International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2011 
International Tables for Crystallography (2011). Vol. A1, ch. 2.1, pp. 7296
https://doi.org/10.1107/97809553602060000797 Chapter 2.1. Guide to the subgroup tables and graphs^{a}Institut für Kristallographie, Universität, D76128 Karlsruhe, Germany, and ^{b}Departamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E48080 Bilbao, Spain This chapter is the users' guide to the subgroup tables and graphs of Part 2 of this volume. Sections 2.1.1 and 2.1.2 refer to the contents, arrangement and structure of the subgroup tables of Chapters 2.2 and 2.3 . In the next three sections, the data listed for the maximal translationengleiche subgroups (Section 2.1.3), maximal klassengleiche subgroups (Section 2.1.4) and for the series of the (infinitely many) maximal isomorphic subgroups (Section 2.1.5) are described in detail with many examples. Section 2.1.6 describes the data listed for minimal supergroups. In Section 2.1.7 the procedures for the derivation of the data for minimal supergroups from the (complete) data for maximal subgroups are developed. Such procedures exist for the minimal supergroups of all space groups with the exception of minimal tsupergroups of triclinic and monoclinic space groups. The last section, Section 2.1.8, familiarizes the user with the subgroup graphs of the space groups listed in Chapters 2.4 and 2.5 . The chapter is supplemented by the subgroup table of the space group R3 , which is printed inside the front cover of the volume with some explanatory text. 
In this chapter, the subgroup tables, the subgroup graphs and their general organization are discussed. In the following sections, the different types of data are explained in detail. For every plane group and every space group there is a separate table of maximal subgroups and minimal supergroups. The subgroup data are listed either individually, or as members of (infinite) series, or both. The supergroup data are not as complete as the subgroup data. However, most of them can be obtained by proper evaluation of the subgroup data, as shown in Section 2.1.7. In addition, there are graphs of translationengleiche and klassengleiche subgroups which contain for each space group all kinds of subgroups, not just the maximal ones.
The presentation of the planegroup and spacegroup data in the tables of Chapters 2.2 and 2.3 follows the style of the tables of Parts 6 (plane groups) and 7 (space groups) in Vol. A of International Tables for Crystallography (2005), henceforth abbreviated as IT A. The data comprise:
For the majority of groups, the data can be listed completely on one page. Sometimes two pages are needed. If the data extend less than half a page over one full page and data for a neighbouring spacegroup table `overflow' to a similar extent, then the two overflows are displayed on the same page. Such deviations from the standard sequence are indicated on the relevant pages by a remark Continued on…. The two overflows are separated by a solid line and are designated by their headlines.
The sequence of the plane groups and space groups in this volume follows exactly that of the tables of Part 6 (plane groups) and Part 7 (space groups) in IT A. The format of the subgroup tables has also been chosen to resemble that of the tables of IT A as far as possible. Graphs for translationengleiche and klassengleiche subgroups are found in Chapters 2.4 and 2.5 . Examples of graphs of subgroups can also be found in Section 10.1.4.3 of IT A, but only for subgroups of point groups. The graphs for the space groups are described in Section 2.1.8.
Some basic data in these tables have been repeated from the tables of IT A in order to allow the use of the subgroup tables independently of IT A. These data and the main features of the tables are described in this section.
The headline contains the specification of the space group for which the maximal subgroups are considered. The headline lists from the outside margin inwards:
As in IT A, for each plane group and space group a set of symmetry operations is listed under the heading `Generators selected'. From these group elements, can be generated conveniently. The generators in this volume are the same as those in IT A. They are explained in Section 2.2.10 of IT A and the choice of the generators is explained in Section 8.3.5 of IT A.
The generators are listed again in this present volume because many of the subgroups are characterized by their generators. These (often nonconventional) generators of the subgroups can thus be compared with the conventional ones without reference to IT A.
Like the generators, the general position has also been copied from IT A, where an explanation can be found in Section 2.2.11 . The general position in IT A is the first block under the heading `Positions', characterized by its site symmetry of 1. The elements of the general position have the following meanings:
Many of the subgroups in these tables are characterized by the elements of their general position. These elements are specified by numbers which refer to the corresponding numbers in the general position of . Other subgroups are listed by the numbers of their generators, which again refer to the corresponding numbers in the general position of . Therefore, the listing of the general position of as well as the listing of the generators of is essential for the structure of these tables. For examples, see Sections 2.1.3 and 2.1.4.
All 17 planegroup types^{1} and 230 spacegroup types are listed and described in IT A. However, whereas each planegroup type is represented exactly once, 44 spacegroup types, i.e. nearly 20%, are represented twice. This means that the conventional setting of these 44 spacegroup types is not uniquely determined and must be specified. The same settings underlie the data of this volume, which follows IT A as much as possible.
There are three reasons for listing a spacegroup type twice:
If there is a choice of setting for the space group , the chosen setting is indicated under the HM symbol in the headline. If a subgroup belongs to one of these 44 spacegroup types, its `conventional setting' must be defined. The rules that are followed in this volume are explained in Section 2.1.2.5.
As in the subgroup data of IT A, the sequence of the maximal subgroups is as follows: subgroups of the same kind are collected in a block. Each block has a heading. Compared with IT A, the blocks have been partly reorganized because in this volume all maximal isomorphic subgroups are listed, whereas in IT A only a few of them are described. In addition, the subgroups are described here in more detail.
The sequence of the subgroups within each block follows the value of the index; subgroups of lowest index are listed first. Subgroups having the same index are listed according to their lattice relations to the lattice of the original group , cf. Section 2.1.4.3. Subgroups with the same lattice relations are listed in decreasing order of spacegroup number.
Conjugate subgroups have the same index and the same spacegroup number. They are grouped together and connected by a brace on the lefthand side. Conjugate classes of maximal subgroups and their lengths are therefore easily recognized. In the series of maximal isomorphic subgroups, braces are inapplicable so there the conjugacy classes are stated explicitly.
The block designations are:
The multiple listing of 44 spacegroup types has implications for the subgroup tables. If a subgroup belongs to one of these types, its `conventional setting' must be defined. In many cases there is a natural choice; sometimes, however, such a choice does not exist, and the appropriate conventions have to be stated.
The three reasons for listing a space group twice will be discussed in this section, cf. Section 2.1.2.3.

Remarks (see also the following examples):

The necessary adjustment is performed through a coordinate transformation, i.e. by a change of the basis and by an origin shift, see Section 2.1.3.3.
Example 2.1.2.5.1
, No. 10; unique axis b.
II Maximal klassengleiche subgroups, Enlarged unit cell

Example 2.1.2.5.2
, No. 10; unique axis c.
II Maximal klassengleiche subgroups, Enlarged unit cell

Example 2.1.2.5.3
, No. 50; origin choice 1.
I Maximal (monoclinic) translationengleiche subgroups

Altogether, 24 orthorhombic, tetragonal and cubic space groups with inversions are listed twice in IT A. There are three kinds of possible ambiguities for such groups with two origin choices:

The seven trigonal space groups with a rhombohedral lattice are often called rhombohedral space groups. Their HM symbols begin with the lattice letter R and they are listed with both hexagonal axes and rhombohedral axes.
Rules

Remarks

Example 2.1.2.5.7
, No. 146. Maximal klassengleiche subgroups of index 2 and 3. Comparison of the data for the settings `hexagonal axes' and `rhombohedral axes'. The data for the general position and the generators are omitted.
The sequence of the blocks has priority over the classification by increasing index. Therefore, in the setting `hexagonal axes', the subgroups of index 3 precede the subgroup of index 2.
In the tables, the lattice relations are simpler for the setting `hexagonal axes'.
The complete general position is listed for the maximal ksubgroups of index 3 in the setting `hexagonal axes'; only the generator is listed for rhombohedral axes.
In this block, all maximal tsubgroups of the plane groups and the space groups are listed individually. Maximal tsugroups are always nonisomorphic.
For the sequence of the subgroups, see Section 2.1.2.4. There are no lattice relations for tsubgroups because the lattice is retained. Therefore, the sequence is determined only by the rising value of the index and by the decreasing spacegroup number.
The listing is similar to that of IT A and presents on one line the following information for each subgroup : Conjugate subgroups are listed together and are connected by a left brace.
The symbols have the following meaning:

Remarks

In general, the numbers in the list `Sequence' of follow the order of the numbers in the group , i.e. they rise monotonically. Sometimes this sequence is modified because those entries which have the same additional translations are joined together, see, e.g. the maximal ksubgroups of with `Loss of centring translations'. In addition, in a class of conjugate subgroups, the monotonically rising order may be obeyed only for the first member of the conjugacy class. The order of the other members is then determined by the conjugation of the first member. (In IT A the monotonically rising order of the numbers is kept in all conjugate subgroups.)
Example 2.1.3.2.1
, No. 221, tetragonal tsubgroups
I Maximal translationengleiche subgroupsComments:
If is the order of the first sequence, then the second sequence follows the order , , . Here the C means a threefold rotation and is the conjugating element; for the second subgroup of the general position of ; for the third subgroup . In this example the columns w of the symmetry operations (and thus of the conjugating elements) are the zero columns o and could be omitted.
The description of the subgroups can be explained by the following four examples.
Example 2.1.3.2.4
, No. 137, ORIGIN CHOICE 2
I Maximal translationengleiche subgroups
…
(59, ) 1; 2; 5; 6; 9; 10; 13; 14
Comments:

Example 2.1.3.2.5
, No. 151
I Maximal translationengleiche subgroupsComments:

