Maximal k subgroups

Bertaut, E. F.

doi:10.1107/97809553602060000509

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

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International Tables for Crystallography (2006). Vol. A. ch. 4.3, p. 74

Section 4.3.5.6.1. Maximal k subgroups

E. F. Bertaut^a ^‡

^a Laboratoire de Cristallographie, CNRS, Grenoble, France

4.3.5.6.1. Maximal k subgroups

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Maximal k subgroups of index [3] are obtained by `decentring' the triple cells R (hexagonal description), D and H in the trigonal system, H in the hexagonal system. Any one of the three centring points may be taken as origin of the subgroup.

(i) Trigonal system

Examples

(1) P3m1 (156) (cell a, b, c) is equivalent to H31m (a′, b′, c). Decentring of the H cell yields maximal non-isomorphic k subgroups of type P31m. Similarly, P31m (157) has maximal subgroups of type P3m1; thus, one can construct infinite chains of subgroup relations of index [3], tripling the cell at each step: $[P3m1 \rightarrow P31m \rightarrow P3m1 \ldots]$
(2) R3 (146), by decentring the triple hexagonal R cell $[{\bf a}',{\bf b}',{\bf c}']$ , yields the subgroups P3, $[P3_{1}]$ and $[P3_{2}]$ of index [3].
(3) Likewise, decentring of the triple rhombohedral cells $[D_{1}]$ and $[D_{2}]$ gives rise, for each cell, to the rhombohedral subgroups of a trigonal P group, again of index [3].

Combining (2) and (3), one may construct infinite chains of subgroup relations, tripling the cell at each step: $[P3 \rightarrow R3 \rightarrow P3 \rightarrow R3 \ldots]$ These chains illustrate best the connections between rhombohedral and hexagonal lattices.
(4) Special care must be applied when secondary or tertiary symmetry elements are present. Combining (1), (2) and (3), one has for instance: $[P31c \rightarrow R3c \rightarrow P3c1 \rightarrow P31c \rightarrow R3c \ldots]$
(5) Rhombohedral subgroups, found by decentring the triple cells $[D_{1}]$ and $[D_{2}]$ , are given under block IIb and are referred there to hexagonal axes, $[{\bf a}',{\bf b}',{\bf c}]$ as listed below. Examples are space groups P3 (143) and $[P\bar{3}1c]$ (163) $[\matrix{{\bf a}' = \phantom{2}{\bf a} - {\bf b}, &{\bf b}' = \phantom{-}{\bf a} + 2{\bf b}, &{\bf c}' = 3{\bf c};\cr {\bf a}' = 2{\bf a} + {\bf b}, &{\bf b}' = -{\bf a} + \phantom{2}{\bf b}, &{\bf c}' = 3{\bf c}.\cr}]$

(ii) Hexagonal system

Examples

(1) $[P\bar{6}2c]$ (190) is described as $[H\bar{6}c2]$ in the triple cell $[{\bf a}',{\bf b}',{\bf c}']$ ; decentring yields the non-isomorphic subgroup $[P\bar{6}c2]$ .
(2) (192) (cell a, b, c) keeps the same symbol in the H cell and, consequently, gives rise to the maximal isomorphic subgroup with cell $[{\bf a}',{\bf b}',{\bf c}']$ . An analogous result applies whenever secondary and tertiary symmetry elements in the Hermann–Mauguin symbol are the same and also to space groups of classes 6, $[\bar{6}]$ and .

References

International Tables for Crystallography (2006). Vol. A. ch. 4.3, p. 74