Space group
Overview
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematics, the other being the study of numbers ....
, a space group is a symmetry group
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
, usually for three dimensions, that divides space into discrete repeatable domains.
In three dimensions, there are 219 unique types, or counted as 230 if chiral
Chirality (mathematics)
In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...
copies are considered distinct. Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach
Ludwig Bieberbach
Ludwig Georg Elias Moses Bieberbach was a German mathematician.Biography:Born in Goddelau, near Darmstadt, he studied at Heidelberg and under Felix Klein at Göttingen, receiving his doctorate in 1910. His dissertation was titled On the theory of automorphic functions...
groups, and are discrete cocompact group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
s of isometries of an oriented Euclidean space.
In crystallography
Crystallography
Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...
, they are also called the crystallographic or Fedorov
Yevgraf Fyodorov
Yevgraf Stepanovich Fyodorov, sometimes spelled Evgraf Stepanovich Fedorov , was a Russian mathematician, crystallographer, and mineralogist....
groups, and represent a description of the symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
of the crystal.
Unanswered Questions
Encyclopedia
In mathematics
and geometry
, a space group is a symmetry group
, usually for three dimensions, that divides space into discrete repeatable domains.
In three dimensions, there are 219 unique types, or counted as 230 if chiral
copies are considered distinct. Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach
groups, and are discrete cocompact group
s of isometries of an oriented Euclidean space.
In crystallography
, they are also called the crystallographic or Fedorov
groups, and represent a description of the symmetry
of the crystal. A definitive source regarding 3dimensional space groups is the International Tables for Crystallography .
s which have been known for several centuries.
The space groups in 3 dimensions were first enumerated by , and shortly afterwards were independently enumerated by and . These first enumerations all contained several minor mistakes, and the correct list of 230 space groups was found during correspondence between Fyodorov and Schönflies.
s with the 14 Bravais lattices, each of the latter belonging to one of 7 lattice systems. This results in a space group being some combination of the translational symmetry of a unit cell including lattice centering, the point group symmetry operations of reflection
, rotation
and improper rotation
(also called rotoinversion), and the screw axis
and glide plane
symmetry operations. The combination of all these symmetry operations results in a total of 230 unique space groups describing all possible crystal symmetries.
s.
is a reflection in a plane, followed by a translation parallel with that plane. This is noted by a, b or c, depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is a fourth of the way along either a face or space diagonal of the unit cell. The latter is called the diamond glide plane as it features in the diamond
structure.
is a rotation about an axis, followed by a translation along the direction of the axis. These are noted by a number, n, to describe the degree of rotation, where the number is how many operations must be applied to complete a full rotation (e.g., 3 would mean a rotation one third of the way around the axis each time). The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. So, 2_{1} is a twofold rotation followed by a translation of 1/2 of the lattice vector.
y = M.x + D
where M is its matrix, D is its vector, and where the element transforms point x into point y. In general, D = D(lattice
) + D(M), where D(M) is a unique function of M that is zero for M being the identity. The matrices M form a point group
that is a basis of the space group; the lattice must be symmetric under that point group.
The lattice dimension can be less than the overall dimension, resulting in a "subperiodic" space group. For (overall dimension, lattice dimension):
gave another classification of the space groups, called a fibrifold notation, according to the fibrifold
structures on the corresponding orbifold
. They divided the 219 affine space groups into reducible and irreducible groups. The reducible groups fall into 17 classes corresponding to the 17 wallpaper group
s, and the remaining 35 irreducible groups are the same as the cubic groups and are classified separately.
It is essential in Bieberbach's theorems to assume that the group acts as isometries; the theorems do not generalize to discrete cocompact groups of affine transformations of Euclidean space. A counterexample is given by the 3dimensional Heisenberg group of the integers acting by translations on the Heisenberg group of the reals, identified with 3dimensional Euclidean space. This is a discrete cocompact group of affine transformations of space, but does not contain a subgroup Z^{3}.
s that contain ordered unpaired spins, i.e. ferro
, ferri
or antiferromagnetic
structures as studied by neutron diffraction
. The time reversal element flips a magnetic spin while leaving all other structure the same and it can be combined with a number of other symmetry elements. Including time reversal there are 1651 magnetic space groups in 3D . It has also been possible to construct magnetic versions for other overall and lattice dimensions (Daniel Litvin's papers, ). Frieze groups are magnetic 1D line groups and layer groups are magnetic wallpaper groups, and the axial 3D point groups are magnetic 2D point groups. Number of original and magnetic groups by (overall, lattice) dimension:
Note. An e plane is a double glide plane, one having glides in two different directions. They are found in five space groups, all in the orthorhombic system and with a centered lattice. The use of the symbol e became official with .
