International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 3.2, pp. 720737

A point group^{1} is a group of symmetry operations all of which leave at least one point unmoved. Thus, all operations containing translations are excluded. Point groups can be subdivided into crystallographic and noncrystallographic point groups. A crystallographic point group is a point group that maps a point lattice onto itself. Consequently, rotations and rotoinversions are restricted to the well known crystallographic cases 1, 2, 3, 4, 6 and (cf. Section 1.2.1 ); matrices for these symmetry operations are listed in Tables 1.2.2.1 and 1.2.2.2 . No such restrictions apply to the noncrystallographic point groups.
The numbers of the crystallographic point groups are finite: 2 for one dimension, 10 for two dimensions and 32 for three dimensions. The numbers of noncrystallographic point groups for dimensions are infinite. The two and threedimensional crystallographic point groups and their crystal systems are summarized in Tables 3.2.1.1 and 3.2.1.2. They are described in detail in Section 3.2.1.2. The two onedimensional point groups are discussed in Section 2.1.3.16 . The noncrystallographic point groups are treated in Section 3.2.1.4.


Crystallographic point groups occur:
General point groups, i.e. crystallographic and noncrystallographic point groups, occur as:
A (geometric) crystal class (pointgroup type) is the set of all crystals having the same pointgroup symmetry. The word `class', therefore, denotes a classificatory pigeonhole and should not be used as synonymous with the point group of a particular crystal. The symbol of a crystal class is that of the common point group. (For geometric and arithmetic crystal classes of space groups, see Sections 1.3.4.2 and 1.3.4.4 .)
Of particular importance for the structure determination of crystals are the 11 centrosymmetric crystallographic point groups, because they describe the possible symmetries of the diffraction record of a crystal: ; ; mmm; ; ; ; ; ; ; ; . This is due to Friedel's rule, which states that, provided anomalous dispersion is neglected, the diffraction record of any crystal is centrosymmetric, even if the crystal is noncentrosymmetric. The symmetry of the diffraction record determines the Laue class of the crystal; this is further explained in Chapter 1.6 . For a given crystal, its Laue class is obtained if a symmetry centre is added to its point group, as shown in Table 3.2.2.1.
In two dimensions, six `centrosymmetric' crystallographic point groups and hence six twodimensional Laue classes exist: 2; 2mm; 4; 4mm; 6; 6mm. These point groups are, for instance, the only possible symmetries of zerolayer Xray photographs.
Among the centrosymmetric crystallographic point groups in three dimensions, the seven lattice point groups (holohedral point groups, holohedries) are of special importance because they constitute the possible point symmetries of lattices, i.e. the site symmetries of their nodes. In three dimensions, the seven holohedries are: ; ; mmm; ; ; ; . Note that is the point symmetry of the rhombohedral lattice and the point symmetry of the hexagonal lattice; both occur in the hexagonal crystal family (cf. Chapter 2.1 ). Point groups that are, within a crystal family, subgroups of a holohedry are called merohedries; they are called specifically hemihedries for subgroups of index 2, tetartohedries for index 4 and ogdohedries for index 8.
In two dimensions, four holohedries exist: 2; 2mm; 4mm; 6mm. Note that the hexagonal crystal family in two dimensions contains only one lattice type, with point symmetry 6mm.
Another classification of the crystallographic point groups is that into isomorphism classes. Here all those point groups that have the same kind of group table appear in one class. These isomorphism classes are also known under the name of abstract point groups.
There are 18 abstract crystallographic point groups in three dimensions: the point groups in each of the following lines are isomorphous and belong to the same abstract group: In two dimensions, the ten crystallographic point groups form nine abstract groups; the groups 2 and m are isomorphous and belong to the same abstract group, the remaining eight point groups correspond to one abstract group each.
In crystallography, point groups usually are described
Descriptions (i) through (iii) are given in this section, whereas for crystal drawings and actual photographs reference is made to textbooks of crystallography and mineralogy [Buerger (1956, ch. 10) and Phillips (1971, chs. 3, 4 and 6) are particularly rich in pictures of crystal morphologies]; this also applies to the construction and the properties of the stereographic projection.
In Tables 3.2.3.1 and 3.2.3.2, the two and threedimensional crystallographic point groups are listed and described. The tables are arranged according to crystal systems and Laue classes. Within each crystal system and Laue class, the sequence of the point groups corresponds to that in the spacegroup tables of this volume: pure rotation groups are followed by groups containing reflections, rotoinversions and inversions. The holohedral point group is always given last.
In Tables 3.2.3.1 and 3.2.3.2, some point groups are described in two or three versions, in order to bring out the relations to the corresponding space groups (cf. Section 2.1.3.2 ):
The presentation of the point groups is similar to that of the space groups in Part 2 . The headline contains the short Hermann–Mauguin and the Schoenflies symbols. The full Hermann–Mauguin symbol, if different, is given below the short symbol. No Schoenflies symbols exist for twodimensional groups. For an explanation of the symbols see Sections 1.4.1 and 2.1.3.4 , and Chapter 3.3 .
Next to the headline, a pair of stereographic projections is given. The diagram on the left displays a general crystal or point form, that on the right shows the `framework of symmetry elements'. Except as noted below, the c axis is always normal to the plane of the figure, the a axis points down the page and the b axis runs horizontally from left to right. For the five trigonal point groups, the c axis is normal to the page only for the description with `hexagonal axes'; if described with `rhombohedral axes', the direction [111] is normal and the positive a axis slopes towards the observer. The conventional coordinate systems used for the various crystal systems are listed in Table 2.1.1.1 and illustrated in Figs. 2.1.3.1 to 2.1.3.10 .
In the righthand projection, the graphical symbols of the symmetry elements are the same as those used in the spacegroup diagrams; they are listed in Chapter 2.1 . Note that the symbol of a symmetry centre, a small circle, is also used for a face pole in the lefthand diagram. Mirror planes are indicated by heavy solid lines or circles; thin lines are used for the projection circle, for symmetry axes in the plane and for some special zones in the cubic system.
In the lefthand projection, the projection circle and the coordinate axes are indicated by thin solid lines, as are again some special zones in the cubic system. The dots and circles in this projection can be interpreted in two ways.

