Group–subgroup relations

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, pp. 70-71

Section 4.3.3.2. Group–subgroup relations

E. F. Bertaut^a ^‡

^a Laboratoire de Cristallographie, CNRS, Grenoble, France

4.3.3.2. Group–subgroup relations

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The present section emphasizes the use of the extended and full symbols for the derivation of maximal subgroups of types I and IIa; maximal orthorhombic subgroups of types IIb and IIc cannot be recognized by inspection of the synoptic Table 4.3.2.1.

4.3.3.2.1. Maximal non-isomorphic k subgroups of type IIa (decentred)

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(i) Extended symbols of centred groups A, B, C, I

By convention, the second line of the extended space-group symbol is the result of the multiplication of the first line by the centring translation (cf. Table 4.1.2.3 ). As a consequence, the product of any two terms in one line is equal to the product of the corresponding two terms in the other line.

(a) Class 222

The extended symbol of I222 (23) is $[\matrix{\noalign{\vskip 13pt}I 2\ 2\ 2{\hbox{;}}\hfill\cr 2_{1} 2_{1} 2_{1}\hfill\cr}]$ the twofold axes intersect and one obtains $[2_{x}\;{\times\; 2_{y} = 2_{z} = 2_{1x}}\; {\times\; 2_{1y}}]$ .

Maximal k subgroups are P222 and $[P2_{1}2_{1}2]$ (plus permutations) but not $[P2_{1}2_{1}2_{1}]$ .

The extended symbol of $[I2_{1}2_{1}2_{1}\;(24)]$ is $[\matrix{\noalign{\vskip 13pt}I 2_{1} 2_{1} 2_{1}\hfill\cr 222\hfill\cr}]$ , where one obtains $[2_{1x} \times 2_{1y} = 2_{1z} = 2_{x} \times 2_{y}]$ ; the twofold axes do not intersect. Thus, maximal non-isomorphic k subgroups are $[P2_{1}2_{1}2_{1}]$ and $[P222_{1}]$ (plus permutations), but not P222.
(b) Class mm2

The extended symbol of Aea2 (41) is $[\matrix{\noalign{\vskip 12pt}Aba2{\hbox{;}}\hfill\cr cn2_{1}\hfill\cr}]$ the following relations hold: $[b \times a = 2 = c \times n]$ and $[b \times n = 2_{1} = c \times a]$ .

Maximal k subgroups are Pba2; Pcn2 (Pnc2); $[Pbn2_{1}]$ ( $[Pna2_{1}]$ ); $[Pca2_{1}]$ .
(c) Class mmm

By convention, the first line of the extended symbol contains those symmetry elements for which the coordinate triplets are explicitly printed under Positions. From the two-line symbols, as defined in the example below, one reads not only the eight maximal k subgroups P of class mmm but also the location of their centres of symmetry, by applying the following rules:

If in the symbol of the P subgroup the number of symmetry planes, chosen from the first line of the extended symbol, is odd (three or one), the symmetry centre is at 0, 0, 0; if it is even (two or zero), the symmetry centre is at $[{1 \over 4},{1 \over 4},0]$ for the subgroups of C groups and at $[{1 \over 4},{1 \over 4},{1 \over 4}]$ for the subgroups of I groups (Bertaut, 1976).

Examples

(1) According to these rules, the extended symbol of Cmce (64) is $[\!\matrix{\noalign{\vskip 13pt}C m c b\hfill\cr b n a\hfill\cr}]$ (see above). The four k subgroups with symmetry centres at 0, 0, 0 are Pmcb (Pbam); Pmna; Pbca; Pbnb (Pccn); those with symmetry centres at $[{1 \over 4},{1 \over 4},0]$ are Pbna (Pbcn); Pmca (Pbcm); Pmnb (Pnma); Pbcb (Pcca). These rules can easily be transposed to other settings.
(2) The extended symbol of Ibam (72) is $[\matrix{\noalign{\vskip 12pt}I b a m\hfill\cr c c n\hfill\cr}]$ . The four subgroups with symmetry centre at 0, 0, 0 are Pbam; Pbcn; Pcan (Pbcn); Pccm;

those with symmetry centre at $[{1 \over 4},{1 \over 4},{1 \over 4}]$ are Pccn; Pcam (Pbcm); Pbcm; Pban.

(ii) Extended symbols of F-centred space groups

Maximal k subgroups of the groups F222, Fmm2 and Fmmm are C, A and B groups. The corresponding centring translations are $[w = t({1 \over 2},{1 \over 2},0), u = t(0,{1 \over 2},{1 \over 2})]$ and $[v = w \times u = t({1 \over 2},0,{1 \over 2})]$ .

The (four-line) extended symbols of these groups can be obtained from the following scheme: $[\halign to \hsize{\hskip0pc# &$#$\hfill\hskip2.7pc &$#$\hfill\hskip2.7pc &$#$\hfill\hskip2.7pc &$#$\hfill\cr &&F222\ (22) &Fmm2\ (42) &Fmmm\ (69)\cr \noalign{\vskip3pt}\noalign{\hrule}\cr &1 &222 &mm2 &mmm\cr &w &2_{1}2_{1}2^{w} &ba2^{w} &ban\cr &u &2^{u}2_{1}^{v}2_{1} &nc2_{1} &ncb\cr &v &2_{1}^{u}2^{v}2_{1}^{w} &cn2_{1}^{w} &cna\cr\noalign{\vskip 3pt}\noalign{\hrule}\cr}]$ The second, third, and fourth lines are the result of the multiplication of the first line by the centring translations w, u and v, respectively.

