International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 2.2, pp. 35-38

## Section 2.2.15. Maximal subgroups and minimal supergroups

Th. Hahna* and A. Looijenga-Vosb

aInstitut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and bLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands
Correspondence e-mail:  hahn@xtl.rwth-aachen.de

### 2.2.15. Maximal subgroups and minimal supergroups

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The present section gives a brief summary, without theoretical explanations, of the sub- and supergroup data in the space-group tables. The theoretical background is provided in Section 8.3.3 and Part 13 . Detailed sub- and supergroup data are given in International Tables for Crystallography Volume A1 (2004).

#### 2.2.15.1. Maximal non-isomorphic subgroups6

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The maximal non-isomorphic subgroups of a space group are divided into two types: For practical reasons, type II is subdivided again into two blocks:

IIa the conventional cells of and are the same

IIb the conventional cell of is larger than that of . 7

Block IIa has no entries for space groups with a primitive cell. For space groups with a centred cell, it contains those maximal subgroups that have lost some or all centring translations of but none of the integral translations (decentring' of a centred cell).

Within each block, the subgroups are listed in order of increasing index [i] and in order of decreasing space-group number for each value of i.

• (i) Blocks I and IIa

In blocks I and IIa, every maximal subgroup of a space group is listed with the following information:

[i] HMS1 (HMS2, No.) Sequence of numbers.

The symbols have the following meaning:

[i]: index of in (cf. the footnote to Section 8.1.6);

HMS1: Hermann–Mauguin symbol of , referred to the coordinate system and setting of ; this symbol may be unconventional;

(HMS2, No.): conventional short Hermann–Mauguin symbol of , given only if HMS1 is not in conventional short form, and the space-group number of .

Sequence of numbers: coordinate triplets of retained in . The numbers refer to the numbering scheme of the coordinate triplets of the general position of (cf. Section 2.2.9). The following abbreviations are used:

 Block I (all translations retained): Number Coordinate triplet given by Number, plus those obtained by adding all centring translations of . (Numbers) The same, but applied to all Numbers between parentheses. Block IIa (not all translations retained): Coordinate triplet obtained by adding the translation to the triplet given by Number. The same, but applied to all Numbers between parentheses.

In blocks I and IIa, sets of conjugate subgroups are linked by left-hand braces. For an example, see space group (148) below.

#### Examples

 (1) where the numbers have the following meaning: (2) where the numbers have the following meaning: (3) Operations and c, occurring in the Hermann–Mauguin symbol of , are lacking in . In the unconventional tetragonal version' of the symbol of stands for two sets of (along the two orthogonal secondary symmetry directions), implying that is orthorhombic. In the conventional orthorhombic version', the full symbol of reads and the short symbol Pmmn.

• (ii) Block IIb

Whereas in blocks I and IIa every maximal subgroup of is listed, this is no longer the case for the entries of block IIb. The information given in this block is:

[i] HMS1 (Vectors) (HMS2, No.)

The symbols have the following meaning:

[i]: index of in ;

HMS1: Hermann–Mauguin symbol of , referred to the coordinate system and setting of ; this symbol may be unconventional;8

(Vectors): basis vectors a′, b′, c′ of in terms of the basis vectors a, b, c of . No relations are given for unchanged axes, e.g. is not stated;

(HMS2, No.): conventional short Hermann–Mauguin symbol, given only if HMS1 is not in conventional short form, and the space-group number of .

In addition to the general rule of increasing index [i] and decreasing space-group number (No.), the sequence of the IIb subgroups also depends on the type of cell enlargement. Subgroups with the same index and the same kind of cell enlargement are listed together in decreasing order of space-group number (see example 1 below).

In contradistinction to blocks I and IIa, for block IIb the coordinate triplets retained in are not given. This means that the entry is the same for all subgroups that have the same Hermann–Mauguin symbol and the same basis-vector relations to , but contain different sets of coordinate triplets. Thus, in block IIb, one entry may correspond to more than one subgroup,9 as illustrated by the following examples.

#### Examples

• (1)

Each of the subgroups is referred to its own distinct basis a′, b′, c′, which is different in each case. Apart from the translations of the enlarged cell, the generators of the subgroups, referred to a′, b′, c′, are as follows:

There are thus 2, 1 or 4 actual subgroups that obey the same basis-vector relations. The difference between the several subgroups represented by one entry is due to the different sets of symmetry operations of that are retained in . This can also be expressed as different conventional origins of with respect to .

• (2)

The nine subgroups of type P31m may be described in two ways:

 (i) By partial decentring' of ninetuple cells (, , ) with the same orientations as the cell of the group in such a way that the centring points 0, 0, 0; ; (referred to a′, b′, c′) are retained. The conventional space-group symbol P31m of these nine subgroups is referred to the same basis vectors , , , but to different origins; cf. Section 2.2.15.5. This kind of description is used in the space-group tables of this volume. (ii) Alternatively, one can describe the group with an unconventional H-centred cell (, , ) referred to which the space-group symbol is H31m. Decentring' of this cell results in the conventional space-group symbol P31m for the subgroups, referred to the basis vectors a′, b′, c′. This description is used in Section 4.3.5 .

• (iii) Subdivision of k subgroups into blocks IIa and IIb

The subdivision of k subgroups into blocks IIa and IIb has no group-theoretical background and depends on the coordinate system chosen. The conventional coordinate system of the space group (cf. Section 2.1.3 ) is taken as the basis for the subdivision. This results in a uniquely defined subdivision, except for the seven rhombohedral space groups for which in the space-group tables both rhombohedral axes' (primitive cell) and hexagonal axes' (triple cell) are given (cf. Section 2.2.2). Thus, some k subgroups of a rhombohedral space group are found under IIa (klassengleich, centring translations lost) in the hexagonal description, and under IIb (klassengleich, conventional cell enlarged) in the rhombohedral description.

