Point groups are more important to pure chemistry than materials science, because chemistry often deals with single molecules, while materials science usually deals with crystals that repeat through space (space groups).
However, since space groups are derived from point groups, it’s important to understand the point groups first. I provided a detailed list of all 2D and 3D point groups in this article.
In 3 dimensions, there are 32 point groups. Point groups are mathematical constructs that capture all the non-translation symmetry options that can be performed on an object (reflection, rotation, or rotoinversion).
|Crystal System||Number of|
You can remember the difference between point groups and space groups because point groups don’t involve translation (they stay at a stationary point), while space groups do involve translation through space.
If you’re just looking for a list of point groups, I’ve collected those in a separate article. Check out the full list of 2D and 3D point groups, with alternative notation systems.
To understand point groups, you first need to know what symmetry operations are:
- Reflection is a symmetry operation which causes a set of points to be mirrored across a plane. We call this plane the “mirror plane.” Any point (X,Y,Z) becomes (-X,Y,Z) if there is a mirror axis perpendicular to the X direction.
- Rotation is a symmetry operation which causes a set of points to be rotated around a point. We call this point an “axis of rotation.” In polar coordinates, any point (R, θ, φ) becomes (R, θ + 360º/n, φ) for an n- fold rotation axis perpendicular to θ.
- Inversion is a symmetry operation which pulls every point through an “inversion center” to the other side. Any point (X, Y, Z) becomes (-X, -Y, -Z) if there is an inversion center at the origin. You can combine rotation with inversion to produce the rotoinversion symmetry operation.
- Translation is a symmetry operation that moves a set of points through space. Point groups DO NOT involve translation. Any point (X,Y,Z) becomes (X+a, Y, Z) for translation of length a in the X-direction.
Constructing Point Groups
Point groups show all symmetry relationships in a set of points that don’t move. It turns out that in 3 dimensions, there are only 32 point groups in 3 dimensions. Let’s build them!
First, we can fit each point group to a crystal system: Triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.
When naming the point groups, we typically write the highest-order symmetry on each perpendicular axis. The “/” sign indicates that 2 operations are present in the same axis. For example, 6/mmm can be split into 6/m m m, which means that there is a 6-fold rotation + mirror plane in the x-direction, a mirror plane in the y-direction, and a mirror plane in the z-direction.
Obviously, a six-fold axes of rotation also includes 3-fold, 2-fold, and 1-fold axes of rotation, so we don’t also need to include those if we already know there’s a 6-fold axis of rotation.
If we follow these rules, we can create the 32 point groups with “Hermann-Mauguin” notation.
The triclinic crystal system has the lowest symmetry.
There are 2 triclinic point groups:
- 1: Point group 1 has no symmetry operations besides the trivial 360º rotation from (X, Y, Z) to exactly the same (X, Y, Z). Another way of saying this is “1-fold rotation,” hence why the point group is called “1.”
- -1: Point group -1 (this is pronounced “bar one” and written as when you have fancy formatting available) has a center of inversion (that’s where the minus comes from), but still no mirror planes and only the 1-fold axis of rotation.
The monoclinic crystal system can have at most one 2-fold axis of rotation or one mirror plane.
There are 3 monoclinic point groups:
- 2: Point group 2 has a 2-fold axis of rotation (180º).
- m: Point group m has one mirror plane. “M” stands for mirror plane, and since there are no other symmetry operations, the point group is just “m.”
- 2/m: Point group 2/m has a 2-fold axis of rotation perpendicular to a mirror plane. Technically it’s more accurate to say the axis of rotation is “parallel to the normal” of the mirror plane, because the “/” sign indicates parallel elements, and mirror planes are defined by their perpendicular vector.
Orthorhombic crystal systems can have three 2-fold axes of rotation or one 2-fold axis of rotation with two mirror planes.
There are 3 orthorhombic point groups:
- 222: Point group 222 has three perpendicular 2-fold axes of rotation. This means that any point (X,Y,Z) creates (X,Y,-Z), after a 2-fold rotation around the x-axis (-X,Y,Z), and (-X,Y,-Z) after a 2-fold rotation around the y-axis, and (X,-Y,Z), (X,-Y,-Z), (-X,-Y,Z), and (-X,-Y,-Z) after a 2-fold rotation around the z-axis.