Each tsubgroup is defined by its representatives, listed under `sequence' by numbers each of which designates an element of . These elements form the general position of . They are taken from the general position of and, therefore, are referred to the coordinate system of . In the general position of , however, its elements are referred to the coordinate system of . In order to allow the transfer of the data from the coordinate system of to that of , the tools for this transformation are provided in the columns `matrix' and `shift' of the subgroup tables. The designation of the quantities is that of IT A Part 5 and is repeated here for convenience. The transformation described in this section is not restricted to translationengleiche subgroups but is applied to klassengleiche subgroups as well.
In the following, columns and rows are designated by boldface italic lowercase letters. Point coordinates , translation parts of the symmetry operations and shifts are represented by columns. The sets of basis vectors and are represented by rows [indicated by , which means `transposed']. The quantities with unprimed symbols are referred to the coordinate system of , those with primes are referred to the coordinate system of .
The following columns will be used ( is analogous to w):
The matrices W and of the symmetry operations, as well as the matrix P for a change of basis and its inverse , are designated by boldface italic uppercase letters ( is analogous to W): Let be the row of basis vectors of and the basis of , then the basis is expressed in the basis by the system of equationsor
The column p of coordinates of the origin of is referred to the coordinate system of and is called the origin shift. The matrix–column pair (P, p) describes the transformation from the coordinate system of to that of , for details, cf. IT A, Part 5 . Therefore, P and p are listed in the subgroup tables in the columns `matrix' and `shift', cf. Section 2.1.3.2. The column `matrix' is empty if there is no change of basis, i.e. if P is the unit matrix I. The column `shift' is empty if there is no origin shift, i.e. if p is the column o consisting of zeroes only.
A change of the coordinate system, described by the matrix–column pair , changes the point coordinates from the column x to the column . The formulae for this change do not contain the pair itself, but the related pair :
Not only the point coordinates but also the matrix–column pairs for the symmetry operations are changed by a change of the coordinate system. A symmetry operation is described in the coordinate system of by the system of equations^{2}ori.e. by the matrix–column pair (W, w). The symmetry operation will be described in the coordinate system of the subgroup by the equation and thus by the pair . This pair can be calculated from the pair by the equations and
These equations are rather complicated and unpleasant. They become simple when using augmented matrices and columns. In this case the formulae are reduced formally to normal matrix multiplication [the formalism is simpler but the necessary calculations are not, because the inversion of a (4 × 4) matrix is tedious if done by hand].
The matrices P, Q, W and W′ may be combined with the corresponding columns p, q, w and w′ to form (4 × 4) matrices (callled augmented matrices):^{3}The coefficients of these augmented matrices are integer, rational or real numbers.
The (3 × 1) rows (a)^{T} and (a′)^{T} must be augmented to (4 × 1) rows by appending some vectors and , respectively, as fourth entries in order to enable matrix multiplication with the augmented matrices:with . As the vector one can take the zero vector , which results in , i.e.The relation between and is given by equation (2.1.3.10), which replaces equation (2.1.3.3),
Analogously, the (3 × 1) columns x and x′ must be augmented to (4 × 1) columns by a `1' in the fourth row in order to enable matrix multiplication with the augmented matrices:The three equations (2.1.3.4), (2.1.3.8) and (2.1.3.9) are replaced by the two equationsand
Example 2.1.3.3.1
Consider the data listed for the tsubgroups of , No. 31:This means that the transformation matrices and origin shifts are
The first subgroup is monoclinic, the symmetry direction is the b axis, which is standard. However, the glide direction is nonconventional. Therefore, the basis of is transformed to a basis of the subgroup such that the b axis is retained, the glide direction becomes the axis and the axis is chosen such that the basis is a righthanded one, the angle and the transformation matrix P is simple. This is done by the chosen matrix . The origin shift is the o column.
With equations (2.1.3.8) and (2.1.3.9), one obtains for the glide reflection , which is after standardization by .
For the second monoclinic subgroup, the symmetry direction is the (nonconventional) a axis. The rules of Section 2.1.2.5 require a change to the setting `unique axis b'. A cyclic permutation of the basis vectors is the simplest way to achieve this. The reflection is now described by . Again there is no origin shift.
The third monoclinic subgroup is in the conventional setting `unique axis c', but the origin must be shifted onto the screw axis. This is achieved by applying equation (2.1.3.9) with , which changes of to of .
Example 2.1.3.3.2
Evaluation of the tsubgroup data of the space group , No. 151, started in Example 2.1.3.2.5. The evaluation is now continued with the columns `sequence', `matrix' and `shift'. They are used for the transformation of the elements of to their conventional form. Only the monoclinic tsubgroups are of interest here because the trigonal subgroup is already in the standard setting.
One takes from the tables of subgroups in Chapter 2.3 Designating the three matrices by , , , one obtainswith the corresponding inverse matricesand the origin shiftsFor the three new bases this means All these bases span orthohexagonal cells with twice the volume of the original hexagonal cell because for the matrices holds.
In the general position of , No.151, one findsThese entries represent the matrix–column pairs : Application of equations (2.1.3.8) on the matrices and (2.1.3.9) on the columns of the matrix–column pairs results in All translation vectors of are retained in the subgroups but the volume of the cells is doubled. Therefore, there must be centringtranslation vectors in the new cells. For example, the application of equation (2.1.3.9) with to the translation of with the vector , i.e. , results in the column , i.e. the centring translation of the subgroup. Either by calculation or, more easily, from a small sketch one sees that the vectors for , for (and for ) correspond to the cellcentring translation vectors of the subgroup cells.
Comments:
This example reveals that the conjugation of conjugate subgroups does not necessarily imply the conjugation of the representatives of these subgroups in the general positions of IT A. The three monoclinic subgroups in this example are conjugate in the group by the screw rotation. Conjugation of the representatives (4) and (6) by the screw rotation of results in the column , which is standardized according to the rules of IT A to . Thus, the conjugacy relation is disturbed by the standardization of the representative (5).
The listing of the maximal klassengleiche subgroups (maximal ksubgroups) of the space group is divided into the following three blocks for practical reasons:
Loss of centring translations. Maximal subgroups of this block have the same conventional unit cell as the original space group . They are always nonisomorphic and have index 2 for plane groups and index 2, 3 or 4 for space groups.
Enlarged unit cell. Under this heading, maximal subgroups of index 2, 3 and 4 are listed for which the conventional unit cell has been enlarged. The block contains isomorphic and nonisomorphic subgroups with this property.
Series of maximal isomorphic subgroups. In this block all maximal isomorphic subgroups of a space group are listed in a small number of infinite series of subgroups with no restriction on the index, cf. Sections 2.1.2.4 and 2.1.5.
The description of the subgroups is the same within the same block but differs between the blocks. The partition into these blocks differs from the partition in IT A, where the three blocks are called `maximal nonisomorphic subgroups IIa', `maximal nonisomorphic subgroups IIb' and `maximal isomorphic subgroups of lowest index IIc'.
The kind of listing in the three blocks of this volume is discussed in Sections 2.1.4.2, 2.1.4.3 and 2.1.5 below.
Consider a space group with a centred lattice, a space group whose HM symbol does not start with the lattice letter P but with one of the letters A, B, C, F, I or R. The block contains those maximal subgroups of which have fully or partly lost their centring translations and thus are not tsubgroups. The conventional unit cell is not changed.
Only in space groups with an Fcentred lattice can the centring be partially lost, as is seen in the list of the space group Fmmm, No. 69. On the other hand, for , No. 196, the maximal subgroups , No. 195, or , No. 198, have lost all their centring translations.
For the block `Loss of centring translations', the listing in this volume is the same as that for tsubgroups, cf. Section 2.1.3. The centring translations are listed explicitly where applicable, e.g. for space group , No. 5, unique axis b
In this line, the representatives of the general position are .
The listing differs from that in IT A in only two points:

The sequence of the subgroups in this block is one of decreasing spacegroup number of the subgroups.
Under the heading `Enlarged unit cell', those maximal ksubgroups are listed for which the conventional unit cell is enlarged relative to the unit cell of the original space group . All maximal ksubgroups with enlarged unit cell of index 2, 3 or 4 of the plane groups and of the space groups are listed individually. The listing is restricted to these indices because 4 is the highest index of a maximal nonisomorphic subgroup, and the number of these subgroups is finite. Maximal subgroups of higher indices are always isomorphic to and their number is infinite.
The description of the subgroups with enlarged unit cell is more detailed than in IT A. In the block IIb of IT A, different maximal subgroups of the same spacegroup type with the same lattice relations are represented by the same entry. For example, the eight maximal subgroups of the type , No. 69, of space group , No. 47, are represented by one entry in IT A.
In the present volume, the description of the maximal subgroups in the block `Enlarged unit cell' refers to each subgroup individually and contains for each of them a set of spacegroup generators and the transformation from the setting of the space group to the conventional setting of the subgroup .
Some of the isomorphic subgroups listed in this block may also be found in IT A in the block `Maximal isomorphic subgroups of lowest index IIc'.
Subgroups with the same lattice are collected in small blocks. The heading of each such block consists of the index of the subgroup and the lattice relations of the sublattice relative to the original lattice. Basis vectors that are not mentioned are not changed.
Example 2.1.4.3.1
This example is taken from the table of space group , No. 20.
Enlarged unit cell
The entries mean:
Columns 1 and 2: HM symbol and spacegroup number of the subgroup; cf. Section 2.1.3.2.
Column 3: generators, here the pairsfor the six lines listed in the same order.
Column 4: basis vectors of referred the basis vectors of . means , , ; means , , .
Column 5: origin shifts, referred to the coordinate system of . These origin shifts by o, a and 2a for the first triplet of subgroups and o, b and 2b for the second triplet of subgroups are translations of . The subgroups of each triplet are conjugate, indicated by the left braces.
Often the lattice relations above the data describing the subgroups are the same as the basis vectors in column 4, as in this example. They differ in particular if the sublattice of is nonconventionally centred. Examples are the Hcentred subgroups of trigonal and hexagonal space groups.
The sequence of the subgroups is determined
For sublattices with twice the volume of the unit cell, the sequence of the different cell enlargements is as follows:

With a few exceptions for trigonal, hexagonal and cubic space groups, ksubgroups with enlarged unit cells and index 3 or 4 are isomorphic. To each of the listed sublattices belong either one or several conjugacy classes with three or four conjugate subgroups or one or several normal subgroups. Only the sublattices with the numbers (5)(a)(v), (5)(b)(i), (5)(c)(ii), (6)(iii) and (7)(i) have index 4, all others have index 3. The different cell enlargements are listed in the following sequence:
Maximal subgroups of index higher than 4 have index p, or , where p is prime, are necessarily isomorphic subgroups and are infinite in number. Only a few of them are listed in IT A in the block `Maximal isomorphic subgroups of lowest index IIc'. Because of their infinite number, they cannot be listed individually, but are listed in this volume as members of series under the heading `Series of maximal isomorphic subgroups'. In most of the series, the HM symbol for each isomorphic subgroup will be the same as that of . However, if is an enantiomorphic space group, the HM symbol of will be either that of or that of its enantiomorphic partner.
Example 2.1.5.1.1
Two of the four series of isomorphic subgroups of the space group , No. 76, are (the data for the generators are omitted):
On the other hand, the corresponding data for , No. 78, are
Note that in both tables the subgroups of the type , No. 78, are listed first because of the rules on the sequence of the subgroups.
If an isomorphic maximal subgroup of index is a member of a series, then it is listed twice: as a member of its series and individually under the heading `Enlarged unit cell'.
Most isomorphic subgroups of index 3 are the first members of series but those of index 2 or 4 are rarely so. An example is the space group , No. 77, with isomorphic subgroups of index 2 (not in any series) and 3 (in a series); an exception is found in space group , No. 75, where the isomorphic subgroup for is the first member of the series .
The conventional basis of the unit cell of each isomorphic subgroup in the series has to be defined relative to the basis of the original space group. For this definition the prime p is frequently sufficient as a parameter.
Example 2.1.5.2.1
The isomorphic subgroups of the space group , No. 93, can be described by two series with the bases of their members:
In other cases, one or two positive integers, here called q and r, define the series and often the value of the prime p.
Example 2.1.5.2.2
In space group , No. 174, the series is listed. The values of q and r have to be chosen such that while q > 0, r > 0, is prime.
Example 2.1.5.2.3
In the space group , No. 11, unique axis c, the series is listed. Here p and q are independent and q may take the p values for each value of the prime p.
Each of the sublattices discussed in Section 2.1.4.3.2 is common to a conjugacy class or belongs to a normal subgroup of a given series. The subgroups in a conjugacy class differ by the positions of their conventional origins relative to the origin of the space group . To define the origin of the conventional unit cell of each subgroup in a conjugacy class, one, two or three integers, called u, v or w in these tables, are necessary. For a series of subgroups of index p, or there are p, or conjugate subgroups, respectively. The positions of their origins are defined by the p or or permitted values of u or u, v or u, v, w, respectively.
Example 2.1.5.3.1
The space group , , No. 112, has two series of maximal isomorphic subgroups . For one of them the lattice relations are , listed as . The index is . For each value of p there exist exactly conjugate subgroups with origins in the points , where the parameters u and v run independently: and .
In another type of series there is exactly one (normal) subgroup for each index p; the location of its origin is always chosen at the origin of and is thus not indicated as an origin shift.
Example 2.1.5.3.2
Consider the space group , No. 29. Only one subgroup exists for each value of p in the series . This is indicated in the tables by the statement `no conjugate subgroups'.
The generators of the p (or or ) conjugate isomorphic subgroups are obtained from those of by adding translational components. These components are determined by the parameters p (or q and r, if relevant) and u (and v and w, if relevant).
Example 2.1.5.4.1
Space group , No. 198. In the series defined by the lattice relations and the origin shift there exist exactly conjugate subgroups for each value of p. The generators of each subgroup are defined by the parameter p and the triplet in combination with the generators (2), (3) and (5) of . Consider the subgroup characterized by the basis and by the origin shift . One obtains from the generator (2) of the corresponding generator of by adding the translation vector to the translation vector of the generator (2) of and obtains , so that this generator of is written .
For most space groups, there is only one description of their series of the isomorphic subgroups. However, if a space group is described twice in IT A, then there are also two different descriptions of these series. This happens for monoclinic space groups with the settings unique axis b and unique axis c, for some orthorhombic, tetragonal and cubic space groups with origin choices 1 and 2 and for trigonal space groups with rhombohedral lattices with hexagonal axes and rhombohedral axes.
In the monoclinic space groups, the series in the listings `unique axis b' and `unique axis c' are closely related by a simple cyclic permutation of the axes a, b and c, see IT A, Section 2.2.16 .
In trigonal space groups with rhombohedral lattices, the series with hexagonal axes and with rhombohedral axes appear to be rather different. However, the `rhombohedral' series are the exact transcript of the `hexagonal' series by the same transformation formulae as are used for the different monoclinic settings. However, the transformation matrices P and in Part 5 of IT A are more complicated in this case.
Example 2.1.5.5.1
Space group , No. 148. The second series is described with hexagonal axes by the basis transformation , i.e. and the origin shift . We discuss the basis transformation first. It can be written in analogy to Part 5 , IT A. is the row of basis vectors of the conventional hexagonal basis. The matrix X is defined byWith rhombohedral axes, equation (2.1.5.1) would be written with the matrix Y to be determined.
The transformation from hexagonal to rhombohedral axes is described by where the matricesare listed in IT A, Table 5.1.3.1 , see also Figs. 5.1.3.6 (a) and (c) in IT A.
Applying equations (2.1.5.3), (2.1.5.1) and (2.1.5.2), one gets From equation (2.1.5.4) it follows that One obtains Y from equation (2.1.5.5) by matrix multiplication, and from Y for the bases of the subgroups with rhombohedral axesThe column of the origin shift in hexagonal axes must be transformed by . The result is the column in rhombohedral axes.
Space groups with two origin choices are always described in the same basis, but origin 1 is shifted relative to origin 2 by the shift vector s. For most space groups with two origins, the appearance of the two series related by the origin shift is similar; there are only differences in the generators.
Example 2.1.5.5.2
Consider the space group , No. 48, in both origin choices and the corresponding series defined by and . In origin choice 1, the generator (5) of is described by the `coordinates' . The translation part of the third generator of stems from the term in the first `coordinate' of the generator (5) of . Because must be a translation vector of , p is odd. Such a translation part is not found in the generators (2) and (3) of because the term does not appear in the `coordinates' of the corresponding generators of .
The situation is inverted in the description for origin choice 2.
The translation term appears in the first and second generator of and not in the third one because the term occurs in the first `coordinate' of the generators (2) and (3) of but not in the generator (5).
The term appears in both descriptions. It is introduced in order to adapt the generators to the origin shift .
In other space groups described in two origin choices, surprisingly, the number of series is different for origin choice 1 and origin choice 2.
Example 2.1.5.5.3
In the tetragonal space group , No. 141, for origin choice 1 there is one series of maximal isomorphic subgroups of index , p prime, with the bases and origin shifts . For origin choice 2, there are two series with the same bases but with the different origin shifts and . What are the reasons for these results?
For origin choice 1, the term appears in the first and second `coordinates' of all generators (2), (3), (5) and (9) of . This term is the cause of the translation vectors and in the generators of .
For origin choice 2, fractions and appear in all `coordinates' of the generator (3) of . As a consequence, translational parts with vectors and appear if p = 4n − 1. On the other hand, translational parts with vectors are introduced in the generators of if p = 4n + 1 holds.
Another consequence of the fractions and occurring in the generator (3) of is the difference in the origin shifts. They are for p = 4n − 1 and for p = 4n + 1. Thus, the one series in origin choice 1 for odd p is split into two series in origin choice 2 for p = 4n − 1 and p = 4n + 1.^{4}
In the previous sections, the relation between space groups was seen from the viewpoint of the group . In this case, was a subgroup of . However, the same relation may be viewed from the group . In this case, is a supergroup of . As for the subgroups of , cf. Section 1.2.6 , different kinds of supergroups of may be distinguished.
Definition 2.1.6.1.1. Let be a maximal subgroup of . Then is called a minimal supergroup of . If is a translationengleiche subgroup of then is a translationengleiche supergroup (tsupergroup) of . If is a klassengleiche subgroup of , then is a klassengleiche supergroup (ksupergroup) of . If is an isomorphic subgroup of , then is an isomorphic supergroup of . If is a general subgroup of , then is a general supergroup of .
Following from Hermann's theorem, Lemma 1.2.8.1.2 , a minimal supergroup of a space group is either a translationengleiche supergroup (tsupergroup) or a klassengleiche supergroup (ksupergroup). A proper minimal tsupergroup always has an index i, , and is never isomorphic. A minimal ksupergroup with index i, , may be isomorphic or nonisomorphic; for indices i > 4 a minimal ksupergroup can only be an isomorphic ksupergroup. The propositions, theorems and their corollaries of Sections 1.4.6 and 1.4.7 for maximal subgroups are correspondingly valid for minimal supergroups.
Subgroups of space groups of finite index are always space groups again. This does not hold for supergroups. For example, the direct product of a space group with a group of order 2 is not a space group, although is a subgroup of index 2 of . Moreover, supergroups of space groups may be affine groups which are only isomorphic to space groups but not space groups themselves, see Example 2.1.6.2.2. In the following we restrict the considerations to supergroups of a space group which are themselves space groups. This holds, for example, for the symmetry relations between crystal structures when the symmetries of both structures can be described by space groups. Quasicrystals, incommensurate phases etc. are thus excluded. Even under this restriction, supergroups show much more variable behaviour than subgroups do.
One of the reasons for this complication is that the search for subgroups is restricted to the elements of the space group itself, whereas the search for supergroups has to take into account the whole (continuous) group of all isometries. For example, there are only a finite number of subgroups of any space group for any given index i. On the other hand, there may not only be an infinite number of supergroups of a space group for a finite index i but even an uncountably infinite number of minimal supergroups of .
Example 2.1.6.1.2
Let . Then there is an infinite number of tsupergroups of index 2 because there is no restriction for the sites of the centres of inversion and thus of the conventional origin of .
In the tables of this volume, the minimal translationengleiche supergroups of a space group are not listed individually but the type of is listed by index, conventional HM symbol and spacegroup number if is listed as a translationengleiche subgroup of in the subgroup tables. Not listed is the number of supergroups belonging to one entry. Nonisomorphic klassengleiche supergroups are listed individually. For them, nonconventional HM symbols are listed in addition; for klassengleiche supergroups with `Decreased unit cell', the lattice relations are added. More details, such as the representatives of the general position or the generators as well as the transformation matrix and the origin shift, would only duplicate the subgroup data.
In this Section 2.1.6, the kind of listing is described explicitly. The data for maximal subgroups are complete for all space groups . Therefore, it is possible to derive:
In Section 2.1.7 the procedure is described by which the supergroup data can be obtained from the subgroup data. This procedure is not trivial and care has to be taken to really obtain the full set of supergroups. In Section 2.1.7 one can also find the reasons why this procedure is not applicable when the space group belongs to a triclinic or monoclinic point group.
Isomorphic supergroups are not indicated at all because they are implicitly contained in the subgroup data. Their derivation from the subgroup data is discussed in Section 2.1.7.2.
Like the subgroup data, the supergroup data are also partitioned into blocks.
For each space group , under this heading are listed those spacegroup types for which appears as an entry under the heading I Maximal translationengleiche subgroups. The listing consists of the index in brackets […], the conventional HM symbol and the spacegroup number (in parentheses). The space groups are ordered by ascending spacegroup number. If this line is empty, the heading is printed nevertheless and the content is announced by `none', as in , No. 191. Note that the real setting of the supergroup and thus its HM symbol may be nonconventional.
Example 2.1.6.2.1
Let , No. 18. Among the entries of the block one finds the space groups of the crystal class mmm: , No. 55; , No. 56; , No. 57; , No. 58; , No. 59 and , No. 60, designated by their standard HM symbols. However, the full HM symbols , , , , and reveal that only the four HM symbols Pbam, Pccn, Pnnm and Pmmn of these six entries describe supergroups of . The symbols and represent four supergroups of , namely , , and . This is not obvious but will be derived, as well as the origin shift if necessary, with the procedure described in Examples 2.1.7.3.2 and 2.1.7.4.3.
The supergroups listed in this block represent space groups only if the lattice conditions of fulfil the lattice conditions for . This is not a problem if group and supergroup belong to the same crystal family,^{5} cf. Example 2.1.6.2.1. Otherwise the lattice parameters of have to be checked correspondingly, as in Example 2.1.6.2.2.
Example 2.1.6.2.2
Space group , No. 16. For the minimal supergroups , No. 47, Pnnn, No. 48, Pccm, No. 49, and Pban, No. 50, there is no lattice condition because P222 and all these supergroups belong to the same orthorhombic crystal family and thus are space groups. If, however, a space group of the types , No. 89, , No. 93, , No. 111, or , No. 112, is considered as a supergroup of , two of the three independent lattice parameters a, b, c of P222 must be equal (or in crystallographic practice, approximately equal). These must be a and b if c is the tetragonal axis, b and c if a is the tetragonal axis or c and a if b is the tetragonal axis. In the latter two cases, the setting of P222 has to be transformed to the caxis setting of P422. For the cubic supergroup P23, No. 195, all three lattice parameters of P222 must be (approximately) equal. If they are not, elements of are not isometries and is an affine group which is only isomorphic to a space group.
The lattice conditions are useful in the search for supergroups which are space groups, i.e. form the symmetry of crystal structures. Whereas a subgroup does not become noticeable in the lattice parameters of a space group , a space group of another crystal family must be indicated by the lattice parameters of the space group . Thus it may be an important advantage if the conditions of temperature, pressure or composition allow the start of the search for possible phase transitions of the lowsymmetry phase.
As mentioned already, the number of the minimal tsupergroups cannot be taken or concluded from the subgroup tables. It is different in the different cases of Example 2.1.6.2.2 above. The space group P222 has one minimal supergroup of the type Pmmm and one of Pnnn; there are three minimal supergroups of type Pccm, namely Pccm, Pmaa and Pbmb, as well as three minimal supergroups of type Pban, viz. Pban, Pncb and Pcna. There are six minimal supergroups of the type P422 and four minimal supergroups of the type P23; they are space groups if the lattice conditions are fulfilled. The number of different supergroups will be calculated in Examples 2.1.7.3.1 and 2.1.7.4.2 by the procedure described in Section 2.1.7.4.
If is a ksupergroup of , and always belong to the same crystal family and there are no lattice restrictions for .
As mentioned above, in the tables of this volume only nonisomorphic minimal ksupergroups are listed among the supergroup data; no isomorphic minimal supergroups are given. The block II Minimal nonisomorphic klassengleiche supergroups is divided into two subblocks with the headings Additional centring translations and Decreased unit cell.
If both subblocks are empty, only the heading of the block is listed, stating `none' for the content of the block, as in , No. 191.
If at least one of the subblocks is nonempty, then the heading of the block and the headings of both subblocks are listed. An empty subblock is then designated by `none'; in the other subblock the supergroups are listed. The kind of listing depends on the subblock. Examples may be found in the tables of , No. 16, and , No. 228.
As discussed in Section 2.1.7.1, there is exactly one supergroup for each of the nonisomorphic ksupergroup entries of , although often not in the conventional setting. A transformation of the generalposition representatives or of the generators to the conventional setting may be necessary to obtain the standard HM symbol of in the same way as in Examples 2.1.7.3.1 and 2.1.7.3.2, which refer to translationengleiche supergroups.
Under the heading `Additional centring translations', the supergroups are listed by their indices and either by their nonconventional HM symbols, with the spacegroup numbers and the conventional HM symbols in parentheses, or by their conventional HM symbols and only their spacegroup numbers in parentheses. Examples are provided by space group , No. 61, with both subblocks nonempty and by space group , No. 16, with supergroups only under the heading `Additional centring translations'.
Not only the HM symbols but also the centrings themselves may be nonconventional. In this volume, the nonconventional centrings tetragonal c (c4gm as a supergroup of p4gm) and h (h31m as a supergroup of p31m) are used for HM symbols of plane groups, tetragonal C ( as a supergroup of ), `reverse', different from the conventional `obverse' ( as supergroup of P3), and H (H312 as supergroup of P312) are used for HM symbols of space groups.
Under the heading `Decreased unit cell' each supergroup is listed by its index and by its lattice relations, where the basis vectors , and refer to the supergroup and the basis vectors a, b and c to the original group . After these data are listed either the nonconventional HM symbol, followed by the spacegroup number and the conventional HM symbol in parentheses, or the conventional HM symbol with the spacegroup number in parentheses. Examples are provided again by space group , No. 61, with both subblocks occupied and space group , No. 216, with an empty subblock `Additional centring translations' but data under the heading `Decreased unit cell'.
The minimal supergroups of the space groups, in particular the translationengleiche or tsupergroups, are not fully listed in the tables. However, with the exception of the translationengleiche supergroups of triclinic and monoclinic space groups , the listing is sufficient to derive the supergroups with the aid of the subgroup tables. For this derivation the coset decomposition of a space group relative to its translation subgroup as well as the normalizers and of the space groups and play a decisive role. The coset decomposition of a group relative to a subgroup has been defined in Section 1.2.4.2 ; the coset decomposition of a space group relative to its translation subgroup in Sections 1.2.5.1 and 1.2.5.4 . The notions of the affine and the Euclidean normalizer have been introduced in Section 1.2.6.3 .
In the next Sections 2.1.7.1 and 2.1.7.2 the minimal ksupergroups including the isomorphic supergroups will be derived by inversion of the subgroup data. In Section 2.1.7.3 one minimal tsupergroup of each type will be found from the corresponding subgroup data, also by inversion. Starting from this supergroup other minimal tsupergroups can be obtained. This procedure is described in Sections 2.1.7.4 and 2.1.7.5.
All nonisomorphic klassengleiche maximal subgroups of a space group are listed individually, whereas the infinite number of isomorphic ksubgroups of are listed essentially by series. Therefore, it is more transparent to deal with the isomorphic supergroups separately (in Section 2.1.7.2). The nonisomorphic klassengleiche supergroups are considered here.
The procedure for deriving the ksupergroups of a space group is simpler than that for the derivation of the tsupergroups, described in Sections 2.1.7.3, 2.1.7.4 and 2.1.7.5. The data for ksupergroups of are more detailed and, unlike the tsupergroups, there is only one ksupergroup per entry for the supergroup data of .
To show this, one considers the coset decomposition of the group with respect to the normal subgroup of all its translations, cf. Section 8.1.6 of IT A. The set of cosets with respect to this decomposition forms a group, the factor group . Each coset, i.e. each element of the factor group , consists of all those elements (symmetry operations) of which have the same matrix part in common and differ in their translation parts only.
In a klassengleiche supergroup the coset decomposition of is retained; only the set of translations is increased in relative to , . With the additional translations, each coset of is extended to a coset of . The cosets are independent of the chosen coset representatives. Thus, as the coset representatives of always belong to the elements of , the coset representatives of can be taken as the coset representatives of and the elements of are uniquely determined.
It follows that for each maximal ksubgroup which is listed among the subgroups of , is the only minimal ksupergroup for the corresponding extension of the lattice translations (there may be other lattice extensions in addition which result in other supergroups). If different ksubgroups of and their lattice extensions are conjugate under the Euclidean normalizer of , then is the common minimal ksupergroup of these subgroups . This result is independent of whether the minimal ksupergroup is isomorphic to the space group or not. Therefore, the last paragraph of Section 2.1.7.2 also holds for the nonisomorphic ksubgroup pairs of , and among the subgroups of .
Example 2.1.7.1.1
Consider the minimal supergroups of the space group , No. 18. Four entries for `Additional centring translations' and two for `Decreased unit cell' are listed in the supergroup data of ; the missing entry for results in a ksupergroup isomorphic to and is thus not listed among the supergroup data of . The supergroups with `Additional centring translations' shall be looked at in more detail.
The supergroup C222 is obtained directly by adding the Ccentring to the symmetry operations of .
Adding the A and Bcentrings to results in supergroups of the type . For , in the a and b directions and axes alternate, whereas in the c direction there are only axes. Adding the Acentring to results in alternating and axes in the directions b and c but there are only axes in the direction of a; is obtained. Adding the Bcentring to results in alternating and axes in the directions a and c but there are only axes in the b direction; is obtained.
These relations can also be derived in another way. Transformation of the relation with its matrix–column pair as listed in the subgroup table of in Chapter 2.3 results in the relation . On the other hand, the relation is transformed by its matrix–column pair to .
It is often easy to construct the supergroup from the drawing of the original space group in IT A by adding the centring vectors or the additional basis vectors. This happens, for example, for the supergroup I222, No. 23, where the origin shift by is obvious from the comparison of the drawings of and I222. This agrees with the data in the subgroup table of I222.
The completeness of the data for the minimal ksupergroups depends on the completeness of the listed lattice extensions, i.e. on the completeness of the listed possible centrings as well as of the possible decreased unit cells of the lattices. These data are well known for the small indices 2, 3 and 4 occurring in these group–subgroup relations.
It is not necessary to list the isomorphic minimal supergroups, i.e. those minimal supergroups which belong to the spacegroup type of . Therefore, a block `series of isomorphic minimal supergroups' does not occur among the supergroup data.
The derivation of the isomorphic minimal supergroups from the data in the subgroup tables is straightforward. For each index, one looks for the listed isomorphic normal subgroups and for the classes of conjugate isomorphic subgroups in the subgroup table of . If some of these items are conjugate under the Euclidean normalizer of , then only one item of this conjugacy class has to be taken into consideration as a representative. For each of these representatives there is one corresponding supergroup of .
As for the isomorphic maximal subgroups, the indices of the minimal supergroups are p, p^{2} or p^{3}, p prime. However, the large conjugacy classes of isomorphic maximal subgroups always belong to single isomorphic minimal supergroups.
Example 2.1.7.2.1
Consider the p^{2} conjugate isomorphic subgroups of a space group , No. 112, in the series , prime p fixed. The same supergroup belongs to each of these subgroups . The indices u and v, designating the members of a conjugacy class of subgroups , may have any of their admissible values or may be set to zero. Choosing other values of u and/or v means dealing with the same supergroup but transformed by an element of . Because the parameters u and/or v appear only in the translation parts of the (4 × 4) symmetry matrices, this may mean a shift of the origin in the description of . If for practical reasons the origin of will be chosen as the origin of , i.e. at different points of for the different groups , then the (same) group is described relative to different origins. These origins are then chosen in different translationally equivalent points of .
Space groups which are conjugate under the Euclidean normalizer of the supergroup have the supergroup in common if they are complemented by the corresponding conjugate sets of translations. For example, both members of each of the subgroup pairs P222 of index [2] in the subgroup table of the space group P222, No. 16, for or or have their minimal supergroup in common because they are conjugate under the Euclidean normalizer .^{6} If for some temperature, pressure and composition of a substance a = b = c holds for the lattice parameters of P222, , i.e. the Euclidean normalizer is equal to the affine normalizer, making the three supergroups conjugate in the normalizer. Such relations have little importance in practice if the space group describes the symmetry of a substance. This substance has the crystal symmetry P222 independent of the accidentally higher lattice symmetry.
A proper translationengleiche supergroup of a space group cannot be isomorphic to because it belongs to another crystal class. Therefore, it is not necessary to include the word `nonisomorphic' in the header.
The minimal tsupergroups of have indices , ; only the conventional HM symbol of the supergroup together with the index and the spacegroup number are listed in the supergroup data of the group . However, in the subgroup data of the subgroups are explicitly listed with their indices, their (nonconventional and) conventional HM symbols, their spacegroup numbers, their general positions and their transformation matrices P and columns p. Suppose the supergroups are listed on the line `I Minimal translationengleiche supergroups' of the space group . In order to determine all supergroups , one takes one of the listed supergroups , say . In the subgroup table of one finds a subgroup , isomorphic to (at least one must exist, otherwise would not be listed among the minimal supergroups of ). The transformation (P_{1}, p_{1}), listed with the subgroup , transforms the symmetry operations of from the coordinate system of to the standard coordinate system of . Such transformations are described by equations (2.1.3.8) and (2.1.3.9) in Section 2.1.3.3. The matrix–column pair (P_{1}, p_{1}) also transforms the group from its standard description to that referred to the coordinate system of . This transformed group is one supergroup from which one can start to derive other supergroups of this type, if there are any, cf. Sections 2.1.7.4 and 2.1.7.5. The same procedure has to be applied to any other maximal tsubgroup and to the other listed tsupergroups .
The calculations can be verified by viewing the spacegroup diagrams of the corresponding space groups in IT, Volume A.
Example 2.1.7.3.1
Space group , No.16. In continuation of Example 2.1.6.2.2, one finds no data for the matrix P and the column p listed in the entry for the subgroup P222 in the subgroup tables of Pmmm, No. 47, Pnnn, No. 48, origin choice 1, Pban, No. 50, origin choice 1, P422, No. 89, , No. 93, , No. 111, and P23, No. 195. This means P = I and p = o. Thus, P222 is a subgroup of these space groups and the standard settings agree, i.e. the generators of can be taken directly from those of , adding the last generator of . This is confirmed by the spacegroup diagrams. Regarding the tetragonal and cubic supergroups, the lattice restrictions for the space group have to be obeyed, cf. Example 2.1.6.2.2.
Only an origin shift but no transformation matrix P is listed in the subgroup tables of the supergroups Pnnn origin choice 2, ; Pccm, No. 49, ; Pban origin choice 2, and , No. 112, . Thus, the r matrix parts of the r nontranslational generators of , are retained, , and equation (2.1.3.9) is reduced towhere are those (two) generators of which stem from . The third generator of , the inversion or rotoinversion, has to be transformed correspondingly.
The column parts of the r = 2 generators and of the four space groups Pnnn origin choice 2, Pccm, Pban origin choice 2 and are then (normalized to values between ) the same as those of the group , i.e. they are generators of supergroups of . The columns of the inversions arerespectively. This describes the position of or relative to the origin of the P222 framework in the corresponding spacegroup diagrams of IT A. The coordinates of the inversion centre are half the coefficients of the columns. The supergroup Pnnn, origin choice 2, is the same as Pnnn, origin choice 1; only the setting is different. The spacegroup diagrams in IT A display the agreement of the frameworks of the rotation axes in and . The supergroup is a space group only if the corresponding lattice relations for are fulfilled.
Example 2.1.7.3.2
Continued from Example 2.1.6.2.1. For , No. 18, and its orthorhombic types of supergroups, the places for the matrix P and the column p under in the subgroup data of Pbam, No. 55, and Pmmn, No. 59, origin choice 1 are empty. In analogy to the preceding example, is a subgroup of these space groups in the standard setting.
The equation also holds for the subgroup data of Pccn, No. 56, Pnnm, No. 58, and Pmmn, No. 59, origin choice 2, but , and , respectively. Thus, the reduced equation (2.1.7.1) holds for the generators of and the inversion.
Again the column parts of the generators and of the groups Pccn, Pnnm and Pmmn, origin choice 2, are the same as those of the group , if normalized to values between , i.e. they are generators of supergroups of . The columns of the inversions are
respectively. As in the previous example, this describes the position of relative to the origin of the framework in the corresponding spacegroup diagrams of IT A. The coordinates of the inversion centre are half the coefficients of the columns. The supergroup Pmmn, origin choice 2, is the same as Pmmn, origin choice 1; only the setting is different. Again the spacegroup diagrams in IT A display the agreement of the frameworks of rotation and screw rotation axes in and .
Concerning Pbcm, No. 57, in its subgroup table one finds the line of with the representatives 1, 2, 3, 4 of the general position andThe representative 1 is described by ; it is invariant under the transformation. Using equations (2.1.3.8) and (2.1.3.9) and the above values for P and p for the representatives 2, 3 and 4 of the subgroup , these will be transformed to the matrix–column pairs of , and in the standard form of . The inversion is transformed to an inversion with the column , i.e. the centre of inversion has the coordinates . Combining the (screw) rotations with the inversion, one obtains the reflection and the glide reflections, referred to the coordinate system of . With the application of the formulae of IT A, Chapter 11.2, one finds at , at and at . This results in the HM symbol for this supergroup; see also the diagram in IT A.
For Pbcn,A procedure analogous to that applied for Pbcm yields for Pbcn, No. 60, the standard setting of and the inversion centre at (1/4, 0, 1/4). Again by combination of the (screw) rotations with the inversion one gets at , at and at . The nonconventional HM symbol of this supergroup of is thus Pnca or .
Up to now one minimal tsupergroup per entry using the tables of maximal subgroups has been found, see Examples 2.1.7.3.1 of P222 and 2.1.7.3.2 of .
The question arises as to whether this list is complete or whether further tsupergroups exist which belong to the spacegroup type of and are represented by the same entry of the supergroup data. If these supergroups belong to the same spacegroup type then they are isomorphic and are thus conjugate under the group of all affine mappings according to the theorem of Bieberbach. Then there must be an affine mapping such that . To find these mappings, one makes use of the affine normalizers and and considers their intersection (Koch, 1984). One of the two diagrams of Fig. 2.1.7.1 will describe the situation because is a minimal supergroup of . Let be the normalizer of the group in the group , i.e. the set of all elements of which leave invariant. Whereas in general holds, for a minimal supergroup either (right diagram) or (left diagram) holds.
Lemma 2.1.7.4.1. Let be a minimal tsupergroup of a space group , let and be their affine normalizers, and is the intersection of these normalizers. Then minimal supergroups exist, isomorphic to , where is the index of in . If is finite, then the representatives , of the cosets in the decomposition of transform to .
The lemma is proven by coset decomposition of relative to .
If is a subgroup, for the translation groups and of the normalizers always holds (Wondratschek & Aroyo, 2001). Therefore, for tsupergroups there may be translations of which transform the space group into another one, . Transformation of and by an element of will map as well as onto itself. Transformation of and by an element but will map onto itself but will map the supergroup onto another supergroup .
For applications it is transparent to split the index into the index of the translation lattices and of the pointgroup parts, .
For the application of Lemma 2.1.7.4.1, the kind of the normalizers and and in particular the index are decisive. The affine normalizers of the plane groups and space groups are listed in the tables of Chapter 15.2 of IT A. Their pointgroup parts are:
The translation parts of the normalizers are continuous or partly continuous groups for polar space groups and lattices for nonpolar space groups.
The following cases may be distinguished in the use of Lemma 2.1.7.4.1:

The application of Lemma 2.1.7.4.1 will be described by three examples. Example 2.1.7.4.2 is the continuation of Example 2.1.7.3.1, Example 2.1.7.4.3 is the continuation of Example 2.1.7.3.2 and Example 2.1.7.4.4 deals with the minimal tsupergroups of space group , No. 25.
Example 2.1.7.4.2
Application of the normalizers to the minimal tsupergroups of , No. 16; continuation of Example 2.1.7.3.1.
The affine normalizer , cf. IT A, Table 15.2.1.3 . [In the header of this table, only the words Euclidean normalizer are found. It is mentioned in Section 15.2.2 , Affine normalizers of plane and space groups, that the type of affine normalizers corresponds to the type of the highestsymmetry Euclidean normalizers belonging to that space (plane)group type.] The affine normalizers of the supergroups Pmmm, No. 47, and Pnnn, No. 48, are the same as that of P222. The index , there is only one supergroup with the same origin (origin choice 1 for Pnnn) of each of these types.
The affine normalizers of , No. 49, and , No. 50, are such that the index . Coset decomposition of relative to reveals the coset representatives and so that from Pccm (origin shift ) the space groups (in unconventional HM symbols) Pmaa and Pbmb are generated with the origin shifts and ; from Pban (no origin shift for origin choice 1) one obtains Pncb and Pcna with no origin shifts. To all these orthorhombic supergroups there are no translationally equivalent supergroups.
The tetragonal supergroups P422, No. 89, , No. 93, , No. 111, and , No. 112, are space groups if one of the conditions a = b or b = c or c = a holds for the lattice parameters, with the tetragonal axes perpendicular to the tetragonal plane. Otherwise the tetragonal supergroups are affine groups which are only isomorphic to space groups. They all have affine normalizer , such that the index . The supergroups P422 form three pairs; one pair with its tetragonal axes parallel to c and with the origins either coinciding with that of P222 or shifted by (or equivalently by ). The other two pairs point with their tetragonal axes parallel to a and to b, with one origin at the origin of P222 and the other origin shifted by (or equivalently c) and (or equivalently c). The same holds for the six supergroups of type and for the six supergroups of type . The six supergroups of type again form three pairs with their tetragonal axes along c or a or b but their origins are shifted against that of P222 by along the tetragonal axes because of the entry in the subgroup table of , see also Example 2.1.7.3.1.
For the supergroups with the symbol P23, No. 195, which is a space group if a = b = c, the affine normalizer is with pointgroup index but translation index . Thus, there are four such (translationally equivalent) supergroups with their origins at ; ; ; and .
Example 2.1.7.4.3
Application of the normalizers to the supergroups of , No. 18.
The normalizer is the same as those for the minimal tsupergroups Pbam, No. 55, Pccn, No. 56, Pnnm, No. 58, and Pmmn, No. 59. There is one supergroup for each of these types with the origin shifts ; ; ; and (origin choice 1), respectively, according to the p values listed in the subgroup tables of the supergroups. This can also be concluded from the spacegroup diagrams of IT A.
The listed HM symbols Pbcm, No. 57, and Pbcn, No. 60, do not refer to supergroups of , but Pbma and Pnca do. This can be seen either from the full HM symbols, cf. Example 2.1.6.2.1, or by applying the (P, p) data of the supergroups for the subgroup , from which the origin shifts may also be taken, cf. Example 2.1.7.3.2. Both supergroups have normalizer with index . There are two supergroups of each type, the second one, Pmab and Pcnb, generated from that already listed by the fourfold rotation in the normalizer of .
The tetragonal supergroups (they are space groups if for their lattice parameters the equation a = b holds) , No. 90, , No. 94, , No. 113, and , No. 114, all have normalizer , such that the index . There are two translationally equivalent supergroups in each case, one with origin at ; ; ; and , respectively, the other shifted against the first one by the translation or (equivalently) by .
The procedure just described works fine if the symmetry of the group is higher than triclinic or monoclinic. The following example shows that an infinite lattice index is acceptable.
Example 2.1.7.4.4
Application of the normalizers to the tsupergroups of , No. 25.
The affine normalizer is .
The list of tsupergroups of Pmm2 starts with Pmmm, No. 47. The affine normalizer ; its point group is a supergroup of the point group 4/mmm of such that the intersection of the point groups is 4/mmm and . The index , leading to a continuous sequence of supergroups with origins at ; .
The tsupergroup Pmma, No. 51, follows. such that with and . There is a continuous set of supergroups Pmma. Because of the origin shift for Pmm2 in the subgroup table of Pmma, the origins of these supergroups are placed at with . Because , there is a second set of supergroups, rotated by 90° relative to the first set, such that its (unconventional) HM symbol is Pmmb and its origins are placed at , .
The normalizer of the last orthorhombic tsupergroup Pmmn, No. 59, has the same pointgroup part as that of Pmm2 and its translation part differs only in instead of . There are no transformation data for Pmm2 in the subgroup table of Pmmn, origin choice 1. There is one set of supergroups Pmmn with the origins at for .
The tetragonal minimal supergroups P4mm, No. 99, , No. 105, and , No. 115, are space groups if a = b holds for the lattice parameters of Pmm2. The affine and Euclidean normalizers of tetragonal space groups are the same.
The normalizers of P4mm and differ from only in the translation part such that and and the additional translations of relative to may be represented by . There are two supergroups each: P4mm and , with their origins at and of Pmm2.
Finally, the normalizer and there are two continuous sets of supergroups with their origins at and ; .
By the procedure discussed in Section 2.1.7.4, from a supergroup other supergroups could be found which are isomorphic to with the same index. The question arises as to whether there exist further minimal supergroups isomorphic to and of the same index which can not be obtained by consideration of the normalizers.
Suppose is such a supergroup. According to the theorem of Bieberbach for space groups, isomorphism and affine equivalence result in the same classification of the space groups, cf. Section 8.2.2 of IT A. Therefore, there exists an affine mapping in the group of all affine mappings which transforms onto the space group and does not belong to , . Let be obtained from by the inverse of the transformation which transforms to . Then the group is a subgroup of if and only if is a subgroup of . Therefore, if the space group has a subgroup in addition to , then the space group has an additional supergroup which can be found using the transformation of to . This transformation is effective only if it does not belong to the normalizer , otherwise it would transform the space group onto itself. Therefore, only those subgroups have to be taken into consideration which are not conjugate to under the normalizer .
An example of the application of this procedure is given in Section 1.7.1 and Example 1.7.3.2.2 . It refers to minimal ksupergroups; an example for minimal tsubgroups is not known to the authors.
In Sections 2.1.7.4 and 2.1.7.5 only minimal tsupergroups are dealt with. The same considerations can be applied to minimal ksupergroups with corresponding results. It was not necessary to mention the minimal ksupergroups here, as because of the more detailed data in the tables of this volume, the simpler procedure of Sections 2.1.7.1 and 2.1.7.2 could be used to determine the minimal ksupergroups.
The group–subgroup relations between the space groups may also be described by graphs. This way is chosen in Chapters 2.4 and 2.5 . Graphs for the group–subgroup relations between crystallographic point groups have been published, for example, in Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935) and in IT A (2005), Figs. 10.1.3.2 and 10.1.4.3 . Three kinds of graphs for subgroups of space groups have been constructed and can be found in the literature:

A complete collection of graphs of the first two kinds is presented in this volume: in Chapter 2.4 those displaying the translationengleiche or tsubgroup relations and in Chapter 2.5 those for the klassengleiche or ksubgroup relations. Neither type of graph is restricted to maximal subgroups but both contain t or ksubgroups of higher indices, with the exception of isomorphic subgroups, cf. Section 2.1.8.3 below.
The group–subgroup relations are direct relations between the space groups themselves, not between their types. However, each such relation is valid for a pair of space groups, one from each of the types, and for each space group of a given type there exists a corresponding relation. In this sense, one can speak of a relation between the spacegroup types, keeping in mind the difference between space groups and spacegroup types, cf. Section 1.2.5.3 .
The space groups in the graphs are denoted by the standard HM symbols and the spacegroup numbers. In each graph, each spacegroup type is displayed at most once. Such graphs are called contracted graphs here. Without this contraction, the more complex graphs would be much too large for the page size of this volume.
The symbol of a space group is connected by uninterrupted straight lines with the symbols of all its maximal nonisomorphic subgroups or minimal nonisomorphic supergroups. In general, the maximal subgroups of are drawn on a lower level than ; in the same way, the minimal supergroups of are mostly drawn on a higher level than . For exceptions see Section 2.1.8.3. Multiple lines may occur in the graphs for tsubgroups. They are explained in Section 2.1.8.2. No indices are attached to the lines. They can be taken from the corresponding subgroup tables of Chapter 2.3 , and are also provided by the general formulae of Section 1.2.8 . For the ksubgroup graphs, they are further specified at the end of Section 2.1.8.3.
Let be a space group and () the normal subgroup of all its translations. Owing to the isomorphism between the factor group and the point group , see Section 1.2.5.4 , according to the first isomorphism theorem, Ledermann & Weir (1996), tsubgroup graphs are the same (up to the symbols) as the corresponding graphs between point groups. However, in this volume, the graphs are not complete but are contracted, displaying each spacegroup type at most once. This contraction may cause the graphs to look different from the pointgroup graphs and also different for different space groups of the same point group, cf. Example 2.1.8.2.1.
One can indicate the connections between a space group and its maximal subgroups in different ways. In the contracted tsubgroup graphs one line is drawn for each conjugacy class of maximal subgroups of . Thus, a line represents the connection to an individual subgroup only if this is a normal maximal subgroup of , otherwise it represents the connection to more than one subgroup. The conjugacy relations are not necessarily transferable to nonmaximal subgroups, cf. Example 2.1.8.2.2. On the other hand, multiple lines are possible, see the examples. Although it is not in general possible to reconstruct the complete graph from the contracted one, the content of information of such a graph is higher than that of a graph which is drawn with simple lines only.
The graph for the space group at its top also contains the contracted graphs for all subgroups which occur in it, see Example 2.1.8.2.3.
Owing to lack of space for the large graphs, in all graphs of tsubgroups the group , No. 1, and its connections have been omitted. Therefore, to obtain the full graph one has to supplement the graphs by at the bottom and to connect by one line to each of the symbols that have no connection downwards.
Within the same graph, symbols on the same level indicate subgroups of the same index relative to the group at the top. The distance between the levels indicates the size of the index. For a more detailed discussion, see Example 2.1.8.2.2. For the sequence and the numbers of the graphs, see the paragraph after Example 2.1.8.2.2.
Example 2.1.8.2.1
Compare the tsubgroup graphs in Figs. 2.4.4.2 , 2.4.4.3 and 2.4.4.8 of , No. 52, , No. 53, and , No. 64, respectively. The complete (uncontracted) graphs would have the shape of the graph of the point group with at the top (first level), seven point groups^{7} (, , , , , and ) in the second level, seven point groups (, , , , , and ) in the third level and the point group at the bottom (fourth level). The group is connected to each of the seven subgroups at the second level by one line. Each of the groups of the second level is connected with three groups of the third level by one line. All seven groups of the third level are connected by one line each with the point group 1 at the bottom.
The contracted graph of the point group would have at the top, three pointgroup types (, and ) at the second level and three pointgroup types (, m and ) at the third level. The point group 1 at the bottom would not be displayed (no fourth level). Single lines would connect with 222, with 2, with 2, with m and with ; a double line would connect with m; triple lines would connect with , with and 222 with 2.
The number of fields in a contracted tsubgroup graph is between the numbers of fields in the complete and in the contracted pointgroup graphs. The graph in Fig. 2.4.4.2 of , No. 52, has six spacegroup types at the second level and four spacegroup types at the third level. For the graph in Fig. 2.4.4.3 of , No. 53, these numbers are seven and five and for the graph in Fig. 2.4.4.8 of , No. 64 (formerly ), the numbers are seven and six. However, in all these graphs the number of connections is always seven from top to the second level and three from each field of the second level downwards to the ground level, independent of the amount of contraction and of the local multiplicity of lines.
Example 2.1.8.2.2
Compare the tsubgroup graphs shown in Fig. 2.4.1.1 for , No. 221, and Fig. 2.4.1.5 , , No. 225. These graphs are contracted from the pointgroup graph . There are altogether nine levels (without the lowest level of ). The indices relative to the top space groups and are 1, 2, 3, 4, 6, 8, 12, 16 and 24, corresponding to the pointgroup orders 48, 24, 16, 12, 8, 6, 4, 3 and 2, respectively. The height of the levels in the graphs reflects the index; the distances between the levels are proportional to the logarithms of the indices but are slightly distorted here in order to adapt to the density of the lines.
From the top spacegroup symbol there are five lines to the symbols of maximal subgroups: The three symbols at the level of index 2 are those of cubic normal subgroups, the one (tetragonal) symbol at the level of index 3 represents a conjugacy class of three, the symbol , No. 166, at the level of index 4 represents a conjugacy class of four subgroups.
The graphs differ in the levels of the indices 12 and 24 (orthorhombic, monoclinic and triclinic subgroups) by the number of symbols (nine and seven for index 12, five and three for index 24). The number of lines between neighbouring connected levels depends only on the number and kind of symbols in the upper level.
However, for nonmaximal subgroups the conjugacy relations may not hold. For example, in Fig. 2.4.1.1 , the space group has three normal maximal subgroups of type and is thus connected to its symbol by a triple line, although these subgroups are conjugate subgroups of the nonminimal supergroup .
Example 2.1.8.2.3
The tsubgroup graphs in Figs. 2.4.1.1 and 2.4.1.5 contain the tsubgroups of the summits and and their relations. In addition, the tsubgroup graph of includes the tsubgroup graphs of the cubic summits , , and , those of the tetragonal summits , , , , , , and , those of the rhombohedral summits , , , and etc., as well as the tsubgroup graphs of several orthorhombic and monoclinic summits. The graph of the summit includes analogously the graphs of the cubic summits , , and , of the tetragonal summits , etc., also the graphs of rhombohedral summits etc. Thus, many other graphs are included in the two basic graphs and can be extracted from them. The same holds for the other graphs displayed in Figs. 2.4.1.2 to 2.4.4.8 : each of them includes the contracted graphs of all its subgroups as summits. For this reason one does not need 229 or 218 different graphs to cover all tsubgroup graphs of the 229 spacegroup types but only 37 ( can be excluded as trivial).
The preceding Example 2.1.8.2.3 suggests that one should choose the graphs in such a way that their number can be kept small. It is natural to display the `big' graphs first and later those smaller graphs that are still missing. This procedure is behind the sequence of the tsubgroup graphs in this volume.

