Point groups in three dimensions

In geometry a point group in 3D 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) × { I , −I }

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 × { I , −I }

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) × { − I } )

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 C_k 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 g ∈ G 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 or 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 subgroup 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.

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 = $\infty$ they correspond to the frieze groups. Schönflies notation is used, and, in parentheses, Conway's 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:

C_n (nn ) of order n - n-fold rotational symmetry (abstract group Z_n ); for n = 1: no symmetry (trivial group)
C_nh (n* ) of order 2n (for odd n abstract group Z_2n = Z_n × Z₂ , for even n abstract group Z_n × Z₂ )
C_nv (*nn ) of order 2n - pyramidal symmetry (abstract group Dih_n ); in biology C_2v is called biradial symmetry.
D_n (22n ) of order 2n - dihedral symmetry (abstract group Dih_n )
S_2n (nx ) of order 2n (not to be confused with symmetric groups, for which the same notation is used; abstract group Z_2n )
D_nh (*22n ) of order 4n - prismatic symmetry (for odd n abstract group Dih_2n = Dih_n × Z₂ ; for even n abstract group Dih_n × Z₂ )
D_nd (or D_nv ) (2*n ) - antiprismatic symmetry of order 4n (abstract group Dih_2n )

The terms horizontal (h) and vertical (v) are used with respect to a vertical axis of rotation.

Involutional symmetry (abstract group Z₂ ):

C_i - inversion symmetry
C₂ - 2-fold rotational symmetry
C_s - reflection symmetry, also called bilateral symmetry.

Patterns on a cylindrical band illustrating the case n = 6 for each of the 7 infinite families of point groups. The symmetry group of each pattern is the indicated group.

The second of these is the first of the uniaxial groups (cyclic groups) C_n 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 C_nh of order 2n, or a set of n mirror planes containing the axis, giving the group C_nv, also of order 2n. The latter is the symmetry group for a regular n-sided pyramid.

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 D_nh. Its subgroup of rotations is the dihedral group D_n 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 D_n 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 D_nd (or D_nv), 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. D_nh is the symmetry group for a regular n-sided prisms and also for a regular n-sided bipyramid. D_nd is the symmetry group for a regular n-sided antiprism, and also for a regular n-sided trapezohedron. D_n is the symmetry group of a partially rotated prism.

S_n 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, C_nh of order 2n, and therefore the notation S_n is not needed; however, for n even it is distinct, and of order n. Like D_nd 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:

C_1h and C_1v: group of order 2 with a single reflection (C_s )
C₂ and D₁: group of order 2 with a single 180° rotation
D_1h and C_2v: group of order 4 with a reflection in a plane and a 180° rotation through a line in that plane
D_1d and C_2h: 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 (C_i )

"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. S_2n is algebraically isomorphic with Z_2n.

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 C_n to denote an axis of rotation through 360°/n and S_n 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.

T_d (*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. T_d is isomorphic to S₄, the symmetric group on 4 letters. See also the isometries of the regular tetrahedron.

T_h (3*2) of order 24 - pyritohedral symmetry. This group has the same rotation axes as T, with mirror planes parallel to the cubes faces. The C₃ axes become S₆ axes, and there is inversion symmetry. T_h 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 [1], 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.

O_h (*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 T_d and T_h. 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 I_h. 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).

I_h (*532) of order 120 - full icosahedral symmetry; the symmetry group of the icosahedron and the dodecahedron. The group I_h is isomorphic to A₅ × C₂. The group contains 10 versions of D_3d and 6 versions of D_5d (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 C_n (the rotation group of a regular pyramid), the dihedral groups D_n (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 C_n, C_nh, C_nv or S_2n has rotation group C_n.
An object with symmetry group D_n, D_nh, or D_nd has rotation group D_n.
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:

C_nh and D_nh for even n
S_2n and D_nd for odd n (S₂ = C_i is the group generated by inversion; D_1d = C_2h)
T_h, O_h, and I_h

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

C_nh for even n and S_2n for odd n correspond to C_n
D_nh for even n and D_nd for odd n correspond to D_n
T_h, O_h, and I_h correspond to T, O, and I, respectively.

