# Point groups in three dimensions explained

In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries which leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries.

Symmetry groups of objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible symmetries. All isometries of a bounded 3D object have one or more common fixed points. We choose the origin as one of them.

The symmetry group of an object is sometimes also called full symmetry group, as opposed to its rotation group or proper symmetry group, the intersection of its full symmetry group and the rotation group SO(3) of the 3D space itself. The rotation group of an object is equal to its full symmetry group if and only if the object is chiral.

## Group structure

SO(3) is a subgroup of E+(3), which consists of direct isometries, i.e., isometries preserving orientation; it contains those which leave the origin fixed.

O(3) is the direct product of SO(3) and the group generated by inversion (denoted by its matrix -I):

O(3) = SO(3) ×

Thus there is a 1-to-1 correspondence between all direct isometries and all indirect isometries, through inversion. Also there is a 1-to-1 correspondence between all groups of direct isometries H and all groups K of isometries which contain inversion:

K = H ×

H = K ∩ SO(3)

If a group of direct isometries H has a subgroup L of index 2, then, apart from the corresponding group containing inversion there is also a corresponding group that contains indirect isometries but no inversion:

M = L ∪ ((H \ L) ×)where isometry (A , I) is identified with A.

Thus M is obtained from H by inverting the isometries in H \ L. This group M is as abstract group isomorphic with H. Conversely, for all isometry groups which contain indirect isometries but no inversion we can obtain a rotation group by inverting the indirect isometries. This is clarifying when categorizing isometry groups, see below.

In 2D the cyclic group of k-fold rotations Ck is for every positive integer k a normal subgroup of O(2,R) and SO(2,R). Accordingly, in 3D, for every axis the cyclic group of k-fold rotations about that axis is a normal subgroup of the group of all rotations about that axis, and also of the group obtained by adding reflections in planes through the axis.

## 3D isometries which leave the origin fixed

The isometries of R3 which leave the origin fixed, forming the group O(3,R), can be categorized as follows:

• SO(3,R):
• identity
• rotation about an axis through the origin by an angle not equal to 180°
• rotation about an axis through the origin by an angle of 180°
• the same with inversion (x is mapped to -x), i.e. respectively:
• inversion
• rotation about an axis by an angle not equal to 180°, combined with reflection in the plane through the origin which is perpendicular to the axis
• reflection in a plane through the origin

The 4th and 5th in particular, and in a wider sense the 6th also, are called improper rotations.

See also the similar overview including translations.

## Conjugacy

When comparing the symmetry type of two objects, the origin is chosen for each separately, i.e. they need not have the same center. Moreover, two objects are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of O(3) (two subgroups H1, H2 of a group G are conjugate, if there exists gG such that H1 = g-1H2g).

Thus two 3D objects have the same symmetry type:

• if both have mirror symmetry, but with respect to a different mirror plane
• if both have 3-fold rotational symmetry, but with respect to a different axis.

In the case of multiple mirror planes and/or axes of rotation, two symmetry groups are of the same symmetry type if and only if there is a single rotation mapping this whole structure of the first symmetry group to that of the second. The conjugacy definition would also allow a mirror image of the structure, but this is not needed, the structure itself is achiral. For example, if a symmetry group contains a 3-fold axis of rotation, it contains rotations in two opposite directions. (The structure is chiral for 11 pairs of space groups with a screw axis.)

## Infinite isometry groups

We restrict ourselves to isometry groups which are closed as topological subgroups of O(3). This excludes for example the group of rotations by an irrational number of turns about an axis.

The whole O(3) is the symmetry group of spherical symmetry; SO(3) is the corresponding rotation group. The other infinite isometry groups consist of all rotations about an axis through the origin, and those with additionally reflection in the planes through the axis, and/or reflection in the plane through the origin, perpendicular to the axis. Those with reflection in the planes through the axis, with or without reflection in the plane through the origin, perpendicular to the axis, are the symmetry groups for the two types of cylindrical symmetry.

See also rotational symmetry with respect to any angle.

## Finite isometry groups

For point groups, being finite corresponds to being discrete; infinite discrete groups as in the case of translational symmetry and glide reflectional symmetry do not apply.

Symmetries in 3D that leave the origin fixed are fully characterized by symmetries on a sphere centered at the origin. For finite 3D point groups, see also spherical symmetry groups.

