be parsed and interpreted in a basis before it means anything; a number is an index into the classification and means the same to everyone. Given operations rather than a name, the number is derived exactly as the symbol was derived above — transform to the standard setting, walk the symmetry directions, and look up what comes out.

The failure this prevents is a counting error, and it is the same shape as the one a symbol used as an identity produces elsewhere. A survey that tallies published structures by their printed symbol splits its own largest population three ways: P2₁/c, P2₁/n and P2₁/a are one group, and any statement about how common that group is has to add them back together before it means anything. The mistake is invisible in the output — three plausible bars on a chart, no missing data, nothing to notice.

And the number alone is not enough either, which is the reverse error and the rarer one. A coordinate list is meaningless without knowing which setting it was measured in, because the coordinates are in a basis and the number does not name one. A structure reported as “number 14” with a cell whose β is near 90° and no transformation given has published half of what it knows. The number fixes which group; the setting fixes which description; a reader needs both, and the two together are the whole of what a name is for.

The two failures are worth stating together, because they are opposite and a report can commit either. A symbol without a number invites a reader to compare descriptions and conclude that two settings are two groups. A number without a setting leaves a coordinate list that cannot be placed in a cell. Neither is a small omission and neither shows up as an error in the file.

What the Tables do about it #

The International Tables handle this by listing settings explicitly. The 230 space groups have, in the Tables’ arrangement, 530 settings — a group appears once as a group and several times as a description, with the transformations between them written out.

That arrangement carries two obligations for anybody publishing a structure, and both are routinely met and occasionally not.

State the setting, not only the symbol. P2₁/n is not ambiguous as a name, but the cell parameters that go with it are a different cell from the one P2₁/c would use, and a reader who reindexes without noticing gets a structure that will not refine.

Give the transformation if a non-standard setting is used. A matrix relating the reported cell to the standard one is four lines and removes every ambiguity; a database that has it can reindex automatically, and a reader who has it can compare two structures without guessing. The same discipline applies to a lattice’s own description, where the conventional cell is a choice made for readability and the primitive one is always available underneath.

And prefer the standard setting unless there is a reason. The reason is usually a good one — a cell whose β angle is nearer 90° refines more stably, so P2₁/n is often chosen deliberately over P2₁/c — and saying so takes one sentence.

Why P2₁/c is a third of all structures #

There is a fact worth explaining while the group is in front of the reader, because the explanation is a symmetry argument and the fact is a survey.

Something like a third of published organic crystal structures are in P2₁/c, and something like three quarters are in one of five groups. That distribution is not what a uniform choice among 230 would give, and the reason is packing rather than accident: molecules pack most efficiently when they can be related by operations that bring them face to face without requiring them to be symmetric themselves. A screw axis and a glide plane do exactly that, an inversion centre does it too, and P2₁/c has all three.

Groups with mirrors do badly by comparison, because a mirror requires either a symmetric molecule or two molecules in the asymmetric unit; and chiral molecules cannot use them at all, which is why P2₁2₁2₁ dominates among chiral structures the way P2₁/c does among everything.

None of that is derivable from the symmetry alone — it is a statement about how molecules fit, and this site computes no packing energies. What symmetry supplies is the list of operations available to bring one molecule against another, and the observation that the popular groups are the ones whose operations are all of that kind.

Where the exactness stops #

Three limits.

Six cell choices is a convention, not a theorem. The bases enumerated here are the ones the Tables treat as standard for a monoclinic cell with b unique. A monoclinic lattice has infinitely many bases, most of which give cells nobody would use; restricting to the conventional six is a choice made for reasons of convenience, and a group has as many symbols as there are bases somebody is willing to write down.

The derivation covers the groups this site defines. Forty-five of them, chosen to exhibit every kind of screw and glide, which is enough to exercise every branch of the walk. Extending it to all 230 would need the rhombohedral and cubic settings in full, and this site does not enumerate the 230 in the first place.

Settings are not the same thing as enantiomorphs. P4₁ and P4₃ are two groups, not two names for one — no change of basis with positive determinant relates them, and the pair is a genuine entry in the count of 230. A basis change with negative determinant does relate them, which is why they are one entry in the affine classification’s 219. Settings, enantiomorphs and affine classes are three different equivalences and the numbers differ accordingly.

And the check compares a derivation against a table. The forty-five names were entered by hand from the International Tables, so the agreement says the derivation reproduces the Tables’ convention — not that the convention is right, which is not a question with an answer. What the check catches is an entry that disagrees with its own operations, which is exactly what it caught.

Where the ladder goes next #

The choice of origin is the other half of the same subject and is quietly more dangerous. A group with an inversion centre has two conventional origins — one on the centre, one at a point of higher site symmetry — and coordinates published in one are meaningless in the other. The Tables call them origin choice 1 and origin choice 2, structure reports name which they used about as often as they name their setting, and the failure mode is a structure that looks fine and is displaced by a quarter of a cell.