InternationalSymmetry relations between space groupsTables for Crystallography Volume A1 Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2011 |
International Tables for Crystallography (2011). Vol. A1, ch. 2.1, pp. 82-84
## Section 2.1.5. Series of maximal isomorphic subgroups Y. Billiet
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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

**to be determined.**

*Y*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

**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.**

*Y*

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* = 4*n* − 1. On the other hand, translational parts with vectors are introduced in the generators of if *p* = 4*n* + 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* = 4*n* − 1 and for *p* = 4*n* + 1. Thus, the one series in origin choice 1 for odd *p* is split into two series in origin choice 2 for *p* = 4*n* − 1 and *p* = 4*n* + 1.^{4}