Up to conjugacy the set of finite 3D point groups consists of:

• 7 infinite series with at most one more-than-2-fold rotation axis; they are the finite symmetry groups on an infinite cylinder, or equivalently, those on a finite cylinder.
• 7 point groups with multiple 3-or-more-fold rotation axes; they can also be characterized as point groups with multiple 3-fold rotation axes, because all 7 include these axes; with regard to 3-or-more-fold rotation axes the possible combinations are:
• 4×3
• 4×3 and 3×4
• 10×3 and 6×5

A selection of point groups is compatible with discrete translational symmetry: 27 from the 7 infinite series, and 5 of the 7 others, the 32 so-called crystallographic point groups. See also the crystallographic restriction theorem.

## The seven infinite series

The infinite series have an index n, which can be any integer; in each series, the nth symmetry group contains n-fold rotational symmetry about an axis, i.e. symmetry with respect to a rotation by an angle 360°/n. n=1 covers the cases of no rotational symmetry at all. There are four series with no other axes of rotational symmetry, see cyclic symmetries, and three with additional axes of 2-fold symmetry, see dihedral symmetry.

For n = ∞ they correspond to the frieze groups. Schönflies notation is used, and, in parentheses, orbifold notation; the latter is not only conveniently related to its properties, but also to the order of the group, see below; it is a unified notation, also applicable for wallpaper groups and frieze groups.

The 7 infinite series are:

• Cn (nn) of order n - n-fold rotational symmetry (abstract group Zn); for n = 1: no symmetry (trivial group)
• Cnh (n*) of order 2n (for odd n abstract group Z2n = Zn × Z2 , for even n abstract group Zn × Z2)
• Cnv (*nn) of order 2n - pyramidal symmetry (abstract group Dihn); in biology C2v is called biradial symmetry.
• Dn (22n) of order 2n - dihedral symmetry (abstract group Dihn)
• S2n (nx) of order 2n (not to be confused with symmetric groups, for which the same notation is used; abstract group Z2n)
• Dnh (*22n) of order 4n - prismatic symmetry (for odd n abstract group Dih2n = Dihn × Z2 ; for even n abstract group Dihn × Z2)
• Dnd (or Dnv) (2*n) - antiprismatic symmetry of order 4n (abstract group Dih2n)

The terms horizontal (h) and vertical (v), and the corresponding subscripts, refer to the additional mirror plane, that can be parallel to the rotation axis (vertical) or perpendicular to the rotation axis (horizontal).

Involutional symmetry (abstract group Z2):

The second of these is the first of the uniaxial groups (cyclic groups) Cn of order n (also applicable in 2D), which are generated by a single rotation of angle 360°/n. In addition to this, one may add a mirror plane perpendicular to the axis, giving the group Cnh of order 2n, or a set of n mirror planes containing the axis, giving the group Cnv, also of order 2n. The latter is the symmetry group for a regular n-sided pyramid. A typical object with symmetry group Cn or Dn is a propellor.

If both horizontal and vertical reflection planes are added, their intersections give n axes of rotation through 180°, so the group is no longer uniaxial. This new group of order 4n is called Dnh. Its subgroup of rotations is the dihedral group Dn of order 2n which still has the 2-fold rotation axes perpendicular to the primary rotation axis, but no mirror planes. Note that in 2D Dn includes reflections, which can also be viewed as flipping over flat objects without distinction of front- and backside, but in 3D the two operations are distinguished: the group contains "flipping over", not reflections.

There is one more group in this family, called Dnd (or Dnv), which has vertical mirror planes containing the main rotation axis, but instead of having a horizontal mirror plane it has an isometry which is the combination of a reflection in the horizontal plane and a rotation by an angle 180°/n. Dnh is the symmetry group for a regular n-sided prisms and also for a regular n-sided bipyramid. Dnd is the symmetry group for a regular n-sided antiprism, and also for a regular n-sided trapezohedron. Dn is the symmetry group of a partially rotated prism.

Sn is generated by the combination of a reflection in the horizontal plane and a rotation by an angle 360°/n. For n odd this is equal to the group generated by the two separately, Cnh of order 2n, and therefore the notation Sn is not needed; however, for n even it is distinct, and of order n. Like Dnd it contains a number of improper rotations without containing the corresponding rotations.

All symmetry groups in the 7 infinite series are different, except for the following four pairs of mutually equal ones:

• C1h and C1v: group of order 2 with a single reflection (Cs)
• D1 and C2: group of order 2 with a single 180° rotation
• D1h and C2v: group of order 4 with a reflection in a plane and a 180° rotation through a line in that plane
• D1d and C2h: group of order 4 with a reflection in a plane and a 180° rotation through a line perpendicular to that plane

S2 is the group of order 2 with a single inversion (Ci)

"Equal" is meant here as the same up to conjugacy in space. This is stronger than "up to algebraic isomorphism". For example, there are three different groups of order two in the first sense, but there is only one in the second sense. Similarly, e.g. S2n is algebraically isomorphic with Z2n.

