Subgroups and supergroups of point and space groups

Ulrich Müller

in Symmetry Relationships between Crystal Structures

Published in print April 2013 | ISBN: 9780199669950
Published online December 2013 | e-ISBN: 9780191775086 | DOI:

Series: International Union of Crystallography Texts on Crystallography

Subgroups and supergroups of point and space groups

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Group-subgroup relations can by depicted by graphs in which the symbol for every group is connected with the symbols of its maximal subgroups. Two graphs are sufficient to present all kinds of group-subgroup relations between point groups. In the case of space groups, three kinds of maximal subgroups are distinguished: translationengleiche subgroups which have kept all translations but belong to a lower crystal class; klassengleiche subgroups having fewer translations but the same crystal class; isomorphic subgroups which belong to the same or the enantiomorphic space group type and have fewer translations, they are a special kind of klassengleiche subgroups. The kinds of translationengleiche subgroups can be depicted in thirty-seven graphs, those of klassengleiche subgroups in twenty-nine graphs. The number of isomorphic subgroups is always infinite. Minimal supergroups of space groups are more manifold than maximal subgroups. The symmetry group of an object in three-dimensional space is a layer group if it has translational symmetry only in two dimensions, and a rod group if it has translational symmetry only in one dimension. They are designated by modified Hermann-Mauguin symbols.

Keywords: translationengleiche subgroups; klassengleiche subgroup; isomorphic subgroup; minimal supergroups; layer group; rod group

Chapter.  4695 words.  Illustrated.

Subjects: Crystallography

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