International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.3, pp. 780-790

Section 3.3.3. Properties of the international symbols

H. Burzlaffa and H. Zimmermannb*

aUniversität Erlangen–Nürnberg, Robert-Koch-Strasse 4a, D-91080 Uttenreuth, Germany, and bInstitut für Angewandte Physik, Lehrstuhl für Kristallographie und Strukturphysik, Universität Erlangen–Nürnberg, Bismarckstrasse 10, D-91054 Erlangen, Germany
Correspondence e-mail:  helmuth.zimmermann@krist.uni-erlangen.de

3.3.3. Properties of the international symbols

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3.3.3.1. Derivation of the space group from the short symbol

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Because the short international symbol contains a set of generators, it is possible to deduce the space group from it. With the same distinction between generators and indicators as for point groups, the modified point-group symbol directly gives the rotation parts W of the generating operations (W, w).

The modified symbols of the generators determine the glide/screw parts [{\bi w}_{g}] of w. To find the location parts [{\bi w}_{l}] of w, it is necessary to inspect the product relations of the group. The deduction of the set of complete generating operations can be summarized in the following rules:

  • (i) The integral translations are included in the set of generators. If the unit cell has centring points, the centring operations are generators.

  • (ii) The location parts of the generators can be set to zero except for the two cases noted under (iii)[link] and (iv)[link].

  • (iii) For non-cubic rotation groups with indicators in the symbol, the location part of the first generator can be set to zero. The location part of the second generator is [{\bi w}_{l} = \pmatrix{0\cr 0\cr -m/n}]; the intersection parameter [-m/n] is derived from the indicator [n_{m}] in the [001] direction [cf. example (3) below].

  • (iv) For cubic rotation groups, the location part of the threefold rotation can be set to zero. For space groups related to the point group 23, the location part of the twofold rotation is [{\bi w}_{l} = \pmatrix{-m/n\cr 0\cr 0}] derived from the symbol [n_{m}] of the twofold operation itself. For space groups related to the point group 432, the location part of the twofold generating rotation is [{\bi w}_{l} = \pmatrix{-m/n\cr m/n\cr m/n}] derived from the indicator [n_{m}] in the [001] direction [cf. examples (4) and (5) below].

The origin that is selected by these rules is called the `origin of the symbol' (Burzlaff & Zimmermann, 1980[link]). It is evident that the reference to the origin of the symbol allows a very short and unique notation of all desirable origins by appending the components of the origin of the symbol [{\bf q} = ( q_{1},q_{2},q_{3})] to the short space-group symbol, thus yielding the so-called expanded Hermann–Mauguin symbol. The shift of origin can be performed easily, for only the translation parts have to be changed. The components of the transformed translation part can be obtained by [cf. Section 1.5.2.3[link] and equation (1.5.2.13)[link] ][{\bi w}' = {\bi w} + ({\bi W} - {\bi I}) {\bi q}.] Applications can be found in Burzlaff & Zimmermann (2002[link]).

Examples: Deduction of the generating operations from the short symbol

Some examples for the use of these rules are now described in detail. It is convenient to describe the symmetry operation (W, w) by the corresponding coordinate triplets, i.e. using the so-called shorthand notation, cf. Section 1.2.2[link] . The coordinate triplets can be interpreted as combinations of two constituents: the first one consists of the coordinates of a point in general position after the application of W on x, y, z, while the second corresponds to the translation part w of the symmetry operation. The coordinate triplets of the symmetry operations are tabulated as the general position in the space-group tables (in some cases a shift of origin is necessary). If preference is given to full matrix notation, Table 1.2.2.1[link] may be used. The following examples contain, besides the description of the symmetry operations, references to the numbering of the general positions in the space-group tables of this volume; cf. Sections 2.1.3.9[link] and 2.1.3.11[link] . Centring translations are written after the numbers, if necessary.

  • (1) [Pccm = D_{2h}^{3}\ (49)]

    Besides the integral translations, the generators, as given in the symbol, are according to rule (ii)[link]:[\let\normalbaselines\relax\openup3pt\matrix{\hbox{glide reflection} \ c_{100} {:}\hfill & \overline{x}, y, z+ \textstyle{1\over 2}\hfill &(8)\hfill\ \cr \hbox{glide reflection} \ c_{010} {:}\hfill & x,\overline{y}, z+ \textstyle{1\over 2} \hfill&(7)\hfill \cr \hbox{reflection} \phantom{glide\ } m_{001} {:}\hfill &x,y,\overline{z}\hfill&(6).\hfill}]No shift of origin is necessary. The expanded symbol is [Pccm \langle 000 \rangle].

  • (2) [Ibam = D_{2h}^{26}\ (72)]

    According to rule (i),[link] the I centring is an additional generating translation. Thus, the generators are:[\let\normalbaselines\relax\openup3pt\matrix{I \hbox{ centring}{:}\hfill &x+\textstyle{1\over 2},y+ \textstyle{1\over 2},z+\textstyle{1\over 2}\hfill & (1)+(\textstyle{1\over 2}, \textstyle{1\over 2}, \textstyle{1\over 2})\hfill\cr \hbox{glide reflection } b_{100}{:}\hfill & \overline{x},y+ \textstyle{1\over 2},z\hfill &(8)\hfill \cr \hbox{glide reflection } a_{010}{:}\hfill & x+\textstyle{1\over 2},\overline{y},z \hfill &(7)\hfill \cr \hbox{reflection } \phantom{tion\,\,}\ m_{001}{:}\hfill & x,y,\overline{z} \hfill &(6).\hfill}]To obtain the tabulated general position, a shift of origin by [(-{\textstyle{1\over 4}},\! -{\textstyle{1\over 4}}, 0)] is necessary, the expanded symbol is [Ibam \langle - \textstyle{1\over 4} - \textstyle{1\over 4} 0 \rangle].