- mmm: Point group mmm has three perpendicular mirror axes. This means that any point (X,Y,Z) creates (-X,Y,Z) after a reflection through the mirror axis normal to the x-direction, (X,-Y,Z) and (-X,-Y,Z) after a reflection through the mirror axis normal to the y-direction, and (X,Y,-Z), (-X,Y,-Z), (X,-Y,-Z) and (-X,-Y,-Z) after a reflection through the mirror axis normal to the z-direction
- mm2: Point group mm2 has two perpendicular axes of rotation and one 2-fold axis of rotation, all three operations are perpendicular to each other
Tetragonal crystal systems have one 4-fold axis of rotation.
There are 7 teragonal point groups:
- 4: Point group 4 has a single 4-fold axis of rotation.
- -4: Point group -4 has 4-fold rotoinversion (a center of inversion plus 4-fold axis of rotation)
- 4/m: Point group 4/m has 4-fold rotation perpendicular to a mirror plane
- 422: Point group 422 has a 4-fold rotation and two 2-fold rotations, all perpendicular to each other
- 4mm: Point group 4mm has a 4-fold rotation and two mirror planes.
- -42: Point group -42 has a 4-fold rotoinversion and perpendicular 2-fold rotation
- 4/mmm: Point group 4/mmm has a 4-fold rotation perpendicular to a mirror plane, and two more mirror planes all perpendicular to each other.
Trigonal crystal systems have at one 3-fold axis of rotation.
There are 5 trigonal point groups:
- 3: Point group 3 has a 3-fold axis of rotation
- -3: Point group -3 has a 3-fold rotoinversion
- 32: Point group 32 has a 3-fold axis of rotation, and a 2-fold axis of rotation in a different axis. Note that this may sometimes be written as point group “321” to completely the third axis, although of course “1” doesn’t convey any extra information
- 3m: Point group 3m has a 3-fold axis of rotation, and a mirror plane
- -3m: Point group -3m has a 3-fold rotoinversion, and a mirror plane
Hexagonal crystal systems have one 6-fold axis of rotation.
There are 7 hexagonal point groups:
- 6: Point group has a 6-fold axis of rotation
- -6: Point group has a 6-fold rotoinversion
- 6/m: Point group has a 6-fold axis of rotation perpendicular to a mirror plane
- 622: Point group has a 6-fold axis of rotation perpendicular to two other 2-fold axes of rotations
- 6mm: Point group 6mm has a 6-fold axis of rotation perpendicular to the normal of two mirror planes
- -62m: Point group -62m has a 6-fold rotoinversion, a 2-fold axis of rotation, and a mirror plane
- 6/mmm: Point group 6/mmm has a 6-fold axis of rotation perpendicular to a mirror plane, and 2 other perpendicular mirror planes
Cubic crystal systems have four 3-fold axes of rotations (the 3-fold axes of rotations are along the body diagonals of a cube).
There are 5 cubic point groups:
- 23: Point group has a 2-fold axis of rotation, and a 3-fold axis of rotation
- m-3: Point group has a mirror plane and 3-fold rotoinversion
- 432: Point group has a 4-fold axis of rotation, 3-fold axis of rotation, and 2-fold axis of rotation
- -43m: Point group -43m has a 4-fold rotoinversion, 3-fold axis of rotation, and a mirror plane
- m-3m: Point group m-3m has a mirror plane, 3-fold rotoinversion, and another mirror plane.
Here’s a table showing how each of these point groups fits in a crystal system. Note that the main difference between crystal system and crystal family is that the hexagonal and trigonal crystal systems both belong to the hexagonal crystal family.
|Crystal System||Number of|
Special Properties due to Asymmetry
Crystal symmetry is important because it can tell you certain potential properties about the material., Neumann’s Principle says that if the crystal structure is identical along certain symmetries, then the crystal properties are also identical along those symmetries.
This can “forbid” certain properties which require asymmetry to exist.
