Graphs for translationengleiche subgroups

Gramlich, V.; Wondratschek, H.

doi:10.1107/97809553602060000545

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. 2.4, pp. 395-414 | 1 | 2 |
https://doi.org/10.1107/97809553602060000545

Chapter 2.4. Graphs for translationengleiche 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 graphs for the translationengleiche subgroups of the space groups. In order to fit each graph in the available space, the graphs are `contracted', i.e. each space-group type, specified by its short Hermann–Mauguin symbol and its space-group number, is displayed in a graph at most once. Each graph has a space group at the summit. The field of each space-group type is connected by downward lines to the fields of all space-group types to which its maximal subgroups belong. The chapter is partitioned into four sections: 11 graphs have a cubic summit (Section 2.4.1 ), 13 graphs have a tetragonal summit (Section 2.4.2 ), five graphs have a hexagonal summit (Section 2.4.3 ) and eight graphs have an orthorhombic summit (Section 2.4.4 ). All other possible graphs are contained within these graphs.

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

2.4.1. Graphs of the translationengleiche subgroups with a cubic summit

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Graphs of the translationengleiche subgroups of the following space groups are shown: $[Pm\overline{3}m]$ , $[Pn\overline{3}n]$ , $[Pm\overline{3}n]$ , $[Pn\overline{3}m]$ , $[Fm\overline{3}m]$ , $[Fm\overline{3}c]$ , $[Fd\overline{3}m]$ , $[Fd\overline{3}c]$ , $[Im\overline{3}m]$ , $[Ia\overline{3}d]$ , [P4_132] , [P4_332] and $[Pa\overline{3}]$ . For an explanation of these graphs, see Section 2.1.7.2 (p. 54).

Figure 2.4.1.1| top | pdf |

Graph of the translationengleiche subgroups of the space group $[Pm\overline{3}m]$ .

Figure 2.4.1.2| top | pdf |

Graph of the translationengleiche subgroups of the space group $[Pn\overline{3}n]$ .

Figure 2.4.1.3| top | pdf |

Graph of the translationengleiche subgroups of the space group $[Pm\overline{3}n]$ .

Figure 2.4.1.4| top | pdf |

Graph of the translationengleiche subgroups of the space group $[Pn\overline{3}m]$ .

Figure 2.4.1.5| top | pdf |

Graph of the translationengleiche subgroups of the space group $[Fm\overline{3}m]$ .

Figure 2.4.1.6| top | pdf |

Graph of the translationengleiche subgroups of the space group $[Fm\overline{3}c]$ .

Figure 2.4.1.7| top | pdf |

Graph of the translationengleiche subgroups of the space group $[Fd\overline{3}m]$ .

Figure 2.4.1.8| top | pdf |

Graph of the translationengleiche subgroups of the space group $[Fd\overline{3}c]$ .

Figure 2.4.1.9| top | pdf |

Graph of the translationengleiche subgroups of the space group $[Im\overline{3}m]$ .

Figure 2.4.1.10| top | pdf |

Graph of the translationengleiche subgroups of the space group $[Ia\overline{3}d]$ .

Figure 2.4.1.11| top | pdf |

Graph of the translationengleiche subgroups of the space groups [P4_132] , [P4_332] and $[Pa\overline{3}]$ .

2.4.2. Graphs of the translationengleiche subgroups with a tetragonal summit

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Graphs of the translationengleiche subgroups of the following space groups are shown: [P4/mcc] , [P4/nbm] , [P4/mbm] , [P4/mnc] , [P4/nmm] , [P4/ncc] , [P4_2/mcm] , [P4_2/nbc] , [P4_2/mbc] , [P4_2/mnm] , [P4_2/nmc] , [P4_2/ncm] , [P4_122] and [P4_322] . For an explanation of these graphs, see Section 2.1.7.2 (p. 54).

Figure 2.4.2.1| top | pdf |

Graph of the translationengleiche subgroups of the space group [P4/mcc] .