#### Example:

Hexagonal axes

Rhombohedral axes

Apart from the change from IIa to IIb, the above example demonstrates again the restricted character of the IIb listing, discussed above. The three conjugate subgroups of index [3] are listed under IIb by one entry only, because for all three subgroups the basis-vector relations between and are the same. Note the brace for the IIa subgroups, which unites conjugate subgroups into classes.

#### 2.2.15.2. Maximal isomorphic subgroups of lowest index (cf. Part 13 )

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Another set of klassengleiche subgroups are the isomorphic subgroups listed under IIc, i.e. the subgroups which are of the same or of the enantiomorphic space-group type as . The kind of listing is the same as for block IIb. Again, one entry may correspond to more than one isomorphic subgroup.

As the number of maximal isomorphic subgroups of a space group is always infinite, the data in block IIc are restricted to the subgroups of lowest index. Different kinds of cell enlargements are presented. For monoclinic, tetragonal, trigonal and hexagonal space groups, cell enlargements both parallel and perpendicular to the main rotation axis are listed; for orthorhombic space groups, this is the case for all three directions, a, b and c. Two isomorphic subgroups and of equal index but with cell enlargements in different directions may, nevertheless, play an analogous role with respect to . In terms of group theory, and then are conjugate subgroups in the affine normalizer of , i.e. they are mapped onto each other by automorphisms of .10 Such subgroups are collected into one entry, with the different vector relationships separated by or' and placed within one pair of parentheses; cf. example (4).

#### Examples

 (1) The first subgroup of index [3] entails an enlargement of the c axis, the second one of index [4] an enlargement of the mesh size in the a,b plane. (2) It seems surprising that [27] is the lowest index listed, even though another isomorphic subgroup of index [8] exists. The latter subgroup, however, is not maximal, as chains of maximal non-isomorphic subgroups can be constructed as follows: (3) Note that the isomorphic subgroup of index [4] with is not listed, because it is not maximal. This is apparent from the chain (4) For , the x and y directions are analogous, i.e. they may be interchanged by automorphisms of . Such an automorphism does not exist for because this space group contains glide reflections a but not b.

#### 2.2.15.3. Minimal non-isomorphic supergroups

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If is a maximal subgroup of a group , then is called a minimal supergroup of . Minimal non-isomorphic supergroups are again subdivided into two types, the translationengleiche or t supergroups I and the klassengleiche or k supergroups II. For the minimal t supergroups I of , the listing contains the index [i] of in , the conventional Hermann–Mauguin symbol of and its space-group number in parentheses.

There are two types of minimal k supergroups II: supergroups with additional centring translations (which would correspond to the IIa type) and supergroups with smaller conventional unit cells than that of (type IIb). Although the subdivision between IIa and IIb supergroups is not indicated in the tables, the list of minimal supergroups with additional centring translations (IIa) always precedes the list of IIb supergroups. The information given is similar to that for the non-isomorphic subgroups IIb, i.e., where applicable, the relations between the basis vectors of group and supergroup are given, in addition to the Hermann–Mauguin symbols of and its space-group number. The supergroups are listed in order of increasing index and increasing space-group number.

The block of supergroups contains only the types of the non-isomorphic minimal supergroups of , i.e. each entry may correspond to more than one supergroup . In fact, the list of minimal supergroups of should be considered as a backwards reference to those space groups for which appears as a maximal subgroup. Thus, the relation between and can be found in the subgroup entries of .

#### Example:

Minimal non-isomorphic supergroupsBlock I lists, among others, the entry [2] Pnma (62). Looking up the subgroup data of Pnma (62), one finds in block I the entry [2] . This shows that the setting of Pnma does not correspond to that of but rather to that of . To obtain the supergroup referred to the basis of , the basis vectors b and c must be interchanged. This changes Pnma to Pnam, which is the correct symbol of the supergroup of .

Note on R supergroups of trigonal P space groups: The trigonal P space groups Nos. 143–145, 147, 150, 152, 154, 156, 158, 164 and 165 each have two rhombohedral supergroups of type II. They are distinguished by different additional centring translations which correspond to the obverse' and reverse' settings of a triple hexagonal R cell; cf. Chapter 1.2 . In the supergroup tables of Part 7 , these cases are described as [3] R3 (obverse) (146); [3] R3 (reverse) (146) etc.

#### 2.2.15.4. Minimal isomorphic supergroups of lowest index

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No data are listed for isomorphic supergroups IIc because they can be derived directly from the corresponding data of subgroups IIc (cf. Part 13 ).

#### 2.2.15.5. Note on basis vectors

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In the subgroup data, a′, b′, c′ are the basis vectors of the subgroup of the space group . The latter has the basis vectors a, b, c. In the supergroup data, a′, b′, c′ are the basis vectors of the supergroup and a, b, c are again the basis vectors of . Thus, a, b, c and a′, b′, c′ exchange their roles if one considers the same group–subgroup relation in the subgroup and the supergroup tables.

#### Examples

 (1) Listed under subgroups IIb, one finds, among other entries, ; thus, . Under supergroups II of , the corresponding entry reads ; thus . (2) Tetragonal k space groups with P cells. For index [2], the relations between the conventional basis vectors of the group and the subgroup read (cf. Fig. 5.1.3.5 ) Thus, the basis vectors of the supergroup are An alternative description is (3) Hexagonal k space groups. For index [3], the relations between the conventional basis vectors of the sub- and supergroup read (cf. Fig 5.1.3.8 ) Thus, the basis vectors of the supergroup are An alternative description is

### References

International Tables for Crystallography (2004). Vol. A1, edited by H. Wondratschek & U. Müller. Dordrecht: Kluwer Academic Publishers.