## The seven remaining point groups

The remaining point groups are said to be of very high or polyhedral symmetry because they have more than one rotation axis of order greater than 2. Using Cn to denote an axis of rotation through 360°/n and Sn to denote an axis of improper rotation through the same, the groups are:

• T (332) of order 12 - chiral tetrahedral symmetry. There are four C3 axes, each through two vertices of a cube (body diagonals) or one of a regular tetrahedron, and three C2 axes, through the centers of the cube's faces, or the midpoints of the tetrahedron's edges. This group is isomorphic to A4, the alternating group on 4 elements, and is the rotation group for a regular tetrahedron.
• Td (*332) of order 24 - full tetrahedral symmetry. This group has the same rotation axes as T, but with six mirror planes, each containing two edges of the cube or one edge of the tetrahedron, a single C2 axis and two C3 axes. The C2 axes are now actually S4 axes. This group is the symmetry group for a regular tetrahedron. Td is isomorphic to S4, the symmetric group on 4 letters. See also the isometries of the regular tetrahedron.
• Th (3*2) of order 24 - pyritohedral symmetry. This group has the same rotation axes as T, with mirror planes parallel to the cube faces. The C3 axes become S6 axes, and there is inversion symmetry. Th is isomorphic to A4 × C2. It is the symmetry of a cube with on each face a line segment dividing the face into two equal rectangles, such that the line segments of adjacent faces do not meet at the edge. The symmetries correspond to the even permutations of the body diagonals and the same combined with inversion. It is also the symmetry of a pyritohedron, which is similar to the cube described, with each rectangle replaced by a pentagon with one symmetry axis and 4 equal sides and 1 different side (the one corresponding to the line segment dividing the cube's face); i.e., the cube's faces bulge out at the dividing line and become narrower there. It is a subgroup of the full icosahedral symmetry group (as isometry group, not just as abstract group), with 4 of the 10 3-fold axes.
• O (432) of order 24 - chiral octahedral symmetry. This group is like T, but the C2 axes are now C4 axes, and additionally there are 6 C2 axes, through the midpoints of the edges of the cube. This group is also isomorphic to S4, and is the rotation group of the cube and octahedron.
• Oh (*432) of order 48 - full octahedral symmetry. This group has the same rotation axes as O, but with mirror planes, comprising both the mirror planes of Td and Th. This group is isomorphic to S4 × C2, and is the symmetry group of the cube and octahedron. See also the isometries of the cube.
• I (532) of order 60 - chiral icosahedral symmetry; the rotation group of the icosahedron and the dodecahedron. It is a normal subgroup of index 2 in the full group of symmetries Ih. The group I is isomorphic to A5, the alternating group on 5 letters. The group contains 10 versions of D3 and 6 versions of D5 (rotational symmetries like prisms and antiprisms).
• Ih (*532) of order 120 - full icosahedral symmetry; the symmetry group of the icosahedron and the dodecahedron. The group Ih is isomorphic to A5 × C2. The group contains 10 versions of D3d and 6 versions of D5d (symmetries like antiprisms).

## Relation between orbifold notation and order

The order of each group is 2 divided by the orbifold Euler characteristic; the latter is 2 minus the sum of the feature values, assigned as follows:

• n without or before * counts as (n−1)/n
• n after * counts as (n−1)/(2n)
• * and x count as 1

This can also be applied for wallpaper groups and frieze groups: for them, the sum of the feature values is 2, giving an infinite order; see orbifold Euler characteristic for wallpaper groups

## Rotation groups

The rotation groups, i.e. the finite subgroups of SO(3), are: the cyclic groups Cn (the rotation group of a regular pyramid), the dihedral groups Dn (the rotation group of a regular prism, or regular bipyramid), and the rotation groups T, O and I of a regular tetrahedron, octahedron/cube and icosahedron/dodecahedron.

In particular, the dihedral groups D3, D4 etc. are the rotation groups of plane regular polygons embedded in three-dimensional space, and such a figure may be considered as a degenerate regular prism. Therefore it is also called a dihedron (Greek: solid with two faces), which explains the name dihedral group.

• An object with symmetry group Cn, Cnh, Cnv or S2n has rotation group Cn.
• An object with symmetry group Dn, Dnh, or Dnd has rotation group Dn.
• An object with one of the other seven symmetry groups has as rotation group the corresponding one without subscript: T, O or I.

The rotation group of an object is equal to its full symmetry group if and only if the object is chiral. In other words, the chiral objects are those with their symmetry group in the list of rotation groups.