For the index of a maximal tsubgroup, Lemma 1.2.8.2.3 is repeated: the index of a maximal nonisomorphic subgroup is always 2 for oblique, rectangular and square plane groups and for triclinic, monoclinic, orthorhombic and tetragonal space groups . The index is 2 or 3 for hexagonal plane groups and for trigonal and hexagonal space groups . The index is 2, 3 or 4 for cubic space groups .
There are 29 graphs for klassengleiche or ksubgroups, one for each crystal class with the exception of the crystal classes 1, and with only one spacegroup type each: , No. 1, , No. 2, and , No. 174, respectively. The sequence of the graphs is determined by the sequence of the point groups in IT A, Table 2.1.2.1 , fourth column. The graphs of , and are nearly trivial, because to these crystal classes only two spacegroup types belong. The graphs of with 22, of with 28 and of with 20 spacegroup types are the most complicated ones.
Isomorphic subgroups are special cases of ksubgroups. With the exception of both partners of the enantiomorphic spacegroup types, isomorphic subgroups are not displayed in the graphs. The explicit display of the isomorphic subgroups would add an infinite number of lines from each field for a space group back to this field, or at least one line (e.g. a circle) implicitly representing the infinite number of isomorphic subgroups, see the tables of maximal subgroups of Chapter 2.3 .^{8} Such a line would have to be attached to every spacegroup symbol. Thus, there would be no more information.
Nevertheless, connections between isomorphic space groups are included indirectly if the group–subgroup chain encloses a space group of another type. In this case, a space group may be a subgroup of a space group and a subgroup of , where and belong to the same spacegroup type. The subgroup chain is then – – . The two space groups and are not identical but isomorphic. Whereas in general the label for the subgroup is positioned at a lower level than that for the original space group, for such relations the symbols for and can only be drawn on the same level, connected by horizontal lines. If this happens at the top of a graph, the top level is occupied by more than one symbol (the number of symbols in the top level is the same as the number of symmorphic spacegroup types of the crystal class).
Horizontal lines are drawn as left or right arrows depending on the kind of relation. The arrow is always directed from the supergroup to the subgroup. If the relation is twosided, as is always the case for enantiomorphic spacegroup types, then the relation is displayed by a pair of horizontal lines, one of them formed by a right and the other by a left arrow. In the graph in Fig. 2.5.1.5 for crystal class , the connections of with and with are displayed by doubleheaded arrows instead. Furthermore, some arrows in Fig. 2.5.1.5 , crystal class , and Fig. 2.5.1.6 , , are dashed or dotted in order to better distinguish the different lines and to increase clarity.
The different kinds of relations are demonstrated in the following examples.
Example 2.1.8.3.1
In the graph in Fig. 2.5.1.1 , crystal class 2, a space group may be a subgroup of index 2 of a space group by `Loss of centring translations'. On the other hand, subgroups of in the block `Enlarged unit cell', belong to the type , see the tables of maximal subgroups in Chapter 2.3 . Therefore, both symbols are drawn at the same level and are connected by a pair of arrows pointing in opposite directions. Thus, the top level is occupied twice. In the graph in Fig. 2.5.1.2 of crystal class m, both the top level and the bottom level are each occupied by the symbols of two spacegroup types.
Example 2.1.8.3.2
There are four symbols at the top level of the graph in Fig. 2.5.1.4 , crystal class 222. Their relations are rather complicated. Whereas one can go (by index 2) from directly to a subgroup of type and vice versa, the connection from directly to is oneway. One always has to pass on the way from to a subgroup of the types or . Thus, the only maximal subgroup of among these groups is . One can go directly from to but not to etc.
Because of the horizontal connecting arrows, it is clear that there cannot be much correspondence between the level in the graphs and the subgroup index. However, in no graph is a subgroup positioned at a higher level than the supergroup.
Example 2.1.8.3.3
Consider the graph in Fig. 2.5.1.6 for crystal class . To the space group , No. 65, belong maximal nonisomorphic subgroups of the 11 spacegroup types (from left to right) , No. 72, , No. 63, , No. 74, , No. 59, , No. 55, , No. 50, , No. 51, , No. 53, , No. 66, , No. 47, and , No. 71. Although all of them have index 2, their symbols are positioned at very different levels of the graph.
The table for the subgroups of in Chapter 2.3 lists 22 nonisomorphic ksubgroups of index 2, because some of the spacegroup types mentioned above are represented by two or four different subgroups. This multiplicity cannot be displayed by multiple lines because the density of the lines in some of the ksubgroup graphs does not permit this kind of presentation, e.g. for . The multiplicity may be taken from the subgroup tables in Chapter 2.3 , where each nonisomorphic subgroup is listed individually.
Consider the connections from , No. 65, to , No. 55. There are among others: the direct connection of index 2, the connection of index 4 over , No. 72, the connection of index 8 over , No. 74, and , No. 51. Thus, starting from the same space group of type one arrives at different space groups of the type with different unit cells but all belonging to the same spacegroup type and thus represented by the same field of the graph.
The index of a ksubgroup is restricted by Lemma 1.2.8.2.3 and by additional conditions. For the following statements one has to note that enantiomorphic space groups are isomorphic.

