International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

International Tables for Crystallography (2006). Vol. A1, ch. 2.1, pp. 53-54   | 1 | 2 |

Section 2.1.6. Minimal supergroups

Hans Wondratscheka* and Mois I. Aroyob

aInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany, and bDepartamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain
Correspondence e-mail:  wondra@physik.uni-karlsruhe.de

2.1.6. Minimal supergroups

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2.1.6.1. General description

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In the previous sections, the relation [{\cal H} \,\lt\, {\cal G}] was seen from the viewpoint of the group [{\cal G}]. In this case, [{\cal H}] was a subgroup of [{\cal G}]. However, the same relation may be viewed from the group [{\cal H}]. In this case, [{\cal G}> {\cal H}] is a supergroup of [{\cal H}]. As for the subgroups of [{\cal G}], cf. Section 1.2.6[link] , different kinds of supergroups of [{\cal H}] may be distinguished. The following definitions are obvious.

Definition 2.1.6.1.1. Let [{\cal H} \,\lt\, {\cal G}] be a maximal subgroup of [{\cal G}]. Then [{\cal G}> {\cal H}] is called a minimal supergroup of [{\cal H}]. If [{\cal H}] is a translationengleiche subgroup of [{\cal G}] then [{\cal G}] is a translationen­glei­che supergroup (t-supergroup) of [{\cal H}]. If [{\cal H}] is a klassengleiche subgroup of [{\cal G}], then [{\cal G}] is a klassengleiche supergroup (k-supergroup) of [{\cal H}]. If [{\cal H}] is an isomorphic subgroup of [{\cal G}], then [{\cal G}] is an isomorphic supergroup of [{\cal H}]. If [{\cal H}] is a general subgroup of [{\cal G}], then [{\cal G}] is a general supergroup of [{\cal H}].

The search for supergroups of space groups is much more difficult than the search for subgroups. One of the reasons for this difficulty is that the search for subgroups [{\cal H} \,\lt\, {\cal G}] is restricted to the elements of the space group [{\cal G}] itself, whereas the search for supergroups [{\cal G}> {\cal H}] has to take into account the whole (continuous) group [{\cal E}] of all isometries. For example, there are only a finite number of subgroups [{\cal H}] of any space group [{\cal G}] for any given index i. On the other hand, there may not only be an infinite number of supergroups [{\cal G}] of a space group [{\cal H}] for a finite index i but even an uncountably infinite number of supergroups of [{\cal H}].

Example 2.1.6.1.2

Let [{\cal H}=P1]. Then there is an infinite number of t-supergroups [P\overline{1}] of index 2 because there is no restriction for the sites of the centres of inversion and thus of the conventional origin of [P\overline{1}].

In the tables of this volume, a supergroup [{\cal G}] of a space group [{\cal H}] is listed by its type if [{\cal H}] is listed as a subgroup of [{\cal G}]. The entry contains at least the index of [{\cal H}] in [{\cal G}], the conventional HM symbol of [{\cal G}] and its space-group number. Additional data may be given for klassengleiche supergroups. More details, e.g. 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. The number of supergroups belonging to one entry can neither be concluded from the subgroup data nor is it listed among the supergroup data.

Like the subgroup data, the supergroup data are also partitioned into blocks.

2.1.6.2. I Minimal translationengleiche supergroups

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For each space group [{\cal H}], under this heading are listed those space-group types [{\cal G}] for which [{\cal H}] appears as an entry under the heading I Maximal translationengleiche subgroups. The listing consists of the index in brackets […], the conventional HM symbol and (in parentheses) the space-group number (…). The space groups are ordered by ascending space-group number. If this line is empty, the heading is printed nevertheless and the content is announced by `none', as in [P6/mmm], No. 191.

The supergroups listed on the line I Minimal translationen­glei­che supergroups are realized only if the lattice conditions of [{\cal H}] fulfil the lattice conditions for [{\cal G}]. For example, if [{\cal G}=P422], No. 89, is a supergroup of [{\cal H}=P222], No. 16, 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 adapted to the conventional c-axis setting of [P422]. For the cubic supergroup [P23], No. 195, all three lattice parameters of [P222] must be (approximately) equal. Such conditions are always to be taken into consideration if the t-supergroup belongs to a different crystal family4 to the original group. Therefore, for [{\cal H}=P222] there is no lattice condition for the supergroup [{\cal G}=Pmmm] because [P222] and [Pmmm] belong to the same crystal family.

2.1.6.3. II Minimal non-isomorphic klassengleiche supergroups

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Klassengleiche supergroups [{\cal G}> {\cal H}] always belong to the crystal family of [{\cal H}]. Therefore, there are no restrictions for the lattice parameters of [{\cal H}].

The block II Minimal non-isomorphic 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 [P6/mmm], No. 191.

If at least one of the subblocks is non-empty, 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 [P222], No. 16, and [Fd\overline{3}c], No. 228.

Under the heading `Additional centring translations', the supergroups are listed by their indices and either by their nonconventional HM symbols, with the space-group numbers and the standard HM symbols in parentheses, or by their conventional HM symbols and only their space-group numbers in parentheses. Examples are provided by space group [Pbca], No. 61, with both subblocks non-empty and by space group [P222], No. 16, with supergroups only under the heading `Additional centring translations'.

Under the heading `Decreased unit cell' each supergroup is listed by its index and by its lattice relations, where the basis vectors [{\bf a}'], [{\bf b}'] and [{\bf c}'] refer to the supergroup [{\cal G}] and the basis vectors a, b and c to the original group [{\cal H}]. After these data are listed either the nonconventional HM symbol, followed by the space-group number and the conventional HM symbol in parentheses, or the conventional HM symbol with the space-group number in parentheses. Examples are provided again by space group [Pbca], No. 61, with both subblocks occupied and space group [F\overline{4}3m], No. 216, with an empty subblock `Additional centring translations' but data under the heading `Decreased unit cell'.

2.1.6.4. Isomorphic supergroups

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Each space group [{\cal G}] has an infinite number of isomorphic subgroups [{\cal H}] because the number of primes is infinite. For the same reason, each space group [{\cal H}] has an infinite number of isomorphic supergroups [{\cal G}]. They are not listed in the tables of this volume because they are implicitly listed among the subgroup data.








































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