The other groups contain indirect isometries, but not inversion:

C_nv
C_nh and D_nh for odd n
S_2n and D_nd for even n
T_d

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:

C_n is subgroup of D_n of index 2, giving C_nv
C_n is subgroup of C_2n of index 2, giving C_nh for odd n and S_2n for even n
D_n is subgroup of D_2n of index 2, giving D_nh for odd n and D_nd for even n
T is subgroup of O of index 2, giving T_d

Maximal symmetries

There are two discrete point groups with the property that no discrete point group has it as proper subgroup: O_h and I_h. Their largest common subgroup is T_h. 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 T_h.

There are two crystallographic point groups with the property that no crystallographic point group has it as proper subgroup: O_h and D_6h. Their maximal common subgroups, depending on orientation, are D_3d and D_2h.

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 , C_i , C_s. 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 C_n; its abstract group type is cyclic group Z_n , which is also denoted by C_n. However, there are two more infinite series of symmetry groups with this abstract group type:

For even order 2n there is the group S_2n (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 C_i is used; it is generated by inversion.
For any order 2n where n is odd, we have C_nh; 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 C_1h the notation C_s 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:

Order	Isometry groups	Abstract group	# of order 2 elements
1	C₁	Z₁	0
2	C₂ , C_i , C_s	Z₂	1
3	C₃	Z₃	0
4	C₄ , S₄	Z₄	1
5	C₅	Z₅	0
6	C₆ , S₆ , C_3h	Z₆ = Z₃ × Z₂	1
7	C₇	Z₇	0
8	C₈ , S₈	Z₈	1
9	C₉	Z₉	0
10	C₁₀ , S₁₀ , C_5h	Z₁₀ = Z₅ × Z₂	1

etc.

Symmetry groups in 3D which are dihedral as abstract group

In 2D dihedral group D_n 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 D_n contains n 2-fold axes perpendicular to the n-fold axis, not reflections. D_n 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 Dih_n, which is also denoted by D_n. However, there are three more infinite series of symmetry groups with this abstract group type:

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

Note the following property:

Dih_4n+2

\cong

Dih_2n+1 × Z₂

Thus we have, with bolding of the 12 crystallographic point groups, and writing D_1d as the equivalent C_2h:

Order	Isometry groups	Abstract group	# of order 2 elements
4	D₂ , C_2v , C_2h	Dih₂ = Z₂ × Z₂	3
6	D₃ , C_3v	Dih₃	3
8	D₄ , C_4v , D_2d	Dih₄	5
10	D₅ , C_5v	Dih₅	5
12	D₆ , C_6v , D_3d , D_3h	Dih₆ = Dih₃ × Z₂	7
14	D₇ , C_7v	Dih₇	7
16	D₈ , C_8v , D_4d	Dih₈	9
18	D₉ , C_9v	Dih₉	9

etc.

Other

C_2n,h of order 4n is of abstract group type Z_2n × 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:

Order	Isometry group	Abstract group	# of order 2 elements
8	C_4h	Z₄ × Z₂	3
12	C_6h	Z₆ × Z₂ = Z₃ × Z₂ × Z₂ = Z₃ × Dih₂	3
16	C_8h	Z₈ × Z₂	3
20	C_10h	Z₁₀ × Z₂ = Z₅ × Z₂ × Z₂	3

etc.

D_nh of order 4n is of abstract group type Dih_n × Z₂. For odd n this is already covered above, so we have here D<sub>2nh of order 8n, which is of abstract group type Dih_2n × Z₂ (n≥1).

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

Order	Isometry group	Abstract group	# of order 2 elements
8	D_2h	Dih₂ × Z₂	7
16	D_4h	Dih₄ × Z₂	11
24	D_6h	Dih₆ × Z₂	15
32	D_8h	Dih₈ × Z₂	19

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): T_d, O
- of type A₄ × Z₂: T_h .
order 48, of type S₄ × Z₂: O_h
order 60, of type A₅: I
order 120, of type A₅ × Z₂: I_h

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.

Examples

A typical object with symmetry group C_n or D_n is a propellor.

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.

External links

Graphic overview of the 32 crystallographic point groups - form the first parts (apart from skipping n=5) of the 7 infinite series and 5 of the 7 separate 3D point groups
Overview of properties of point groups