There are no graphs for plane groups in this volume. The four graphs for tsubgroups of plane groups are apart from the symbols the same as those for the corresponding space groups: –, –, – and –, where the graphs for the space groups are included in the tsubgroup graphs in Figs. 2.4.1.1 , 2.4.3.1 , 2.4.2.1 and 2.4.2.3 , respectively.
The ksubgroup graphs are trivial for the plane groups , , , , and because there is only one plane group in its crystal class. The graphs for the crystal classes and consist of two plane groups each: and , and . Nevertheless, the graphs are different: the relation is onesided for the tetragonal planegroup pair as it is in the spacegroup pair – and it is twosided for the hexagonal planegroup pair as it is in the spacegroup pair (81)– (82). The graph for the three plane groups of the crystal class m corresponds to the spacegroup graph for the crystal class 2.
Finally, the graph for the four plane groups of crystal class has no direct analogue among the ksubgroup graphs of the space groups. It can be obtained, however, from the graph in Fig. 2.5.1.3 of crystal class by removing the fields of , No. 15, and , No. 11, with all their connections to the remaining fields. The replacements are then: , No. 12, by , No. 9, , No. 10, by , No. 6, , No. 13, by , No. 7, and , No. 14, by , No. 8.
If a subgroup is not maximal then there must be a group–subgroup chain – of maximal subgroups with more than two members which connects with . There are three possibilities: may be a tsubgroup or a ksubgroup or a general subgroup of . In the first two cases, the application of the graphs is straightforward because at least one of the graphs will permit one to find the possible chains directly. If is a ksubgroup of , isomorphic subgroups have to be included if necessary. If is a general subgroup of one has to combine t and ksubgroup graphs.
There is, however, a severe shortcoming to using contracted graphs for the analysis of group–subgroup relations, and great care has to be taken in such investigations. All subgroups with the same spacegroup type are represented by the same field of the graph, but these different nonmaximal subgroups may permit different routes to a common original (super)group.
Example 2.1.8.5.1
An example for translationengleiche subgroups is provided by the group–subgroup chain – of index 12. The contracted graph may be drawn by the program Subgroupgraph from the Bilbao Crystallographic Server, http://www.cryst.ehu.es/ . It is shown in Fig. 2.1.8.1; each field represents all occurring subgroups of a spacegroup type: , No. 139, represents three subgroups, , No. 166, represents four subgroups, and , No. 12, represents nine subgroups belonging to two conjugacy classes. Fig. 2.1.8.1 is part of the contracted total graph of the translationengleiche subgroups of the space group , which is displayed in Fig. 2.4.1.5 . With Subgroupgraph one can also obtain the complete graph between and the set of all nine subgroups of the type . It is too large to be reproduced here.