The general, special and limiting crystal forms and point forms constitute the main part of the table for each point group. The theoretical background is given below in Section 3.2.1.2.2, Crystal and point forms and an explanation of the listed data is to be found in Section 3.2.1.2.3, Description of crystal and point forms.
The last entry for each point group contains the Symmetry of special projections, i.e. the plane point group that is obtained if the threedimensional point group is projected along a symmetry direction. The special projection directions are the same as for the space groups; they are listed in Section 2.1.3.14 . The relations between the axes of the threedimensional point group and those of its twodimensional projections can easily be derived with the help of the stereographic projection. No projection symmetries are listed for the twodimensional point groups.
Note that the symmetry of a projection along a certain direction may be higher than the symmetry of the crystal face normal to that direction. For example, in point group all faces have face symmetry 1, whereas projections along any direction have symmetry 2; in point group 422, the faces (001) and have face symmetry 4, whereas the projection along [001] has symmetry 4mm.
For a point group a crystal form is a set of all symmetryequivalent faces; a point form is a set of all symmetryequivalent points. Crystal and point forms in point groups correspond to `crystallographic orbits' in space groups; cf. Sections 1.1.7 and 1.4.4.1 .
Two kinds of crystal and point forms with respect to can be distinguished. They are defined as follows:
General and special crystal and point forms can be represented by their sets of equivalent Miller indices and point coordinates x, y, z. Each set of these `triplets' stands for infinitely many crystal forms or point forms which are obtained by independent variation of the values and signs of the Miller indices h, k, l or the point coordinates x, y, z.
It should be noted that for crystal forms, owing to the well known `law of rational indices', the indices h, k, l must be integers; no such restrictions apply to the coordinates x, y, z, which can be rational or irrational numbers.
Example
In point group 4, the general crystal form {hkl} stands for the set of all possible tetragonal pyramids, pointing either upwards or downwards, depending on the sign of l; similarly, the general point form x, y, z includes all possible squares, lying either above or below the origin, depending on the sign of z. For the limiting cases l = 0 or z = 0, see below.
In order to survey the infinite number of possible forms of a point group, they are classified into Wyckoff positions of crystal and point forms, for short Wyckoff positions. This name has been chosen in analogy to the Wyckoff positions of space groups; cf. Sections 1.4.4.2 and 2.1.3.11 . In point groups, the term `position' can be visualized as the position of the face poles and points in the stereographic projection. Each `Wyckoff position' is labelled by a Wyckoff letter.
Definition
A `Wyckoff position of crystal and point forms' consists of all those crystal forms (point forms) of a point group for which the face poles (points) are positioned on the same set of conjugate symmetry elements of ; i.e. for each face (point) of one form there is one face (point) of every other form of the same `Wyckoff position' that has exactly the same face (site) symmetry.
Each point group contains one `general Wyckoff position' comprising all general crystal and point forms. In addition, up to two `special Wyckoff positions' may occur in two dimensions and up to six in three dimensions. They are characterized by the different sets of conjugate face and site symmetries and correspond to the seven positions of a pole in the interior, on the three edges, and at the three vertices of the socalled `characteristic triangle' of the stereographic projection.
Examples
It is instructive to subdivide the crystal forms (point forms) of one Wyckoff position further, into characteristic and noncharacteristic forms. For this, one has to consider two symmetries that are connected with each crystal (point) form:
Examples
The eigensymmetries and the generating symmetries of the 47 crystal forms (point forms) are listed in Table 3.2.1.3. With the help of this table, one can find the various point groups in which a given crystal form (point form) occurs, as well as the face (site) symmetries that it exhibits in these point groups; for experimental methods see Sections 3.2.2.2 and 3.2.2.3. Diagrams of the 47 crystal forms are presented in Fig. 3.2.1.1.