The lattice system can be found as follows. If the crystal system is not trigonal then the lattice system is of the same type. If the crystal system is trigonal, then the lattice system is hexagonal unless the space group is one of the seven in the rhombohedral lattice system consisting of the 7 trigonal space groups in the table above whose name begins with R. (The term rhombohedral system is also sometimes used as an alternative name for the whole trigonal system.) The hexagonal lattice system is larger than the hexagonal crystal system, and consists of the hexagonal crystal system together with the 18 groups of the trigonal crystal system other than the seven whose names begin with R.
The Bravais lattice of the space group is determined by the lattice system together with the initial letter of its name, which for the nonrhombohedral groups is P, I, F, or C, standing for the principal, body centered, face centered, or Cface centered lattices.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematics, the other being the study of numbers ....
, a space group is a symmetry group
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
, usually for three dimensions, that divides space into discrete repeatable domains.
In three dimensions, there are 219 unique types, or counted as 230 if chiral
Chirality (mathematics)
In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...
copies are considered distinct. Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach
Ludwig Bieberbach
Ludwig Georg Elias Moses Bieberbach was a German mathematician.Biography:Born in Goddelau, near Darmstadt, he studied at Heidelberg and under Felix Klein at Göttingen, receiving his doctorate in 1910. His dissertation was titled On the theory of automorphic functions...
groups, and are discrete cocompact group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
s of isometries of an oriented Euclidean space.
In crystallography
Crystallography
Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...
, they are also called the crystallographic or Fedorov
Yevgraf Fyodorov
Yevgraf Stepanovich Fyodorov, sometimes spelled Evgraf Stepanovich Fedorov , was a Russian mathematician, crystallographer, and mineralogist....
groups, and represent a description of the symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
of the crystal. A definitive source regarding 3dimensional space groups is the International Tables for Crystallography .
History
Space groups in 2 dimensions are the 17 wallpaper groupWallpaper group
A wallpaper group is a mathematical classification of a twodimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
s which have been known for several centuries.
The space groups in 3 dimensions were first enumerated by , and shortly afterwards were independently enumerated by and . These first enumerations all contained several minor mistakes, and the correct list of 230 space groups was found during correspondence between Fyodorov and Schönflies.
Elements of a space group
The space groups in three dimensions are made from combinations of the 32 crystallographic point groupCrystallographic point group
In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind...
s with the 14 Bravais lattices, each of the latter belonging to one of 7 lattice systems. This results in a space group being some combination of the translational symmetry of a unit cell including lattice centering, the point group symmetry operations of reflection
Reflection (mathematics)
In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...
, rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A threedimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
and improper rotation
Improper rotation
In 3D geometry, an improper rotation, also called rotoreflection or rotary reflection is, depending on context, a linear transformation or affine transformation which is the combination of a rotation about an axis and a reflection in a plane perpendicular to the axis.Equivalently it is the...
(also called rotoinversion), and the screw axis
Screw axis
The screw axis of an object is a line that is simultaneously the axis of rotation and the line along which a translation occurs...
and glide plane
Glide plane
In crystallography, a glide plane is symmetry operation describing how a reflection in a plane, followed by a translation parallel with that plane, may leave the crystal unchanged....
symmetry operations. The combination of all these symmetry operations results in a total of 230 unique space groups describing all possible crystal symmetries.
Elements fixing a point
The elements of the space group fixing a point of space are rotations, reflections, the identity element, and improper rotations.Translations
The translations form a normal abelian subgroup of rank 3, called the Bravais lattice. There are 14 possible types of Bravais lattice. The quotient of the space group by the Bravais lattice is a finite group which is one of the 32 possible point groupPoint group
In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...
s.