The following abbreviations are used: $[2_{z}^{w} = w \times 2_{z}{\hbox{;}} \quad 2_{1z}^{w} = w \times 2_{1z}{\hbox{;}}\ etc.]$ For the location of the symmetry elements in the above scheme, see Table 4.1.2.3 . In Table 4.3.2.1, the centring translations and the superscripts u, v, w have been omitted. The first two lines of the scheme represent the extended symbols of C222, Cmm2 and Cmmm. An interchange of the symmetry elements in the first two lines does not change the group. To obtain further maximal C subgroups, one has to replace symmetry elements of the first line by corresponding elements of the third or fourth line. Note that the symbol `e' is not used in the four-line symbols for Fmm2 and Fmmm in order to keep the above scheme transparent.

Examples

(1) F222 (22). In the first line replace $[2_{x}]$ by $[2_{x}^{u}]$ (third line, same column) and keep $[2_{y}]$ . Complete the first line by the product $[2_{x}^{u} \times 2_{y} = 2_{1z}]$ and obtain the maximal C subgroup $[C2^{u}22_{1}]$ .

Similarly, in the first line keep $[2_{x}]$ and replace $[2_{y}]$ with $[2_{y}^{v}]$ (fourth line, same column). Complete the first line by the product $[2_{x} \times 2_{y}^{v} = 2_{1z}]$ and obtain the maximal C subgroup $[C22^{v}2_{1}]$ .

Finally, replace $[2_{x}]$ and $[2_{y}]$ by $[2_{x}^{u}]$ and $[2_{y}^{v}]$ and form the product $[2_{x}^{u} \times 2_{y}^{v} = 2_{z}^{w}]$ , to obtain the maximal C subgroup $[C2^{u}2^{v}2^{w}]$ (where $[2^{w}]$ can be replaced by 2). Note that C222 and $[C2^{u}2^{v}2]$ are two different subgroups, as are $[C2^{u}22_{1}]$ and $[C22^{v}2_{1}]$ .
(2) Fmm2 (42). A similar procedure leads to the four maximal k subgroups $[Cmm2{\hbox{;}}\; Cmc2_{1}{\hbox{;}}\; Ccm2_{1}^{w}\ \hbox{(}Cmc2_{1}\hbox{)}]$ ; and Ccc2.
(3) Fmmm (69). One finds successively the eight maximal k subgroups Cmmm; Cmma; Cmcm; Ccmm (Cmcm); Cmca; Ccma (Cmca); Cccm; and Ccca.

Maximal A- and B-centred subgroups can be obtained from the C subgroups by simple symmetry arguments.

In space groups Fdd2 (43) and Fddd (70), the nature of the d planes is not altered by the translations of the F lattice; for this reason, a two-line symbol for Fdd2 and a one-line symbol for Fddd are sufficient. There exist no maximal non-isomorphic k subgroups for these two groups.

4.3.3.2.2. Maximal t subgroups of type I

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(i) Orthorhombic subgroups

The standard full symbol of a P group of class mmm indicates all the symmetry elements, so that maximal t subgroups can be read at once.

Example

$[P\;2_{1}/m2/m2/a]$ (51) has the following four t subgroups: $[P2_{1}22]$ $[(P222_{1})]$ ; ; $[P2_{1}ma]$ $[(Pmc2_{1})]$ ; Pm2a (Pma2).

From the standard full symbol of an I group of class mmm, the t subgroup of class 222 is read directly. It is either I222 [for Immm (71) and Ibam (72)] or $[I2_{1}2_{1}2_{1}]$ [for Ibca (73) and Imma (74)]. Use of the two-line symbols results in three maximal t subgroups of class mm2.

Example

$[\matrix{\noalign{\vskip 13pt}I b a m\hfill\cr c c n\hfill\cr}]$ (72) has the following three maximal t subgroups of class mm2: ; $[Ib2_{1}m]$ ; $[I2_{1}am]$ .

From the standard full symbol of a C group of class mmm, one immediately reads the maximal t subgroup of class 222, which is either $[C222_{1}]$ [for Cmcm (63) and Cmce (64)] or C222 (for all other cases). For the three maximal t subgroups of class mm2, the two-line symbols are used.

Example

$[\matrix{\noalign{\vskip 12pt}C m c e \hfill\cr b n a\hfill\cr}]$ (64) has the following three maximal t subgroups of class mm2: $[Cmc2_{1}]$ ; (); ().

Finally, Fmmm (69) has maximal t subgroups F222 and Fmm2 (plus permutations), whereas Fddd (70) has F222 and Fdd2 (plus permutations).

(ii) Monoclinic subgroups

These subgroups are obtained by substituting the symbol `l' in two of the three positions. Non-standard centred cells are reduced to primitive cells.

Examples

(1) $[C222_{1}]$ (20) has the maximal t subgroups C211 (C2), C121 (C2) and $[C112_{1}]$ . The last one reduces to $[P112_{1}]$ ( $[P2_{1}]$ ).
(2) Ama2 (40) has the maximal t subgroups Am11, reducible to Pm, A1a1 (Cc) and A112 (C2).
(3) Pnma (62) has the standard full symbol $[P2_{1}/n2_{1}/m2_{1}/a]$ , from which the maximal t subgroups $[P2_{1}/n11]$ $[(P2_{1}/c)]$ , $[P12_{1}/m1]$ $[(P2_{1}/m)]$ and $[P112_{1}/a]$ $[(P2_{1}/c)]$ are obtained.
(4) Fddd (70) has the maximal t subgroups , and , each one reducible to .

References

Bertaut, E. F. (1976). Study of principal subgroups and of their general positions in C and I groups of class mmm–D_zh. Acta Cryst. A32, 380–387.Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 4.3, pp. 70-71