^{†}These limiting forms occur in three or two nonequivalent orientations (different types of limiting forms); cf. Table 3.2.3.2.
^{‡}In point groups and , the tetragonal prism and the hexagonal prism occur twice, as a `basic special form' and as a `limiting special form'. In these cases, the point groups are listed twice, as `' and as `'. 
With the help of the two groups and , each crystal or point form occurring in a particular point group can be assigned to one of the following two categories:
The importance of this classification will be apparent from the following examples.
Examples
The general forms of the 13 point groups with no, or only one, symmetry direction (`monoaxial groups') , are always noncharacteristic, i.e. their eigensymmetries are enhanced in comparison with the generating point groups. The general positions of the other 19 point groups always contain characteristic crystal forms that may be used to determine the point group of a crystal uniquely (cf. Section 3.2.2).^{4}
So far, we have considered the occurrence of one crystal or point form in different point groups and different Wyckoff positions. We now turn to the occurrence of different kinds of crystal or point forms in one and the same Wyckoff position of a particular point group.
In a Wyckoff position, crystal forms (point forms) of different eigensymmetries may occur; the crystal forms (point forms) with the lowest eigensymmetry (which is always well defined) are called basic forms (German: Grundformen) of that Wyckoff position. The crystal and point forms of higher eigensymmetry are called limiting forms (German: Grenzformen) (cf. Table 3.2.1.3). These forms are always noncharacteristic.
Limiting forms^{5} occur for certain restricted values of the Miller indices or point coordinates. They always have the same multiplicity and oriented face (site) symmetry as the corresponding basic forms because they belong to the same Wyckoff position. The enhanced eigensymmetry of a limiting form may or may not be accompanied by a change in the topology^{6} of its polyhedra, compared with that of a basic form. In every case, however, the name of a limiting form is different from that of a basic form.
The face poles (or points) of a limiting form lie on symmetry elements of a supergroup of the point group that are not symmetry elements of the point group itself. There may be several such supergroups.
Examples
Whereas basic and limiting forms belonging to one `Wyckoff position' are always clearly distinguished, closer inspection shows that a Wyckoff position may contain different `types' of limiting forms. We need, therefore, a further criterion to classify the limiting forms of one Wyckoff position into types: A type of limiting form of a Wyckoff position consists of all those limiting forms for which the face poles (points) are located on the same set of additional conjugate symmetry elements of the holohedral point group (for the trigonal point groups, the hexagonal holohedry has to be taken). Different types of limiting forms may have the same eigensymmetry and the same topology, as shown by the examples below. The occurrence of two topologically different polyhedra as two `realizations' of one type of limiting form in point groups 23, and 432 is explained below in Section 3.2.1.2.4, Notes on crystal and point forms, item (viii).
Examples
Not considered in this volume are limiting forms of another kind, namely those that require either special metrical conditions for the axial ratios or irrational indices or coordinates (which always can be closely approximated by rational values). For instance, a rhombic disphenoid can, for special axial ratios, appear as a tetragonal or even as a cubic tetrahedron; similarly, a rhombohedron can degenerate to a cube. For special irrational indices, a ditetragonal prism changes to a (noncrystallographic) octagonal prism, a dihexagonal pyramid to a dodecagonal pyramid or a crystallographic pentagondodecahedron to a regular pentagondodecahedron. These kinds of limiting forms are listed by A. Niggli (1963).
In conclusion, each general or special Wyckoff position always contains one set of basic crystal (point) forms. In addition, it may contain one or more sets of limiting forms of different types. As a rule,^{7} each type comprises polyhedra of the same eigensymmetry and topology and, hence, of the same name, for instance `ditetragonal pyramid'. The name of the basic general forms is often used to designate the corresponding crystal class, for instance `ditetragonalpyramidal class'; some of these names are listed in Table 3.2.1.4.