Glide planes
A glide planeGlide plane
In crystallography, a glide plane is symmetry operation describing how a reflection in a plane, followed by a translation parallel with that plane, may leave the crystal unchanged....
is a reflection in a plane, followed by a translation parallel with that plane. This is noted by a, b or c, depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is a fourth of the way along either a face or space diagonal of the unit cell. The latter is called the diamond glide plane as it features in the diamond
Diamond
In mineralogy, diamond is an allotrope of carbon, where the carbon atoms are arranged in a variation of the facecentered cubic crystal structure called a diamond lattice. Diamond is less stable than graphite, but the conversion rate from diamond to graphite is negligible at ambient conditions...
structure.
Screw axes
A screw axisScrew axis
The screw axis of an object is a line that is simultaneously the axis of rotation and the line along which a translation occurs...
is a rotation about an axis, followed by a translation along the direction of the axis. These are noted by a number, n, to describe the degree of rotation, where the number is how many operations must be applied to complete a full rotation (e.g., 3 would mean a rotation one third of the way around the axis each time). The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. So, 2_{1} is a twofold rotation followed by a translation of 1/2 of the lattice vector.
General formula
The general formula for the action of an element of a space group isy = M.x + D
where M is its matrix, D is its vector, and where the element transforms point x into point y. In general, D = D(lattice
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
) + D(M), where D(M) is a unique function of M that is zero for M being the identity. The matrices M form a point group
Point group
In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...
that is a basis of the space group; the lattice must be symmetric under that point group.
The lattice dimension can be less than the overall dimension, resulting in a "subperiodic" space group. For (overall dimension, lattice dimension):
 (1,1): Onedimensional line groupLine groupA line group is a mathematical way of describing symmetries associated with moving along a line. These symmetries include repeating along that line, making that line a onedimensional lattice...
s  (2,1): Twodimensional line groupLine groupA line group is a mathematical way of describing symmetries associated with moving along a line. These symmetries include repeating along that line, making that line a onedimensional lattice...
s: frieze groupFrieze groupA frieze group is a mathematical concept to classify designs on twodimensional surfaces which are repetitive in one direction, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
s  (2,2): Wallpaper groupWallpaper groupA wallpaper group is a mathematical classification of a twodimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
s  (3,1): Threedimensional line groupLine groupA line group is a mathematical way of describing symmetries associated with moving along a line. These symmetries include repeating along that line, making that line a onedimensional lattice...
s; with the 3D crystallographic point groups, the rod groupRod groupIn mathematics, a rod group is a threedimensional line group whose point group is one of the axial crystallographic point groups. This constraint means that the point group must be the symmetry of some threedimensional lattice....
s  (3,2): Layer groupLayer groupIn mathematics, a layer group is a threedimensional extension of a wallpaper group, with reflections in the third dimension. It is a space group with a twodimensional lattice, meaning that it is symmetric over repeats in the two lattice directions...
s  (3,3): The space groups discussed in this article
Notation for space groups
There are at least eight methods of naming space groups. Some of these methods can assign several different names to the same space group, so altogether there are many thousands of different names. Number. The International Union of Crystallography publishes tables of all space group types, and assigns each a unique number from 1 to 230. The numbering is arbitrary, except that groups with the same crystal system or point group are given consecutive numbers.
 International symbol or Hermann–Mauguin notation. The Hermann–Mauguin (or international) notation describes the lattice and some generators for the group. It has a shortened form called the international short symbol, which is the one most commonly used in crystallography, and usually consists of a set of four symbols. The first describes the centering of the Bravais lattice (P, A, B, C, I, R or F). The next three describe the most prominent symmetry operation visible when projected along one of the high symmetry directions of the crystal. These symbols are the same as used in point groupPoint groupIn geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...
s, with the addition of glide planes and screw axis, described above. By way of example, the space group of quartzQuartzQuartz is the secondmostabundant mineral in the Earth's continental crust, after feldspar. It is made up of a continuous framework of SiO4 silicon–oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall formula SiO2. There are many different varieties of quartz,...
is P3_{1}21, showing that it exhibits primitive centering of the motif (i.e., once per unit cell), with a threefold screw axis and a twofold rotation axis. Note that it does not explicitly contain the crystal systemCrystal systemIn crystallography, the terms crystal system, crystal family, and lattice system each refer to one of several classes of space groups, lattices, point groups, or crystals...
, although this is unique to each space group (in the case of P3_{1}21, it is trigonal).
 In the international short symbol the first symbol (3_{1} in this example) denotes the symmetry along the major axis (caxis in trigonal cases), the second (2 in this case) along axes of secondary importance (a and b) and the third symbol the symmetry in another direction. In the trigonal case there also exists a space group P3_{1}12. In this space group the twofold axes are not along the a and baxes but in a direction rotated by 30^{o}.