For example, piezoelectricity can only occur because of an asymmetry in the crystal cell, which allows the creation of an asymmetrical electric field. If the crystal structure has a center of symmetry, then piezoelectricity is completely impossible.
Of the 32 point groups, 11 have a center of symmetry. Of the remaining 21 point groups, 20 can be piezoelectric (point group 432 can’t be piezoelectric for other reasons).
From this group of 20, 10 are polar, which means the cell can have an electric dipole and lead to “spontaneous electric polarization.” Materials with this property are called pyroelectric.
From the 10 pyroelectric point groups, all 10 have the potential to be ferroelectric, depending on the specifics of the crystal. Ferroelectric materials can have the pyroelectric spontaneous polarization reversed by an external field. Materials which are pyroelectric, but not ferroelectric, cannot have the polarization reversed.
|Center of Symmetry|
(11 Point Groups)
|No Center of Symmetry|
(21 Point Groups)
(20 Point Groups)
(10 Point Groups)
|Ferroelectric||All 10 point groups that are pyroelectric|
have the potential to be ferroelectric.
How to Determine Crystal System from a Point Group?
If you don’t have a handy list of point groups in front of you, you can still determine crystal family based on the point group’s Hermann-Mauguin notation.
- Triclinic crystals have the lowest symmetry, so its point groups are just 1 and -1.
- Monoclinic crystals have slightly higher symmetry with a 2-fold rotation or mirror plane (but still along one axis), so those include 2, m, and 2/m.
- Orthorhombic crystals are like monoclinic (combination of 2-fold rotation and mirror planes) but they have symmetry operations on all 3 axis, so those include 222, mm2, and mmm.
- Tetragonal crystals have 4-fold symmetry, so will include any point group that starts with a 4, which are 4, -4, 4/m, 422, 4mm, -42m, and 4/mmm.
- Trigonal crystals have 3-fold symmetry, so will include any point group that starts with a 3, which are 3, -3, 32, 3m, and -3m
- Hexagonal crystals have 6-fold symmetry, so will include any point group that starts with a 6, which are 6, -6, 6/m 622, 6mm, -62m, and 6/mmm
- Cubic crystals don’t have an easy pattern, but they always have 3-fold rotations, and the 3 is never the first symbol. This includes 23, m-3, 432, -43m, and m-3m.
References and Further Reading
This article was an in-depth explanation of point groups. Check out this list of point groups if you want a quick page to bookmark or print out. That page also includes alternative notation systems besides the Hermann-Mauguin notation used here.
For more notes about symmetry operations, check out this set of notes from the Geoscience department at the University of Massachusetts Amherst.
If you want to construct point group figures (not covered in the article) you can read an explanation here.
If you’re looking for a graduate-level discussion of point groups, you can check out this chapter from the International Tables for Crystallography.
For more on 2D point groups, we recommend you check out these presentations: PDF presentation from ETH Zurich and PDF presentation from University of North Texas.
If you want to see the other crystallography-related articles I’ve written, here is this list, in recommended reading order:
Introduction to Bravais Lattices
What is the Difference Between “Crystal Structure” and “Bravais Lattice”
Atomic Packing Factor
How to Read Miller Indices
How to Read Hexagonal Miller-Bravais Indices
Close-Packed Crystals and Stacking Order
How to Read Crystallography Notation
What are Point Groups
List of Point Groups
What are Space Groups
List of Space Groups
The 7 Crystal Systems
If you are interested in more details about any specific crystal structure, I have written individual articles about simple crystal structures which correspond to each of the 14 Bravais lattices:
1. Simple Cubic
2. Face-Centered Cubic
2a. Diamond Cubic
3. Body-Centered Cubic
4. Simple Hexagonal
4a. Hexagonal Close-Packed
4b. Double Hexagonal Close-Packed (La-type)
5a. Rhombohedral Close-Packed (Sm-type)
6. Simple Tetragonal
7. Body-Centered Tetragonal
7a. Diamond Tetragonal (White Tin)
8. Simple Orthorhombic
9. Base-Centered Orthorhombic
10. Face-Centered Orthorhombic
11. Body-Centered Orthorhombic
12. Simple Monoclinic
13. Base-Centered Monoclinic