Guide to the subgroup tables and graphs

Wondratschek, H.; Aroyo, M. I.

doi:10.1107/97809553602060000797

International Tables for Crystallography (2011). Vol. A1. ch. 2.1, pp. 72-96 | 1 | 2 |
https://doi.org/10.1107/97809553602060000797

Chapter 2.1. Guide to the subgroup tables and graphs

2.1. Guide to the subgroup tables and graphs (pp. 72-96) | html | pdf | chapter contents |
Hans Wondratschek and Mois I. Aroyo
- 2.1.1. Contents and arrangement of the subgroup tables (p. 72) | html | pdf |
- 2.1.2. Structure of the subgroup tables (pp. 72-76) | html | pdf |
  - 2.1.2.1. Headline (p. 72) | html | pdf |
  - 2.1.2.2. Data from IT A (pp. 72-73) | html | pdf |
    - 2.1.2.2.1. Generators selected (p. 72) | html | pdf |
    - 2.1.2.2.2. General position (pp. 72-73) | html | pdf |
  - 2.1.2.3. Specification of the setting (p. 73) | html | pdf |
  - 2.1.2.4. Sequence of the subgroup and supergroup data (p. 73) | html | pdf |
  - 2.1.2.5. Special rules for the setting of the subgroups (pp. 74-76) | html | pdf |
    - 2.1.2.5.1. Monoclinic subgroups (p. 74) | html | pdf |
    - 2.1.2.5.2. Subgroups with two origin choices (pp. 74-75) | html | pdf |
    - 2.1.2.5.3. Space groups with a rhombohedral lattice (pp. 75-76) | html | pdf |
- 2.1.3. I Maximal translationengleiche subgroups (t-subgroups) (pp. 76-79) | html | pdf |
  - 2.1.3.1. Introduction (p. 76) | html | pdf |
  - 2.1.3.2. A description in close analogy with IT A (pp. 76-77) | html | pdf |
  - 2.1.3.3. Basis transformation and origin shift (pp. 77-79) | html | pdf |
- 2.1.4. II Maximal klassengleiche subgroups (k-subgroups) (pp. 79-82) | html | pdf |
  - 2.1.4.1. General description (p. 79) | html | pdf |
  - 2.1.4.2. Loss of centring translations (pp. 79-80) | html | pdf |
  - 2.1.4.3. Enlarged unit cell (pp. 80-82) | html | pdf |
    - 2.1.4.3.1. Enlarged unit cell, index 2 (pp. 80-81) | html | pdf |
    - 2.1.4.3.2. Enlarged unit cell, index 3 or 4 (pp. 81-82) | html | pdf |
- 2.1.5. Series of maximal isomorphic subgroups (pp. 82-84) | html | pdf |
  Y. Billiet
  - 2.1.5.1. General description (p. 82) | html | pdf |
  - 2.1.5.2. Basis transformation (p. 82) | html | pdf |
  - 2.1.5.3. Origin shift (p. 82) | html | pdf |
  - 2.1.5.4. Generators (pp. 82-83) | html | pdf |
  - 2.1.5.5. Special series (pp. 83-84) | html | pdf |
    - 2.1.5.5.1. Monoclinic space groups (p. 83) | html | pdf |
    - 2.1.5.5.2. Trigonal space groups with rhombohedral lattice (p. 83) | html | pdf |
    - 2.1.5.5.3. Space groups with two origin choices (pp. 83-84) | html | pdf |
- 2.1.6. The data for minimal supergroups (pp. 84-85) | html | pdf |
  - 2.1.6.1. Description of the listed data (p. 84) | html | pdf |
  - 2.1.6.2. I Minimal translationengleiche supergroups (pp. 84-85) | html | pdf |
  - 2.1.6.3. II Minimal non-isomorphic klassengleiche supergroups (p. 85) | html | pdf |
- 2.1.7. Derivation of the minimal supergroups from the subgroup tables (pp. 86-90) | html | pdf |
  - 2.1.7.1. Determination of the non-isomorphic minimal k-supergroups by inverting the subgroup data (p. 86) | html | pdf |
  - 2.1.7.2. The isomorphic minimal supergroups (pp. 86-87) | html | pdf |
  - 2.1.7.3. Determination of one minimal t-supergroup by inverting the subgroup data (pp. 87-88) | html | pdf |
  - 2.1.7.4. Derivation of further minimal t-supergroups by using normalizers (pp. 88-90) | html | pdf |
  - 2.1.7.5. Derivation of the remaining minimal t-supergroups (p. 90) | html | pdf |
- 2.1.8. The subgroup graphs (pp. 90-96) | html | pdf |
  - 2.1.8.1. General remarks (pp. 90-91) | html | pdf |
  - 2.1.8.2. Graphs for translationengleiche subgroups (pp. 91-92) | html | pdf |
  - 2.1.8.3. Graphs for klassengleiche subgroups (pp. 92-93) | html | pdf |
  - 2.1.8.4. Graphs for plane groups (p. 93) | html | pdf |
  - 2.1.8.5. Application of the graphs (pp. 93-96) | html | pdf |
- References | html | pdf |
- Figures
  - Fig. 2.1.7.1. In the left-hand diagram, there are i conjugate subgroups $[{\cal H}_k \,\lt\, {\cal G}]$ of $[\cal G]$ if [i] is the index of $[\cal H]$ in $[\cal G]$ ; in the right-hand diagram, $[{\cal H}\triangleleft{\cal G}]$ is a normal subgroup of $[\cal G]$ (p. 88) | html | pdf |
  - Fig. 2.1.8.1. Contracted graph of the group–subgroup chains from $[Fm\overline{3}m]$ (225) to one of those subgroups with index 12 which belong to the space-group type (12) (p. 94) | html | pdf |
  - Fig. 2.1.8.2. Complete graph of the group–subgroup chains from $[Fm\overline{3}m]$ (225) to one representative belonging to those six (12) subgroups with index 12 whose monoclinic axes are along the $[\langle 110 \rangle]$ directions of $[Fm\overline{3}m]$ (p. 94) | html | pdf |
  - Fig. 2.1.8.3. Complete graph of the group–subgroup chains from $[Fm\overline{3}m]$ (225) to , which is one representative of those three (12) subgroups with index 12 whose monoclinic axes are along the $[\langle 001 \rangle]$ directions of $[Fm\overline{3}m]$ (p. 94) | html | pdf |
  - Fig. 2.1.8.4. Complete graph of the group–subgroup chains from perovskite, $[Pm\overline{3}m]$ , here high-temperature SrTiO₃, to the four subgroups of type I4/mcm with their tetragonal axes in the c direction (p. 96) | html | pdf |
  - Fig. 2.1.8.5. 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 (p. 96) | html | pdf |

Chapter 2.1. Guide to the subgroup tables and graphs

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