Figure 2.4.2.2| top | pdf |

Graph of the translationengleiche subgroups of the space group [P4/nbm] .

Figure 2.4.2.3| top | pdf |

Graph of the translationengleiche subgroups of the space group [P4/mbm] .

Figure 2.4.2.4| top | pdf |

Graph of the translationengleiche subgroups of the space group [P4/mnc] .

Figure 2.4.2.5| top | pdf |

Graph of the translationengleiche subgroups of the space group [P4/nmm] .

Figure 2.4.2.6| top | pdf |

Graph of the translationengleiche subgroups of the space group [P4/ncc] .

Figure 2.4.2.7| top | pdf |

Graph of the translationengleiche subgroups of the space group [P4_2/mcm] .

Figure 2.4.2.8| top | pdf |

Graph of the translationengleiche subgroups of the space group [P4_2/nbc] .

Figure 2.4.2.9| top | pdf |

Graph of the translationengleiche subgroups of the space group [P4_2/mbc] .

Figure 2.4.2.10| top | pdf |

Graph of the translationengleiche subgroups of the space group [P4_2/mnm] .

Figure 2.4.2.11| top | pdf |

Graph of the translationengleiche subgroups of the space group [P4_2/nmc] .

Figure 2.4.2.12| top | pdf |

Graph of the translationengleiche subgroups of the space group [P4_2/ncm] .

Figure 2.4.2.13| top | pdf |

Graph of the translationengleiche subgroups of the space groups [P4_122] and [P4_322] .

2.4.3. Graphs of the translationengleiche subgroups with a hexagonal summit

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Graphs of the translationengleiche subgroups of the following space groups are shown: [P6/mmm] , [P6/mcc] , [P6_3/mcm] , [P6_3/mmc] , [P6_122] , [P6_522] , [P6_222] and [P6_422] . For an explanation of these graphs, see Section 2.1.7.2 (p. 54).2.1.7.2 (p. 54).

Figure 2.4.3.1| top | pdf |

Graph of the translationengleiche subgroups of the space group [P6/mmm] .

Figure 2.4.3.2| top | pdf |

Graph of the translationengleiche subgroups of the space group [P6/mcc] .

Figure 2.4.3.3| top | pdf |

Graph of the translationengleiche subgroups of the space group [P6_3/mcm] .

Figure 2.4.3.4| top | pdf |

Graph of the translationengleiche subgroups of the space group [P6_3/mmc] .

Figure 2.4.3.5| top | pdf |

Graph of the translationengleiche subgroups of the space groups [P6_122] , [P6_522] , [P6_222] and [P6_422] .

2.4.4. Graphs of the translationengleiche subgroups with an orthorhombic summit

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Graphs of the translationengleiche subgroups of the following space groups are shown: [Pmma] , [Pnna] , [Pmna] , [Pcca] , [Pbcm] , [Pbcn] , [Pnma] and [Cmce] . For an explanation of these graphs, see Section 2.1.7.2 (p. 54).

Figure 2.4.4.1| top | pdf |

Graph of the translationengleiche subgroups of the space group [Pmma] .

Figure 2.4.4.2| top | pdf |

Graph of the translationengleiche subgroups of the space group [Pnna] .

Figure 2.4.4.3| top | pdf |

Graph of the translationengleiche subgroups of the space group [Pmna] .

Figure 2.4.4.4| top | pdf |

Graph of the translationengleiche subgroups of the space group [Pcca] .

Figure 2.4.4.5| top | pdf |

Graph of the translationengleiche subgroups of the space group [Pbcm] .

Figure 2.4.4.6| top | pdf |

Graph of the translationengleiche subgroups of the space group [Pbcn] .

Figure 2.4.4.7| top | pdf |

Graph of the translationengleiche subgroups of the space group [Pnma] .

Figure 2.4.4.8| top | pdf |

Graph of the translationengleiche subgroups of the space group [Cmce] .

International Tables for Crystallography (2006). Vol. A1. ch. 2.4, pp. 395-414
https://doi.org/10.1107/97809553602060000545