 The international symbols and international short symbols for some of the space groups were changed slightly between 1935 and 2002, so several space groups have 4 different international symbols in use.
 Hall notation. Space group notation with an explicit origin. Rotation, translation and axisdirection symbols are clearly separated and inversion centers are explicitly defined. The construction and format of the notation make it particularly suited to computer generation of symmetry information. For example, group number 3 has three Hall symbols: P 2y (P 1 2 1), P 2 (P 1 1 2), P 2x (P 2 1 1).
 Schönflies notation. The space groups with given point group are numbered by 1, 2, 3, ... (in the same order as their international number) and this number is added as a superscript to the Schönflies symbol for the point group. For example, groups numbers 3 to 5 whose point group is C_{2 have Schönflies symbols C, C, C.}
 Shubnikov symbol
 2D:Orbifold notation and 3D:Fibrifold notation. As the name suggests, the orbifold notation describes the orbifold, given by the quotient of Euclidean space by the space group, rather than generators of the space group. It was introduced by ConwayJohn Horton ConwayJohn Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
and ThurstonWilliam ThurstonWilliam Paul Thurston is an American mathematician. He is a pioneer in the field of lowdimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3manifolds...
, and is not used much outside mathematics. Some of the space groups have several different fibrifolds associated to them, so have several different fibrifold symbols.
 Coxeter notationCoxeter notationIn geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group. It uses a bracketed notation, with modifiers to indicate certain subgroups. The notation is named after H. S. M...
– Spacial and point symmetry groups, represented as modications of the pure reflectional Coxeter groupCoxeter groupIn mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...
s.
Classification systems for space groups
There are (at least) 10 different ways to classify space groups into classes. The relations between some of these are described in the following table. Each classification system is a refinement of the ones below it.(Crystallographic) space group types (230 in three dimensions). Two space groups, considered as subgroups of the group of affine transformations of space, have the same space group type if they are conjugate by an orientationpreserving affine transformation. In three dimensions, for 11 of the affine space groups, there is no orientationpreserving map from the group to its mirror image, so if one distinguishes groups from their mirror images these each split into two cases. So there are 54 + 11 = 65 space group types that preserve orientation.  
Affine space group types (219 in three dimensions). Two space groups, considered as subgroups of the group of affine transformations of space, have the same affine space group type if they are conjugate under an affine transformation. The affine space group type is determined by the underlying abstract group of the space group. In three dimensions there are 54 affine space group types that preserve orientation.  
Arithmetic crystal classes (73 in three dimensions). These are determined by the point group together with the action of the point group on the subgroup of translations. In other words the arithmetic crystal classes correspond to conjugacy classes of finite subgroup of the general linear group GL_{n}(Z) over the integers. A space group is called symmorphic (or split) if there is a point such that all symmetries are the product of a symmetry fixing this point and a translation. Equivalently, a space group is symmorphic if it is a semidirect product Semidirect product In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product... of its point group with its translation subgroup. There are 73 symmorphic space groups, with exactly one in each arithmetic crystal class. There are also 157 nonsymmorphic space group types with varying numbers in the arithmetic crystal classes. 

(geometric) Crystal classes (32 in three dimensions). The crystal class of a space group is determined by its point group: the quotient by the subgroup of translations, acting on the lattice. Two space groups are in the same crystal class if and only if their point groups, which are subgroups of GL_{2}(Z), are conjugate in the larger group GL_{2}(Q).  Bravais flocks (14 in three dimensions). These are determined by the underlying Bravais lattice type. These correspond to conjugacy classes of lattice point groups in GL_{2}(Z), where the lattice point group is the group of symmetries of the underlying lattice that fix a point of the lattice, and contains the point group. 
Crystal systems. (7 in three dimensions) Crystal systems are an ad hoc modification of the lattice systems to make them compatible with the classification according to point groups. They differ from crystal families in that the hexagonal crystal family is split into two subsets, called the trigonal and hexagonal crystal systems. The trigonal crystal system is larger than the rhombohedral lattice system, the hexagonal crystal system is smaller than the hexagonal lattice system, and the remaining crystal systems and lattice systems are the same.  Lattice systems (7 in three dimensions). The lattice system of a space group is determined by the conjugacy class of the lattice point group (a subgroup of GL_{2}(Z)) in the larger group GL_{2}(Q). In three dimensions the lattice point group can have one of the 7 different orders 2, 4, 8, 12, 16, 24, or 48. The hexagonal crystal family is split into two subsets, called the rhombohedral and hexagonal lattice systems. 