The main part of each pointgroup table in Section 3.2.3 describes the general and special crystal and point forms of that point group, in a manner analogous to the positions in a space group. The general Wyckoff position is given at the top, followed downwards by the special Wyckoff positions with decreasing multiplicity. Within each Wyckoff position, the first block of column 6 refers to the basic forms, the subsequent blocks list the various types of limiting form, if any.
The columns, from left to right, contain the following data (further details are to be found below in Section 3.2.1.2.4, Notes on crystal and point forms):
Column 1: Multiplicity of the `Wyckoff position', i.e. the number of equivalent faces and points of a crystal or point form.
Column 2: Wyckoff letter. Each general or special `Wyckoff position' is designated by a `Wyckoff letter', analogous to the Wyckoff letter of a position in a space group (cf. Sections 1.4.4.2 and 2.1.3.11 ).
Column 3: Face symmetry or site symmetry, given in the form of an `oriented pointgroup symbol', analogous to the oriented sitesymmetry symbols of space groups (cf. Sections 1.4.4.2 and 2.1.3.12 ). The face symmetry is also the symmetry of etch pits, striations and other face markings. For the twodimensional point groups, this column contains the edge symmetry, which can be either 1 or m.
Column 4: Coordinates of the symmetryequivalent points of a point form.
Column 5: Name of crystal form. If more than one name is in common use, several are listed. The names of the limiting forms are also given. The crystal forms, their names, eigensymmetries and occurrence in the point groups are summarized in Table 3.2.1.3, which may be useful for determinative purposes, as explained in Sections 3.2.2.2 and 3.2.2.3. There are 47 different types of crystal form. Frequently, 48 are quoted because `sphenoid' and `dome' are considered as two different forms. It is customary, however, to regard them as the same form, with the name `dihedron'.
Name of point form (printed in italics). There exists no general convention on the names of the point forms. Here, only one name is given, which does not always agree with that of other authors. The names of the point forms are also contained in Table 3.2.1.3. Note that the same point form, `line segment', corresponds to both sphenoid and dome.
Column 6: Miller indices (hkl) for the symmetryequivalent faces (edges) of a crystal form. In the trigonal and hexagonal crystal systems, when referring to hexagonal axes, Bravais–Miller indices (hkil) are used, with .
With a few exceptions, the triplets of Miller indices (hkl) and point coordinates are arranged in such a way as to show analogous sequences; they are both based on the same set of generators, as described in Sections 1.4.3 and 2.1.3.10 . For all point groups, except those referred to a hexagonal coordinate system, the correspondence between the (hkl) and the triplets is immediately obvious.^{8}
The sets of symmetryequivalent crystal faces also represent the sets of equivalent reciprocallattice points, as well as the sets of equivalent Xray (neutron) reflections. This important aspect is treated in Klapper & Hahn (2010).
Examples