Contracted graph of the group–subgroup chains from (225) to one of those subgroups with index 12 which belong to the spacegroup type (12). The graph forms part of the total contracted graph of tsubgroups of (Fig. 2.4.1.5 ). 
More instructive are the complete graphs for different single subgroups of the type of . They can also be obtained with the program Subgroupgraph with the exception of the direction indices. In Fig. 2.1.8.2 such a `complete' graph is displayed for one of the six subgroups of type of index 12 whose monoclinic axes point in the directions of . Similarly, in Fig. 2.1.8.3 the complete graph is drawn for one of the three subgroups of of index 12 whose monoclinic axes point in the directions of . It differs markedly from the contracted graph and from the first complete graph. It is easily seen that it may be very misleading to use the contracted graph or the wrong individual complete graph instead of the right individual complete graph.

Complete graph of the group–subgroup chains from (225) to one representative belonging to those six (12) subgroups with index 12 whose monoclinic axes are along the directions of . The direction symbols given as subscripts refer to the basis of . 
In a contracted graph, no basis transformations and origin shifts can be included because they are often ambiguous. In the complete graphs the basis transformations and origin shifts should be listed if these graphs display structural information and not just group–subgroup relations. The group–subgroup relations do not depend on the coordinate systems relative to which the groups are described. On the other hand, the coordinate system is decisive for the coordinates of the atoms of the crystal structures displayed and connected in a Bärnighausen tree. Therefore, for the description of structural relations in a Bärnighausen tree knowledge of the transformations (matrix and column) is essential and great care has to be taken to list them correctly, see Chapter 1.6 and Example 2.1.8.5.4. If one wants to list the transformations in subgroup graphs, one can use the transformations which are presented in the subgroup tables.
The use of the graphs of Chapters 2.4 and 2.5 is advantageous if general subgroups, in particular those of higher indices, are sought. As stated by Hermann's theorem, Lemma 1.2.8.1.2 , a Hermann group always exists and it is uniquely determined for any specific group–subgroup pair . If the subgroup relation is general, the group divides the chain into two subchains, the chain between the translationengleiche space groups and that between the klassengleiche space groups . Thus, however long and complicated the real chain may be, there is always a chain for which only two graphs are needed: a tsubgroup graph for the relation between and and a ksubgroup graph for the relation between and .
For a given pair of spacegroup types and a given index [i], however, there could exist several Hermann groups of different spacegroup types. The graphs of this volume are very helpful in their determination. The index [i] is the product of the index , which is the ratio of the crystal class orders of and , and the index of the lattice reduction from to , . The graphs of tsubgroups (Chapter 2.4 ) are used to find the types of subgroups of with index , belonging to the crystal class of . From the kgraphs (Chapter 2.5 ) it can be seen whether can be a supergroup of with index and thus a possible Hermann group . The following two examples illustrate this twostep procedure for the determination of the spacegroup types of Hermann groups.
Example 2.1.8.5.2
Consider a pair of the spacegroup types with index [i] = 24. The factorization of the index [i] into and follows from the crystal classes of and . From the graph of tsubgroups of (Fig. 2.4.1.1 ) one finds that there are two spacegroup types, namely P2/m and C2/m, of the relevant crystal class (2/m) and index . Checking the graph (Fig. 2.5.1.3 ) of the ksubgroups of the space groups of the crystal class 2/m confirms that space groups of both spacegroup types P2/m and C2/m have (maximal) subgroups, i.e. space groups of both spacegroup types are Hermann groups for the pair of index [i] = 24, depending on the individual space group .
Example 2.1.8.5.3
The determination of the spacegroup types of the Hermann groups for the pair (No. 229) > Cmcm (No. 63) of index [i] = 12 follows the same procedure as in the previous example. The index [i] = 12 is factorized into and taking into account the orders of the point groups of and . The graph of tsubgroups of (Fig. 2.4.1.9 ) shows that the subgroups of the spacegroup types Fmmm and Immm are candidates for Hermann groups. Reference to the graph of ksubgroups of the crystal class mmm (Fig. 2.5.1.6 ) indicates that Immm has no maximal subgroups of Cmcm type, i.e. only space groups of Fmmm type can be Hermann groups for the pair of index [i] = 12.
Apart from the chains that can be found by the above considerations, other chains may exist. In some relatively simple cases, the graphs of this volume may be helpful to find such chains. However, one has to take into account that the tabulated graphs are contracted ones. In particular this means that they contain nothing about the numbers of subgroups of a certain kind and on their relations, for example conjugacy relations.
The following practical example may display the situation. It is based on the combination of the graphs of Chapters 2.4 and 2.5 with the subgroup tables of Chapters 2.2 and 2.3 .
Example 2.1.8.5.4
Let be a space group of type , No. 221. What are its subgroups of type I4/mcm, No. 140, and index 6?
As the order of the crystal class is reduced from cubic (48) to tetragonal (16) by index 3, the reduction of the translation subgroup must have index 2. To find the Hermann group , we look in the subgroup table of for tetragonal tsubgroups and find one class of three conjugate maximal tsubgroups of crystal class 4/mmm: P4/mmm, No. 123. By each of the three conjugate subgroups one of the axes a, b or c is distinguished. As this distinguished direction is kept in the other steps, one can take one of the conjugates as the representative and can continue with the consideration of only this representative. For the representative direction we choose the c axis, because this is the standard setting of P4/mmm. The relations of the other conjugates can then be obtained by replacing c by a or b.
The coordinate systems of and P4/mmm are the same, but is possible for P4/mmm. In the subgroup table of P4/mmm one looks for subgroups of type I4/mcm and index 2 and finds four nonconjugate subgroups with the same basis but different origin shifts. There can be no other subgroups of type I4/mcm because P4/mmm is the only possible Hermann group . Are there other chains from to the subgroups ?
Such a new chain of subgroups must have two steps. The first one leads from to a ksubgroup. In the graph for ksubgroups of one finds two subgroups of index 2, namely and . One finds from the corresponding graphs of tsubgroups or from the subgroup tables that only has subgroups of spacegroup type I4/mcm. The subgroup tables of and show that there are two nonconjugate subgroups of type which each have one conjugacy class of three subgroups of type I4/mcm. For the preferred direction c only one of the conjugate subgroups is relevant. Therefore, there are two subgroups I4/mcm of index 2 belonging to chains passing . It follows that two of the four subgroups obtained from Hermann's group are also subgroups of and two are not.
The two common subgroups are found by comparing their origin shifts from , which must be the same for both ways. The use of (4 × 4) matrices is convenient. The relevant equations for the bases and origin shifts are:leading to an origin shift of andleading to an origin shift of , which is equivalent to because an integer origin shift means only the choice of another conventional origin.
The result is Fig. 2.1.8.4, which is a complete graph, i.e. each field of the graph represents exactly one space group. The names of the substances belonging to the different subgroups show that the occurrence of such unexpected relations is not unrealistic. The crystal structures of KCuF_{3} and LTSrTiO_{3} both belong to the spacegroup type I4/mcm. They can be derived by different distortions from the same ideal perovskite ABX_{3} structure, space group (Figs. 2.1.8.4 and 2.1.8.5).

Complete graph of the group–subgroup chains from perovskite, , here hightemperature SrTiO_{3}, to the four subgroups of type I4/mcm with their tetragonal axes in the c direction. Two of them correspond to KCuF_{3} and lowtemperature SrTiO_{3}. The transformations and origin shifts given in the connecting lines specify the basis vectors and origins of the maximal subgroups in terms of the bases of the preceding space groups (Bärnighausen tree as explained in Section 1.6.3 ). 

Two different subgroups of P4/mmm, both of type I4/mcm, correspond to two kinds of distortions of the coordination octahedra of the perovskite structure. The difference is due to the origin shift by on the left side, resulting in a different selection of the fourfold rotation axes that are retained in the subgroups. 
The subgroup realized by LTSrTiO_{3} is not a subgroup of . The other subgroup, which is realized by KCuF_{3}, is a subgroup of . This cannot be concluded from the contracted graphs, but can be seen from the combination of the graphs with the tables or from the complete graph (Billiet, 1981; Koch, 1984; Wondratschek & Aroyo, 2001).
The remaining two subgroups do not form symmetries of distorted perovskites. The listed orbits of the Wyckoff positions, 4a, 4b, 4c, 4d and 8e, are all extraordinary orbits, i.e. they have more translations than the lattice of I4/mcm, see Engel et al. (1984). In Strukturbericht 1 (1931) the possible space groups for the cubic perovskites are listed on p. 301; there are five possible space groups from P23 to . The latter spacegroup symbol is framed and is considered to be the true symmetry of cubic perovskites. The other space groups are tsubgroups of . They would be taken if for some reason the site symmetries of the orbits would contradict the site symmetries of . Similarly, the true symmetry of these tetragonal perovskite derivatives would be P4/mmm with the original (only tetragonally distorted) lattice and not I4/mcm.^{9} For the two (empty) subgroups I4/mcm a distorted variant of the perovskite structure does not exist. Other special positions or the general position of the original cubic space group have to be occupied if the space group I4/mcm shall be realized. This example shows clearly the difference between the subgroup graphs of group theory and the Bärnighausen trees of crystal chemistry.
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