Crystal families (6 in three dimensions). The point group of a space group does not quite determine its lattice system, because occasionally two space groups with the same point group may be in different lattice systems. Crystal families are formed from lattice systems by merging the two lattice systems whenever this happens, so that the crystal family of a space group is determined by either its lattice system or its point group. In 3 dimensions the only two lattice families that get merged in this way are the hexagonal and rhombohedral lattice systems, which are combined into the hexagonal crystal family. The 6 crystal families in 3 dimensions are called triclinic, monoclinic, orthorhombal, tetragonal, hexagonal, and cubic. Crystal families are commonly used in popular books on crystals, where they are sometimes called crystal systems. 
gave another classification of the space groups, called a fibrifold notation, according to the fibrifold
Fibrifold
In mathematics, a fibrifold is afiber space whose fibers and base spaces are orbifolds. They were introduced by , who introduced a system of notation for 3dimensional fibrifolds and used this to assign names to the 219 affine space group types....
structures on the corresponding orbifold
Orbifold
In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...
. They divided the 219 affine space groups into reducible and irreducible groups. The reducible groups fall into 17 classes corresponding to the 17 wallpaper group
Wallpaper group
A wallpaper group is a mathematical classification of a twodimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
s, and the remaining 35 irreducible groups are the same as the cubic groups and are classified separately.
Bieberbach's theorems
In n dimensions, an affine space group, or Bieberbach group, is a discrete subgroup of isometries of ndimensional Euclidean space with a compact fundamental domain. proved that the subgroup of translations of any such group contains n linearly independent translations, and is a free abelian subgroup of finite index, and is also the unique maximal normal abelian subgroup. He also showed that in any dimension n there are only a finite number of possibilities for the isomorphism class of the underlying group of a space group, and moreover the action of the group on Euclidean space is unique up to conjugation by affine transformations. This answers part of Hilbert's 18th problem. showed that conversely any group that is the extension of Z^{n} by a finite group acting faithfully is an affine space group. Combining these results shows that classifying space groups in n dimensions up to conjugation by affine transformations is essentially the same as classifying isomorphism classes for groups that are extensions of Z^{n} by a finite group acting faithfully.It is essential in Bieberbach's theorems to assume that the group acts as isometries; the theorems do not generalize to discrete cocompact groups of affine transformations of Euclidean space. A counterexample is given by the 3dimensional Heisenberg group of the integers acting by translations on the Heisenberg group of the reals, identified with 3dimensional Euclidean space. This is a discrete cocompact group of affine transformations of space, but does not contain a subgroup Z^{3}.
Classification in small dimensions
This table give the number of space group types in small dimensions.Dimension  Classification  

0  1  1  1  1  Trivial group 
1  1  2  2  2  One is the group of integers and the other is the infinite dihedral group Infinite dihedral group In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups.Definition:... ; see symmetry groups in one dimension Symmetry groups in one dimension A onedimensional symmetry group is a mathematical group that describes symmetries in one dimension .A pattern in 1D can be represented as a function f for, say, the color at position x.... 
2  5  10  17  17  these 2D space groups are also called wallpaper group Wallpaper group A wallpaper group is a mathematical classification of a twodimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art... s or plane groups. 
3  14  32  230  219  In 3D there are 230 crystallographic space group types, which reduces to 219 affine space group types because of some types being different from their mirror image; these are said to differ by "enantiomorphous Chirality (mathematics) In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object... character" (e.g. P3_{1}12 and P3_{2}12). Usually "space group" refers to 3D. They were enumerated independently by , and . 