## Correspondence between rotation groups and other groups

The following groups contain inversion:

• Cnh and Dnh for even n
• S2n and Dnd for odd n (S2 = Ci is the group generated by inversion; D1d = C2h)
• Th, Oh, and Ih

As explained above, there is a 1-to-1 correspondence between these groups and all rotation groups:

• Cnh for even n and S2n for odd n correspond to Cn
• Dnh for even n and Dnd for odd n correspond to Dn
• Th, Oh, and Ih correspond to T, O, and I, respectively.

The other groups contain indirect isometries, but not inversion:

• Cnv
• Cnh and Dnh for odd n
• S2n and Dnd for even n
• Td

They all correspond to a rotation group H and a subgroup L of index 2 in the sense that they are obtained from H by inverting the isometries in H \ L, as explained above:

• Cn is subgroup of Dn of index 2, giving Cnv
• Cn is subgroup of C2n of index 2, giving Cnh for odd n and S2n for even n
• Dn is subgroup of D2n of index 2, giving Dnh for odd n and Dnd for even n
• T is subgroup of O of index 2, giving Td

## Maximal symmetries

There are two discrete point groups with the property that no discrete point group has it as proper subgroup: Oh and Ih. Their largest common subgroup is Th. The two groups are obtained from it by changing 2-fold rotational symmetry to 4-fold, and adding 5-fold symmetry, respectively. Alternatively the two groups are generated by adding for each a reflection plane to Th.

There are two crystallographic point groups with the property that no crystallographic point group has it as proper subgroup: Oh and D6h. Their maximal common subgroups, depending on orientation, are D3d and D2h.

## The groups arranged by abstract group type

Below the groups explained above are arranged by abstract group type.

The smallest abstract groups which are not any symmetry group in 3D, are the quaternion group (of order 8), the dicyclic group Dic3 (of order 12), and 10 of the 14 groups of order 16.

The column "# of order 2 elements" in the following tables shows the total number of isometry subgroups of types C2 , Ci , Cs. This total number is one of the characteristics helping to distinguish the various abstract group types, while their isometry type helps to distinguish the various isometry groups of the same abstract group.

Within the possibilities of isometry groups in 3D, there are infinitely many abstract group types with 0, 1 and 3 elements of order 2, there are two with 2n + 1 elements of order 2, and there are three with 2n + 3 elements of order 2 (for each n ≥ 2). There is never a positive even number of elements of order 2.

### Symmetry groups in 3D which are cyclic as abstract group

The symmetry group for n-fold rotational symmetry is Cn; its abstract group type is cyclic group Zn , which is also denoted by Cn. However, there are two more infinite series of symmetry groups with this abstract group type:

• For even order 2n there is the group S2n (Schoenflies notation) generated by a rotation by an angle 180°/n about an axis, combined with a reflection in the plane perpendicular to the axis. For S2 the notation Ci is used; it is generated by inversion.
• For any order 2n where n is odd, we have Cnh; it has an n-fold rotation axis, and a perpendicular plane of reflection. It is generated by a rotation by an angle 360°/n about the axis, combined with the reflection. For C1h the notation Cs is used; it is generated by reflection in a plane.

Thus we have, with bolding of the 10 cyclic crystallographic point groups, for which the crystallographic restriction applies:

OrderIsometry groupsAbstract group
1. of order 2 elements
1C1Z10
2C2 , Ci , CsZ21
3C3Z30
4C4 , S4Z41
5C5Z50
6C6 , S6 , C3hZ6 = Z3 × Z21
7C7Z70
8C8 , S8Z81
9C9Z90
10C10 , S10 , C5hZ10 = Z5 × Z21
etc.

### Symmetry groups in 3D which are dihedral as abstract group

In 2D dihedral group Dn includes reflections, which can also be viewed as flipping over flat objects without distinction of front- and backside.

However, in 3D the two operations are distinguished: the symmetry group denoted by Dn contains n 2-fold axes perpendicular to the n-fold axis, not reflections. Dn is the rotation group of the n-sided prism with regular base, and n-sided bipyramid with regular base, and also of a regular, n-sided antiprism and of a regular, n-sided trapezohedron. The group is also the full symmetry group of such objects after making them chiral by e.g. an identical chiral marking on every face, or some modification in the shape.

The abstract group type is dihedral group Dihn, which is also denoted by Dn. However, there are three more infinite series of symmetry groups with this abstract group type:

• Cnv of order 2n, the symmetry group of a regular n-sided pyramid
• Dnd of order 4n, the symmetry group of a regular n-sided antiprism
• Dnh of order 4n for odd n. For n = 1 we get D2, already covered above, so n ≥ 3.