Further discussion of the data in Tables 3.2.3.1 and 3.2.3.2 as far as molecular symmetry is concerned can be found in Section 3.2.4.3.

Several different sets of names have been devised for the 32 crystal classes. Their use, however, has greatly declined since the introduction of the international pointgroup symbols. As examples, two sets (both translated into English) that are frequently found in the literature are given in Table 3.2.1.4. To the name of the class the name of the system has to be added: e.g. `tetragonal pyramidal' or `tetragonal tetartohedry'.
Note that Friedel (1926) based his nomenclature on the point symmetry of the lattice. Hence, two names are given for the five trigonal point groups, depending whether the lattice is hexagonal or rhombohedral: e.g. `hexagonal ogdohedry' and `rhombohedral tetartohedry'.
In this section, the sub and supergroup relations between the crystallographic point groups are presented in the form of a `family tree'.^{12} Figs. 3.2.1.2 and 3.2.1.3 apply to two and three dimensions. The sub and supergroup relations between two groups are represented by solid or dashed lines. For a given point group of order the lines to groups of lower order connect with all its maximal subgroups with orders ; the index [i] of each subgroup is given by the ratio of the orders . The lines to groups of higher order connect with all its minimal supergroups with orders ; the index [i] of each supergroup is given by the ratio . In other words: if the diagram is read downwards, subgroup relations are displayed; if it is read upwards, supergroup relations are revealed. The index is always an integer (theorem of Lagrange) and can be easily obtained from the group orders given on the left of the diagrams. The highest index of a maximal subgroup is [3] for two dimensions and [4] for three dimensions.

Maximal subgroups and minimal supergroups of the twodimensional crystallographic point groups. Solid lines indicate maximal normal subgroups; double solid lines mean that there are two maximal normal subgroups with the same symbol. Dashed lines refer to sets of maximal conjugate subgroups. The group orders are given on the left. 

Maximal subgroups and minimal supergroups of the threedimensional crystallographic point groups. Solid lines indicate maximal normal subgroups; double or triple solid lines mean that there are two or three maximal normal subgroups with the same symbol. Dashed lines refer to sets of maximal conjugate subgroups. The group orders are given on the left. Full Hermann–Mauguin symbols are used. 
Two important kinds of subgroups, namely sets of conjugate subgroups and normal subgroups, are distinguished by dashed and solid lines. They are characterized as follows:
The subgroups of a group are conjugate subgroups if are symmetryequivalent in , i.e. if for every pair at least one symmetry operation of exists which maps onto : ; cf. Sections 1.1.5 and 1.1.8 .
Examples

The subgroup of a group is a normal (or invariant) subgroup if no subgroup of exists that is conjugate to in . Note that this does not imply that is also a normal subgroup of any supergroup of . Subgroups of index [2] are always normal and maximal (cf. Section 1.1.5 ). (The role of normal subgroups for the structure of space groups is discussed in Sections 1.3.3 and 1.4.2.3 .)
Examples

Figs. 3.2.1.2 and 3.2.1.3 show that there exist two `summits' in both two and three dimensions from which all other point groups can be derived by `chains' of maximal subgroups. These summits are formed by the square and the hexagonal holohedry in two dimensions and by the cubic and the hexagonal holohedry in three dimensions.
The sub and supergroups of the point groups are useful both in their own right and as a basis of the translationengleiche or t subgroups and supergroups of space groups (cf. Section 1.7.1 ). Tables of the sub and supergroups of the plane groups and space groups are contained in Volume A1 of International Tables for Crystallography (2010). A general discussion of sub and supergroups of crystallographic groups, together with further explanations and examples, is given in Section 1.7.1 .
In Sections 3.2.1.2 and 3.2.1.3, only the 32 crystallographic point groups (crystal classes) are considered. In addition, infinitely many noncrystallographic point groups exist that are of interest as possible symmetries of molecules and of quasicrystals and as approximate local site symmetries in crystals. Crystallographic and noncrystallographic point groups are collected here under the name general point groups. They are reviewed in this section and listed in Tables 3.2.1.5 and 3.2.1.6.
Because of the infinite number of these groups only classes of general point groups (general classes)^{13} can be listed. They are grouped into general systems, which are similar to the crystal systems. The `general classes' are of two kinds: in the cubic, icosahedral, circular, cylindrical and spherical system, each general class contains one point group only, whereas in the 4Ngonal, gonal and gonal system, each general class contains infinitely many point groups, which differ in their principal nfold symmetry axis, with for the 4Ngonal system, for the gonal system and for the gonal system.
Furthermore, some general point groups are of order infinity because they contain symmetry axes (rotation or rotoinversion axes) of order infinity^{14} (∞fold axes). These point groups occur in the circular system (two dimensions) and in the cylindrical and spherical systems (three dimensions).
The Hermann–Mauguin and Schoenflies symbols for the general point groups follow the rules of the crystallographic point groups (cf. Sections 1.4.1 , 2.1.3.4 and 3.3.1 ). This extends also to the infinite groups where symbols like or are immediately obvious.
In two dimensions (Table 3.2.1.5), the eight general classes are collected into three systems. Two of these, the 4Ngonal and the gonal systems, contain only point groups of finite order with one nfold rotation point each. These systems are generalizations of the square and hexagonal crystal systems. The circular system consists of two infinite point groups, with one ∞fold rotation point each.
^{†}A rotating circle has no mirror lines; there exist two enantiomorphic circles with opposite senses of rotation. A stationary circle has infinitely many mirror lines through its centre.