4  64  227  4894  4783  The 4895 4dimensional groups were enumerated by . corrected the number of enantiomorphic groups from 112 to 111, so total number of groups is 4783+111=4894. There are 44 enantiomorphic point groups in 4dimensional space. If we consider enantiomorphic groups as different, the total number of point groups is 227+44=271 
5  189  955  222018  enumerated the ones of dimension 5  
6  841  7104  28934974  28927922  enumerated the ones of dimension 6. Initially published number of 826 Lattice types in was corrected to 841 in . See also 
Magnetic groups and time reversal
In addition to crystallographic space groups there are also magnetic space groups (also called twocolor (black and white) crystallographic groups). These symmetries contain an element known as time reversal. They treat time as an additional dimension, and the group elements can include time reversal as reflection in it. They are of importance in magnetic structureMagnetic structure
The term magnetic structure of a material pertains to the ordered arrangement of magnetic spins, typically within an ordered crystallographic lattice. Its study is a branch of solidstate chemistry.Magnetic structures:...
s that contain ordered unpaired spins, i.e. ferro
Ferromagnetism
Ferromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished...
, ferri
Ferrimagnetism
In physics, a ferrimagnetic material is one in which the magnetic moments of the atoms on different sublattices are opposed, as in antiferromagnetism; however, in ferrimagnetic materials, the opposing moments are unequal and a spontaneous magnetization remains...
or antiferromagnetic
Antiferromagnetism
In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usuallyrelated to the spins of electrons, align in a regular pattern with neighboring spins pointing in opposite directions. This is, like ferromagnetism and ferrimagnetism, a manifestation of ordered magnetism...
structures as studied by neutron diffraction
Neutron diffraction
Neutron diffraction or elastic neutron scattering is the application of neutron scattering to the determination of the atomic and/or magnetic structure of a material: A sample to be examined is placed in a beam of thermal or cold neutrons to obtain a diffraction pattern that provides information of...
. The time reversal element flips a magnetic spin while leaving all other structure the same and it can be combined with a number of other symmetry elements. Including time reversal there are 1651 magnetic space groups in 3D . It has also been possible to construct magnetic versions for other overall and lattice dimensions (Daniel Litvin's papers, ). Frieze groups are magnetic 1D line groups and layer groups are magnetic wallpaper groups, and the axial 3D point groups are magnetic 2D point groups. Number of original and magnetic groups by (overall, lattice) dimension:
 (1,1): 2, 7
 (2,1): 7, 31
 (2,2): 17, 80
 (3,1): 75, 394 (rod groups, not 3D line groups in general)
 (3,2): 80, 528
 (3,3): 230, 1651
 (4,4): 4894, 62227
Table of space groups in 3 dimensions
!Intl >#  Crystal system Crystal system In crystallography, the terms crystal system, crystal family, and lattice system each refer to one of several classes of space groups, lattices, point groups, or crystals... 
Point group Point group In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O... 
Space groups (international short symbol)  

Schönflies  
1  Triclinic (2)  1  C_{1}  P1 
2  C_{i}  P  
3–5  Monoclinic (13)  2  C_{2}  P2, P2_{1}, C2 
6–9  m  C_{s}  Pm, Pc, Cm, Cc  
10–15  2/m  C_{2h}  P2/m, P2_{1}/m, C2/m, P2/c, P2_{1}/c, C2/c  
16–24  Orthorhombic (59)  222  D_{2}  P222, P222_{1}, P2_{1}2_{1}2, P2_{1}2_{1}2_{1}, C222_{1}, C222, F222, I222, I2_{1}2_{1}2_{1} 
25–46  mm2  C_{2v}  Pmm2, Pmc2_{1}, Pcc2, Pma2, Pca2_{1}, Pnc2, Pmn2_{1}, Pba2, Pna2_{1}, Pnn2, Cmm2, Cmc2_{1}, Ccc2, Amm2, Aem2, Ama2, Aea2, Fmm2, Fdd2, Imm2, Iba2, Ima2  
