Graphs for klassengleiche subgroups

Gramlich, V.; Wondratschek, H.

doi:10.1107/97809553602060000546

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

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International Tables for Crystallography (2006). Vol. A1. ch. 2.5, pp. 415-425 | 1 | 2 |
https://doi.org/10.1107/97809553602060000546

Chapter 2.5. Graphs for klassengleiche subgroups

Volker Gramlich^a and Hans Wondratschek^b ^*

^a Laboratorium für Kristallographie, Eidgenössische Technische Hochschule, Wolfgang-Pauli Strasse 10, ETH Hönggerberg HCI, CH-8093 Zürich, Switzerland, and ^bInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: wondra@physik.uni-karlsruhe.de

This chapter presents 29 contracted graphs for the maximal klassengleiche subgroups of the space groups of the 32 crystal classes. The graphs for crystal classes 1, $[\overline{1}]$ and $[\overline{6}]$ are not included, because in these classes there is only one space-group type and their diagrams would be trivial. Isomorphic subgroups are not indicated, because each space group has an infinite number of them. In case of mutual group–subgroup relations, e.g. [P2] – [C2] or [P3_1] – [P3_2] , group and subgroup are connected by horizontal arrows which point from the group to the subgroup.

Keywords: graphs of group–subgroup relations; subgroup graphs; klassengleiche subgroups; maximal subgroups.

2.5.1. Graphs of the klassengleiche subgroups of monoclinic and orthorhombic space groups

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Graphs of the klassengleiche subgroups of the space groups of the following crystal classes are shown: 2 , [m] , [2/m] , [222] , [mm2] and [mmm] . For an explanation of these graphs, see Section 2.1.7.3 (p. 55).

Figure 2.5.1.1| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class 2.

Figure 2.5.1.2| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class [m] .

Figure 2.5.1.3| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class [2/m] .

Figure 2.5.1.4| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class [222] .

Figure 2.5.1.5| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class [mm2] .

Figure 2.5.1.6| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class [mmm] .

2.5.2. Graphs of the klassengleiche subgroups of tetragonal space groups

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Graphs of the klassengleiche subgroups of the space groups of the following crystal classes are shown: 4 , [4/m] , $[\overline4]$ , [422] , [4mm] , $[\overline4 2m]$ and [4/m mm] . For an explanation of these graphs, see Section 2.1.7.3 (p. 55).

Figure 2.5.2.1 | top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class 4.

Figure 2.5.2.3 | top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class [4/m] .

Figure 2.5.2.2 | top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class $[\overline4]$ .

Figure 2.5.2.4 | top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class [422] .

Figure 2.5.2.5 | top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class [4mm] .

Figure 2.5.2.6 | top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class $[\overline4 2m]$ .

Figure 2.5.2.7 | top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class [4/m mm] .

2.5.3. Graphs of the klassengleiche subgroups of trigonal space groups

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Graphs of the klassengleiche subgroups of the space groups of the following crystal classes are shown: 3 , $[\overline3]$ , [32] , [3m] and $[\overline3 m]$ . For an explanation of these graphs, see Section 2.1.7.3 (p. 55).

Figure 2.5.3.1| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class 3.

Figure 2.5.3.2| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class $[\overline3]$ .

Figure 2.5.3.3| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class [32] .

Figure 2.5.3.4| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class [3m] .

Figure 2.5.3.5| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class $[\overline3 m]$ .

2.5.4. Graphs of the klassengleiche subgroups of hexagonal space groups

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Graphs of the klassengleiche subgroups of the space groups of the following crystal classes are shown: 6 , [6/m] , [622] , [6mm] , $[\overline6 2m]$ and [6/mmm] . For an explanation of these graphs, see Section 2.1.7.3 (p. 55).

Figure 2.5.4.1| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class 6.

Figure 2.5.4.2| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class [6/m] .

Figure 2.5.4.3| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class 622.

Figure 2.5.4.4| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class [6mm] .

Figure 2.5.4.5| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class $[\overline6 2m]$ .

Figure 2.5.4.6| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class [6/mmm] .

2.5.5. Graphs of the klassengleiche subgroups of cubic space groups

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Graphs of the klassengleiche subgroups of the space groups of the following crystal classes are shown: 23 , $[m\overline3]$ , [432] , $[\overline4 3m]$ and $[m\overline3 m]$ . For an explanation of these graphs, see Section 2.1.7.3 (p. 55).

Figure 2.5.5.1| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class 23.

Figure 2.5.5.2| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class $[m\overline3]$ .

Figure 2.5.5.3| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class [432] .

Figure 2.5.5.4| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class $[\overline4 3m]$ .

Figure 2.5.5.5| top | pdf |

Graph of the klassengleiche subgroups of the space groups of crystal class $[m\overline3 m]$ .

International Tables for Crystallography (2006). Vol. A1. ch. 2.5, pp. 415-425
https://doi.org/10.1107/97809553602060000546