In three dimensions (Table 3.2.1.6), the 33 general classes are collected into seven systems. Three of these, the 4Ngonal, the gonal and the gonal systems,^{15} contain only point groups of finite order with one principal nfold symmetry axis each. These systems are generalizations of the tetragonal, trigonal and hexagonal crystal systems (cf. Table 3.2.3.2). The five cubic groups are well known as crystallographic groups. The two icosahedral groups of orders 60 and 120, characterized by special combinations of twofold, threefold and fivefold symmetry axes, are discussed in more detail below. The groups of the cylindrical and the spherical systems are all of order infinity; they describe the symmetries of cylinders, cones, rotation ellipsoids, spheres etc.^{16}
^{†}The Hermann–Mauguin symbols of the two icosahedral point groups are often written as 532 and (see text).
^{‡}Rotating and `antirotating' forms in the cylindrical system have no `vertical' mirror planes, whereas stationary forms have infinitely many vertical mirror planes. In classes ∞ and , enantiomorphism occurs, i.e. forms with opposite senses of rotation. Class exhibits no enantiomorphism due to the centre of symmetry, even though the double cone is rotating in one direction. This can be understood as follows: The handedness of a rotating cone depends on the sense of rotation with respect to the axial direction from the base to the tip of the cone. Thus, the rotating double cone consists of two cones with opposite handedness and opposite orientations related by the (single) horizontal mirror plane. In contrast, the `antirotating' double cone in class consists of two cones of equal handedness and opposite orientations, which are related by the (infinitely many) twofold axes. The term `antirotating' means that upper and lower halves of the forms rotate in opposite directions. ^{§}The spheres in class of the spherical system must rotate around an axis with at least two different orientations, in order to suppress all mirror planes. This class exhibits enantiomorphism, i.e. it contains spheres with either righthanded or lefthanded senses of rotation around the axes (cf. Section 3.2.2.4, Optical properties). The stationary spheres in class contain infinitely many mirror planes through the centres of the spheres. Group is sometimes symbolized by ; group by or . The symbols used here indicate the minimal symmetry necessary to generate the groups; they show, furthermore, the relation to the cubic groups. The Schoenflies symbol K is derived from the German name Kugelgruppe. 
It is possible to define the threedimensional point groups on the basis of either rotoinversion axes or rotoreflection axes . The equivalence between these two descriptions is apparent from the following examples: In the present tables, the standard convention of using rotoinversion axes is followed.
Tables 3.2.1.5 and 3.2.1.6 contain for each class its general Hermann–Mauguin and Schoenflies symbols, the group order and the names of the general face form and its dual, the general point form.^{17} Special and limiting forms are not given, nor are `Miller indices' (hkl) and point coordinates x, y, z. They can be derived easily from Tables 3.2.3.1 and 3.2.3.2 for the crystallographic groups.^{18}
The two point groups 235 and of the icosahedral system (orders 60 and 120) are of particular interest among the noncrystallographic groups because of the occurrence of fivefold axes and their increasing importance as symmetries of molecules (viruses), of quasicrystals, and as approximate local site symmetries in crystals (alloys, icosahedron). Furthermore, they contain as special forms the two noncrystallographic platonic solids, the regular icosahedron (20 faces, 12 vertices) and its dual, the regular pentagondodecahedron (12 faces, 20 vertices).
The icosahedral groups (cf. diagrams in Table 3.2.3.3) are characterized by six fivefold axes that include angles of . Each fivefold axis is surrounded by five threefold and five twofold axes, with angular distances of between a fivefold and a threefold axis and of between a fivefold and a twofold axis. The angles between neighbouring threefold axes are , between neighbouring twofold axes . The smallest angle between a threefold and a twofold axis is .
Each of the six fivefold axes is perpendicular to five twofold axes; there are thus six maximal conjugate pentagonal subgroups of types 52 (for 235) and (for ) with index [6]. Each of the ten threefold axes is perpendicular to three twofold axes, leading to ten maximal conjugate trigonal subgroups of types 32 (for 235) and (for ) with index [10]. There occur, furthermore, five maximal conjugate cubic subgroups of types 23 (for 235) and (for ) with index [5].
The two icosahedral groups are listed in Table 3.2.3.3, in a form similar to the cubic point groups in Table 3.2.3.2. Each group is illustrated by stereographic projections of the symmetry elements and the general face poles (general points); the complete sets of symmetry elements are listed below the stereograms. Both groups are referred to a cubic coordinate system, with the coordinate axes along three twofold rotation axes and with four threefold axes along the body diagonals. This relation is well brought out by symbolizing these groups as 235 and instead of the customary symbols 532 and .
The table contains also the multiplicities, the Wyckoff letters and the names of the general and special face forms and their duals, the point forms, as well as the oriented face and sitesymmetry symbols. In the icosahedral `holohedry' , the special `Wyckoff position' 60d occurs in three realizations, i.e. with three types of polyhedra. In 235, however, these three types of polyhedra are different realizations of the limiting general forms, which depend on the location of the poles with respect to the axes 2, 3 and 5. For this reason, the three entries are connected by braces; cf. Section 3.2.1.2.4, Notes on crystal and point forms, item (viii).
Not included are the sets of equivalent Miller indices and point coordinates. Instead, only the `initial' triplets (hkl) and for each type of form are listed. The complete sets of indices and coordinates can be obtained in two steps^{19} as follows:

