Point Group and Space Group Symmetry Identifier
Look up the crystal class behind any space group number, or the full symmetry profile of any of the 32 crystallographic point groups.
🪞 What is a Point Group and Space Group Symmetry Identifier?
A crystallographic point group is the set of symmetry operations that leave at least one point of a crystal fixed: rotations, mirror planes, an inversion centre and rotoinversion axes. Only 32 such groups are compatible with a repeating three-dimensional lattice, which is why they are also called the 32 crystal classes. A space group takes one of those 32 classes, places it on one of the 14 Bravais lattices and allows the extra translation-carrying operations, screw axes and glide planes. That combination produces exactly 230 distinct space groups. This tool moves between the two levels in both directions: give it a space group number and it returns the crystal class, or give it a class and it returns the full symmetry profile plus the block of space group numbers that belong to it.
The lookup matters in day-to-day work. A crystallographer who has just indexed a data set and found systematic absences pointing to space group 19 wants to know immediately that the class is 222, the Laue class is mmm and the group is chiral, so a protein can live there. A solid-state chemist screening candidate materials for a piezoelectric transducer needs to reject every centrosymmetric class before spending time on synthesis. An optics researcher looking for a new second-harmonic crystal applies the same filter, since a centre of symmetry kills the bulk second-order response entirely. A mineralogy student learning Hermann-Mauguin notation just needs a reliable way to check whether m-3m really is order 48.
The single most common misunderstanding is that the space group number encodes the lattice type. It does not. Numbering runs through the crystal classes in order, so space groups 221 through 230 are all in class m-3m, but that block contains primitive, face-centred and body-centred lattices side by side. Only the letter at the front of the Hermann-Mauguin symbol tells you the centring, and only the systematic absences in your diffraction pattern can reveal it experimentally. A second frequent slip is confusing the point group with the Laue class: the Laue class is the point group plus an inversion centre, and it is the symmetry your diffraction pattern shows, which is why 11 Laue classes cover all 32 crystal classes.
This identifier is useful whenever a symbol, a number or a physical property needs translating into the others. It reports the Schoenflies symbol next to the Hermann-Mauguin one, the group order, the Laue class used for merging reflections, and whether the class permits pyroelectricity, piezoelectricity, optical activity and second-harmonic generation. The property table is printable, so it works as a quick reference sheet next to a diffractometer or in a structure-solution session.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Space Group 225, the Rock-Salt Structure
Fm-3m, space group number 225, the setting of NaCl, MgO and copper metal
Example 2 — Space Group 19, the Protein Favourite
P2(1)2(1)2(1), space group number 19, the most common protein space group
Example 3 — Point Group 4mm, the Ferroelectric Class
Class 4mm, the room-temperature class of tetragonal barium titanate
Example 4 — Point Group -43m, the Zincblende Class
Class -43m, the class of GaAs, ZnS zincblende and cubic boron nitride
❓ Frequently Asked Questions
🔗 Related Calculators
What is the difference between a point group and a space group?
A point group collects the symmetry operations that leave at least one point fixed: rotations, mirrors, inversion and rotoinversions. A space group adds the lattice translations plus screw axes and glide planes. There are 32 crystallographic point groups and 230 space groups, and every space group reduces to exactly one point group when the translations are stripped out.
How do I find the point group from a space group number?
Space groups are tabulated in crystal class order, so each point group owns a contiguous range of numbers. Space groups 1 to 2 are triclinic, 3 to 15 monoclinic, 16 to 74 orthorhombic, 75 to 142 tetragonal, 143 to 167 trigonal, 168 to 194 hexagonal and 195 to 230 cubic. Enter the number and this tool returns the class directly.
What is a Laue class?
A Laue class is the point group with an inversion centre added. There are 11 of them, and they are the symmetry actually observed in a diffraction pattern, because Friedel's law makes intensities I(hkl) and I(-h-k-l) equal for ordinary scattering. Laue class is what you use to merge equivalent reflections during data reduction.
How many space groups are centrosymmetric?
Ninety-two of the 230 belong to one of the 11 centrosymmetric crystal classes, that is -1, 2/m, mmm, 4/m, 4/mmm, -3, -3m, 6/m, 6/mmm, m-3 and m-3m. The other 138 are non-centrosymmetric, and of those 65 are Sohncke groups containing only rotations.
What is a Sohncke space group?
A Sohncke group is a space group that contains only orientation-preserving operations, that is rotations and screw axes but no mirror, glide or inversion. There are 65 of them, and they are exactly the space groups whose point group is one of the 11 pure rotation classes. Chiral molecules including proteins can only crystallise in these.
Which point groups are piezoelectric?
Twenty classes: all 21 non-centrosymmetric classes except 432. A centre of symmetry forces the piezoelectric tensor to vanish identically, and in class 432 the four three-fold axes combined with the three four-fold axes also cancel every independent tensor component.
Which point groups are polar?
Ten: 1, 2, m, mm2, 3, 3m, 4, 4mm, 6 and 6mm. A polar class has a direction that is not transformed into any other direction by the symmetry operations, so a permanent dipole can survive. Pyroelectricity and ferroelectricity both require a polar class.
What does the order of a point group mean?
The order is the number of symmetry operations in the group, which equals the number of general equivalent positions it generates. Class 1 has order 1, class mmm has order 8 and class m-3m has order 48, the largest of the crystallographic classes.
What do the bars in symbols like -4 and m-3m mean?
They mark rotoinversion axes, normally printed with an overline. The symbol -4 means a four-fold rotoinversion, a 90 degree rotation followed by inversion through the centre. Plain text writes them as -1, -3, -4, -6, -42m, -43m and m-3m, which is what this calculator uses.
Why does the space group number not tell me the lattice centring?
Because numbering follows the crystal class first. Space groups 221 to 230 all belong to class m-3m but include primitive (Pm-3m), face-centred (Fm-3m) and body-centred (Im-3m) lattices. The centring letter lives in the Hermann-Mauguin symbol, not in the number.
Which space group is the most common for proteins?
P2(1)2(1)2(1), space group number 19, in crystal class 222. It is a Sohncke group, orthorhombic, non-centrosymmetric and non-polar, and roughly a quarter of all deposited protein structures use it. C2 (number 5) and P2(1) (number 4) follow closely behind.
Can I use point group symmetry to predict second-harmonic generation?
Yes, at the first level of screening. Second-harmonic generation requires a non-zero second-order susceptibility tensor, which vanishes in every centrosymmetric class. So a material in one of the 21 non-centrosymmetric classes may be SHG active, while one in a centrosymmetric class cannot be, at least in the bulk electric-dipole approximation.