Note the following property:

Dih4n+2

\cong

Dih2n+1 × Z2

Thus we have, with bolding of the 12 crystallographic point groups, and writing D1d as the equivalent C2h:

OrderIsometry groupsAbstract group
1. of order 2 elements
4D2 , C2v , C2hDih2 = Z2 × Z23
6D3 , C3vDih33
8D4 , C4v , D2dDih45
10D5 , C5vDih55
12D6 , C6v , D3d , D3hDih6 = Dih3 × Z27
14D7 , C7vDih77
16D8 , C8v , D4dDih89
18D9 , C9vDih99
20D10 , C10v , D5h , D5dDih10 = D5 × Z211
etc.

### Other

C2n,h of order 4n is of abstract group type Z2n × Z2. For n = 1 we get Dih2 , already covered above, so n ≥ 2.

Thus we have, with bolding of the 2 cyclic crystallographic point groups:

OrderIsometry groupAbstract group
1. of order 2 elements
Cycle diagram
8C4hZ4 × Z23
12C6hZ6 × Z2 = Z3 × Z2 × Z2 = Z3 × Dih23
16C8hZ8 × Z23
20C10hZ10 × Z2 = Z5 × Z2 × Z23
etc.

Dnh of order 4n is of abstract group type Dihn × Z2. For odd n this is already covered above, so we have here D2nh of order 8n, which is of abstract group type Dih2n × Z2 (n≥1).

Thus we have, with bolding of the 3 dihedral crystallographic point groups:

OrderIsometry groupAbstract group
1. of order 2 elements
Cycle diagram
8D2hDih2 × Z27
16D4hDih4 × Z211
24D6hDih6 × Z215
32D8hDih8 × Z219
etc.

The remaining seven are, with bolding of the 5 crystallographic point groups (see also above):

• order 12: of type A4 (alternating group): T
• order 24:
• of type S4 (symmetric group, not to be confused with the symmetry group with this notation): Td, O
• of type A4 × Z2: Th .
• order 48, of type S4 × Z2: Oh
• order 60, of type A5: I
• order 120, of type A5 × Z2: Ih

See also icosahedral symmetry.

## Impossible discrete symmetries

Since the overview is exhaustive, it also shows implicitly what is not possible as discrete symmetry group. For example:

• a C6 axis in one direction and a C3 in another
• a C5 axis in one direction and a C4 in another
• a C3 axis in one direction and another C3 axis in a perpendicular direction

etc.

## Fundamental domain

The fundamental domain of a point group is a conic solid. An object with a given symmetry in a given orientation is characterized by the fundamental domain. If the object is a surface it is characterized by a surface in the fundamental domain continuing to its radial bordal faces or surface. If the copies of the surface do not fit, radial faces or surfaces can be added. They fit anyway if the fundamental domain is bounded by reflection planes.

For a polyhedron this surface in the fundamental domain can be part of an arbitrary plane. For example, in the disdyakis triacontahedron one full face is a fundamental domain. Adjusting the orientation of the plane gives various possibilities of combining two or more adjacent faces to one, giving various other polyhedra with the same symmetry. The polyhedron is convex if the surface fits to its copies and the radial line perpendicular to the plane is in the fundamental domain.

Also the surface in the fundamental domain may be composed of multiple faces.

## Binary polyhedral groups

The map

Spin(3)\toSO(3)

is the double cover of the rotation group by the spin group in 3 dimensions. (This is the only connected cover of

SO(3)

, since Spin(3) is simply connected.)By the lattice theorem, there is a Galois connection between subgroups of Spin(3) and subgroups of SO(3) (rotational point groups): the image of a subgroup of Spin(3) is a rotational point group, and the preimage of a point group is a subgroup of Spin(3).

The preimage of a finite point group is called a binary polyhedral group, and is called by the same name as its point group, with the prefix binary. For instance, the preimage of the icosahedral group is the binary icosahedral group.

The binary polyhedral groups are:

An

: binary cyclic group of an (n+1)-gon

Dn

: binary dihedral group of an n-gon

E6

: binary tetrahedral group

E7

: binary octahedral group

E8

: binary icosahedral groupThese are classified by the ADE classification, and the quotient of

C2

by the action of a binary polyhedral group is a Du Val singularity[1] .

For point groups that reverse orientation, the situation is more complicated, as there are two pin groups, so there are two possible binary groups corresponding to a given point group.

## Footnotes

1. http://enriques.mathematik.uni-mainz.de/burban/singul.pdf Du Val Singularities, by Igor Burban