47–74  mmm  D_{2h}  Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma, Cmcm, Cmce, Cmmm, Cccm, Cmme, Ccce, Fmmm, Fddd, Immm, Ibam, Ibca, Imma  
75–80  Tetragonal (68)  4  C_{4}  P4, P4_{1}, P4_{2}, P4_{3}, I4, I4_{1} 
81–82  S_{4}  P, I  
83–88  4/m  C_{4h}  P4/m, P4_{2}/m, P4/n, P4_{2}/n, I4/m, I4_{1}/a  
89–98  422  D_{4}  P422, P42_{1}2, P4_{1}22, P4_{1}2_{1}2, P4_{2}22, P4_{2}2_{1}2, P4_{3}22, P4_{3}2_{1}2, I422, I4_{1}22  
99–110  4mm  C_{4v}  P4mm, P4bm, P4_{2}cm, P4_{2}nm, P4cc, P4nc, P4_{2}mc, P4_{2}bc, I4mm, I4cm, I4_{1}md, I4_{1}cd  
111–122  2m  D_{2d}  P2m, P2c, P2_{1}m, P2_{1}c, Pm2, Pc2, Pb2, Pn2, Im2, Ic2, I2m, I2d  
123–142  4/mmm  D_{4h}  P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P4_{2}/mmc, P4_{2}/mcm, P4_{2}/nbc, P4_{2}/nnm, P4_{2}/mbc, P4_{2}/mnm, P4_{2}/nmc, P4_{2}/ncm, I4/mmm, I4/mcm, I4_{1}/amd, I4_{1}/acd  
143–146  Trigonal (25)  3  C_{3}  P3, P3_{1}, P3_{2}, R3 
147–148  S_{6}  P, R  
149–155  32  D_{3}  P312, P321, P3_{1}12, P3_{1}21, P3_{2}12, P3_{2}21, R32  
156–161  3m  C_{3v}  P3m1, P31m, P3c1, P31c, R3m, R3c  
162–167  m  D_{3d}  P1m, P1c, Pm1, Pc1, Rm, Rc,  
168–173  Hexagonal Hexagonal crystal system In crystallography, the hexagonal crystal system is one of the 7 crystal systems, the hexagonal lattice system is one of the 7 lattice systems, and the hexagonal crystal family is one of the 6 crystal families... (27) 
6  C_{6}  P6, P6_{1}, P6_{5}, P6_{2}, P6_{4}, P6_{3} 
174  C_{3h}  P  
175–176  6/m  C_{6h}  P6/m, P6_{3}/m  
177–182  622  D_{6}  P622, P6_{1}22, P6_{5}22, P6_{2}22, P6_{4}22, P6_{3}22  
183–186  6mm  C_{6v}  P6mm, P6cc, P6_{3}cm, P6_{3}mc  
187–190  m2  D_{3h}  Pm2, Pc2, P2m, P2c  
191–194  6/mmm  D_{6h}  P6/mmm, P6/mcc, P6_{3}/mcm, P6_{3}/mmc  
195–199  Cubic Cubic crystal system In crystallography, the cubic crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals.... (36) 
23  T  P23, F23, I23, P2_{1}3, I2_{1}3 
200–206  m  T_{h}  Pm, Pn, Fm, Fd, Im, Pa, Ia  
207–214  432  O  P432, P4_{2}32, F432, F4_{1}32, I432, P4_{3}32, P4_{1}32, I4_{1}32  
215–220  3m  T_{d}  P3m, F3m, I3m, P3n, F3c, I3d  
221–230  mm  O_{h}  Pmm, Pnn, Pmn, Pnm, Fmm, Fmc, Fdm, Fdc, Imm, Iad 
Note. An e plane is a double glide plane, one having glides in two different directions. They are found in five space groups, all in the orthorhombic system and with a centered lattice. The use of the symbol e became official with .
The lattice system can be found as follows. If the crystal system is not trigonal then the lattice system is of the same type. If the crystal system is trigonal, then the lattice system is hexagonal unless the space group is one of the seven in the rhombohedral lattice system consisting of the 7 trigonal space groups in the table above whose name begins with R. (The term rhombohedral system is also sometimes used as an alternative name for the whole trigonal system.) The hexagonal lattice system is larger than the hexagonal crystal system, and consists of the hexagonal crystal system together with the 18 groups of the trigonal crystal system other than the seven whose names begin with R.
The Bravais lattice of the space group is determined by the lattice system together with the initial letter of its name, which for the nonrhombohedral groups is P, I, F, or C, standing for the principal, body centered, face centered, or Cface centered lattices.
External links
 International Union of Crystallography
 Point Groups and Bravais Lattices
 Bilbao Crystallographic Server
 Space Group Info (old)
 Space Group Info (new)
 Crystal Lattice Structures: Index by Space Group
 Full list of 230 crystallographic space groups
 Interactive 3D visualization of all 230 crystallographic space groups
 The Geometry Center: 2.1 Formulas for Symmetries in Cartesian Coordinates (two dimensions)
 The Geometry Center: 10.1 Formulas for Symmetries in Cartesian Coordinates (three dimensions)