This sequence of matrices ensures the same correspondence between the Miller indices and the point coordinates as for the crystallographic point groups in Table 3.2.3.2.
The matrices^{20} are with^{21} These matrices correspond to counterclockwise rotations of 72, 144, 216 and around a fivefold axis parallel to .
The resulting indices h, k, l and coordinates x, y, z are irrational but can be approximated closely by rational (or integral) numbers. This explains the occurrence of almost regular icosahedra or pentagondodecahedra as crystal forms (for instance pyrite) or atomic groups (for instance icosahedron).
Further descriptions (including diagrams) of noncrystallographic groups are contained in papers by Nowacki (1933) and A. Niggli (1963) and in the textbooks by P. Niggli (1941, pp. 78–80, 96), Shubnikov & Koptsik (1974) and Vainshtein (1994). For the geometry of polyhedra, the well known books by H. S. M. Coxeter (especially Coxeter, 1973) are recommended.
In Figs. 3.2.1.4 to 3.2.1.6, the subgroup and supergroup relations between the twodimensional and threedimensional general point groups are illustrated. It should be remembered that the index of a group–subgroup relation between two groups of order infinity may be finite or infinite. For the two spherical groups, for instance, the index is [2]; the cylindrical groups, on the other hand, are subgroups of index [] of the spherical groups.

Subgroups and supergroups of the twodimensional general point groups. Solid lines indicate maximal normal subgroups, double solid lines mean that there are two maximal normal subgroups with the same symbol. Dashed lines refer to sets of maximal conjugate subgroups. For the finite groups, the orders are given on the left. Note that the subgroups of the two circular groups are not maximal and the diagram applies only to a specified value of N (see text). For complete examples see Fig. 3.2.1.5. 

The subgroups of the twodimensional general point groups 16mm (4Ngonal system) and 18mm [gonal system, including the gonal groups]. Compare with Fig. 3.2.1.4 which applies only to one value of N. 

Subgroups and supergroups of the threedimensional general point groups. Solid lines indicate maximal normal subgroups, double solid lines mean that there are two maximal normal subgroups with the same symbol. Dashed lines refer to sets of maximal conjugate subgroups. For the finite groups, the orders are given on the left and on the right. Note that the subgroups of the five cylindrical groups are not maximal and that the diagram applies only to a specified value of N (see text). Only those crystallographic point groups are included that are maximal subgroups of noncrystallographic point groups, cf. Fig. 3.2.1.3. Full Hermann–Mauguin symbols are used. 
Fig. 3.2.1.4 for two dimensions shows that the two circular groups ∞m and ∞ have subgroups of types nmm and n, respectively, each of index []. The order of the rotation point may be or . In the first case, the subgroups belong to the 4Ngonal system, in the latter two cases, they belong to the gonal system. [In the diagram of the gonal system, the gonal groups appear with the symbols and .] The subgroups of the circular groups are not maximal because for any given value of N there exists a group with which is both a subgroup of the circular group and a supergroup of the initial group.
The subgroup relations, for a specified value of N, within the 4Ngonal and the gonal system, are shown in the lower part of the figure. They correspond to those of the crystallographic groups. A finite number of further maximal subgroups is obtained for lower values of N, until the crystallographic groups (Fig. 3.2.1.2) are reached. This is illustrated for both systems in Fig. 3.2.1.5.
Fig. 3.2.1.6 for three dimensions illustrates that the two spherical groups and each have one infinite set of cylindrical maximal conjugate subgroups, as well as one infinite set of cubic and one infinite set of icosahedral maximal finite conjugate subgroups, all of index [].
Each of the two icosahedral groups 235 and has one set of five cubic, one set of six pentagonal and one set of ten trigonal maximal conjugate subgroups of indices [5], [6] and [10], respectively (cf. Section 3.2.1.4.2, The two icosahedral groups); they are listed on the right of Fig. 3.2.1.6. For the crystallographic groups, Fig. 3.2.1.3 applies. The subgroup types of the five cylindrical point groups are shown on the upper left part of Fig. 3.2.1.6. As explained above for two dimensions, these subgroups are not maximal and of index []. Depending upon whether the main symmetry axis has the multiplicity 4N, or , the subgroups belong to the 4Ngonal, gonal or gonal system.
The subgroup and supergroup relations within these three systems are displayed in the lower left part of Fig. 3.2.1.6. They are analogous to the crystallographic groups. To facilitate the use of the diagrams, the gonal and the gonal systems are combined, with the consequence that the five classes of the gonal system now appear with the symbols and . Again, the diagrams apply to a specified value of N. A finite number of further maximal subgroups is obtained for lower values of N, until the crystallographic groups (Fig. 3.2.1.3) are reached (cf. the twodimensional examples in Fig. 3.2.1.5).
References
International Tables for Crystallography (2010). Vol. A1, Symmetry Relations Between Space Groups, 2nd ed., edited by H. Wondratschek & U. Müller. Chichester: Wiley.Buerger, M. J. (1956). Elementary Crystallography, especially chs. 10 and 11. New York: Wiley.
Burzlaff, H. & Zimmermann, H. (1977). Symmetrielehre, especially ch. II.3. Stuttgart: Thieme.
Coxeter, H. S. M. (1973). Regular Polytopes, 3rd ed. New York: Dover.
Fischer, W., Burzlaff, H., Hellner, E. & Donnay, J. D. H. (1973). Space Groups and Lattice Complexes. NBS Monograph No. 134, especially pp. 28–33. Washington, DC: National Bureau of Standards.
Friedel, G. (1926). Leçons de cristallographie. Nancy/Paris/Strasbourg: BergerLevrault. [Reprinted (1964). Paris: Blanchard.]
Klapper, H. & Hahn, Th. (2010). The application of eigensymmetries of face forms to anomalous scattering and twinning by merohedry in Xray diffraction. Acta Cryst. A66, 327–346.
Niggli, A. (1963). Zur Topologie, Metrik und Symmetrie der einfachen Kristallformen. Schweiz. Mineral. Petrogr. Mitt. 43, 49–58.
Niggli, P. (1941). Lehrbuch der Mineralogie und Kristallchemie, 3rd ed. Berlin: Borntraeger.
Nowacki, W. (1933). Die nichtkristallographischen Punktgruppen. Z. Kristallogr. 86, 19–31.
Phillips, F. C. (1971). An Introduction to Crystallography, 4th ed., chs. 3, 4, 6 and 13. London: Longman.
Shubnikov, A. V. & Koptsik, V. A. (1974). Symmetry in Science and Art, especially chs. 2 and 3. New York: Plenum.
Vainshtein, B. K. (1994). Fundamentals of Crystals. Symmetry, and Methods of Structural Crystallography, 2nd ed., especially ch. 2.6 and 3. Berlin: Springer.