  • (3) [P4_{1}2_{1}2 = D_{4}^{4}\ (92)]

    Apart from the translations, the generating elements are:[\let\normalbaselines\relax\openup3pt\matrix{\hbox{screw rotation } 2_{1} \hbox{ in } [100]{:}\hfill& x+\textstyle{1\over 2},\overline{y},\overline{z}\hfill & (6)\hfill\cr \hbox{rotation } \phantom{\ screw} 2\ \hbox{ in }\ [1\overline{1}0]{\hbox to .5pt{}}{:}\hfill& \overline{y},\overline{x},\overline{z}+{\textstyle{1\over 4}} \hfill& (8).\hfill}]

    According to rule (iii)[link], the location part of the first generator, referring to the secondary set of symmetry directions, is equal to zero. For the second generator, the screw part is equal to zero. The location part is [{\bi w}_l = \pmatrix{0\cr 0\cr -\textstyle{1\over 4}}]. The expanded symbol [P4_{1}2_{1}2 \langle \textstyle{1\over 4} -\textstyle{1\over 4} -{3\over8} \rangle] gives the tabulated setting.

  • (4) [P2_{1}3 = T^{4}\ (198)]

    According to rule (iv)[link], the generators are[\let\normalbaselines\relax\openup3pt\matrix{\hbox{rotation 3 in } [111]{:}\hfill & z,x,y \hfill & (5)\hfill\cr \hbox{screw rotation } 2_{1} \hbox{ in } [001]{:}\hfill & \overline{x}+\textstyle{1\over 2},\overline{y},z+\textstyle{1\over 2} \hfill &(2).\hfill}]

    Following rule (iv)[link], the location part of the threefold axis must be set to zero. The screw part of the twofold axis in [001] is [\pmatrix{0\cr 0\cr \textstyle{1\over 2}}], the location part is [{\bi w}_l = \pmatrix{-\textstyle{1\over 2}\cr 0\cr 0} \equiv \pmatrix{\textstyle{1\over 2}\cr 0\cr 0}]. No origin shift is necessary. The expanded symbol is [P2_{1}3 \langle 000 \rangle].

  • (5) [P4_{1}32 = O^{7}\ (213)]

    Besides the integral translations, the generators given by the symbol are:[\let\normalbaselines\relax\openup3pt\matrix{\hbox{rotation 3 in } [111] {:}\hfill& z,x,y\hfill &(5)\hfill\cr \hbox{rotation 2 in } [110]{:}\hfill &y-\textstyle{1\over 4} ,x+\textstyle{1\over 4},\overline{z}+ \textstyle{1\over 4} \hfill& (13).\hfill}]The screw part of the twofold axis is zero. According to rule (iv)[link], the location part is [{\bi w}_l = \let\normalbaselines\relax\openup2pt\pmatrix{-\textstyle{1\over 4}\cr \textstyle{1\over 4}\cr \textstyle{1\over 4}}]. No origin shift is necessary. The expanded symbol is [P4_{1}32 \langle 000 \rangle].

3.3.3.2. Derivation of the full symbol from the short symbol

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If the geometrical point of view is again considered, it is possible to derive the full international symbol for a space group. This full symbol can be interpreted as consisting of symmetry elements. It can be generated from the short symbol with the aid of products between symmetry operations. It is possible, however, to derive the glide/screw parts of the elements in the full symbol directly from the glide/screw parts of the short symbol.

The product of operations corresponding to non-parallel glide or mirror planes generates a rotation or screw axis parallel to the intersection line. The screw part of the rotation is equal to the sum of the projections of the glide components of the planes on the axis. The angle between the planes determines the rotation part of the axis. For 90°, we obtain a twofold, for 60° a threefold, for 45° a fourfold and for 30° a sixfold axis.

Example: [Pbcn = D_{2h}^{14}\ (60)]

The product of b and c generates a screw axis [2_{1}] in the z direction because the sum of the glide components in the z direction is [\textstyle{1\over 2}]. The product of c and n generates a screw axis [2_{1}] in the x direction and the product between b and n produces a rotation axis 2 in the y direction because the y components for b and n add up to [1 \equiv 0] modulo integers.

Thus, the full symbol is[P {2_{1} \over b} {2 \over c} {2_{1} \over n}.]

In most cases, the full symbol is identical with the short symbol; differences between full and short symbols can only occur for space groups corresponding to lattice point groups (holohedries) and to the point group [m\overline{3}]. In all these cases, the short symbol is extended to the full symbol by adding the symbol for the maximal purely rotational subgroup. A special procedure is in use for monoclinic space groups. To indicate the choice of coordinate axes, the full symbol is treated like an orthorhombic symbol, in which the directions without symmetry are indicated by `1', even though they do not correspond to lattice symmetry directions in the monoclinic case.

3.3.3.3. Non-symbolized symmetry elements

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Certain symmetry elements are not given explicitly in the full symbol because they can easily be derived. They are:

  • (i) Rotoinversion axes that are not used to indicate the lattice symmetry directions.

  • (ii) Rotation axes 2 included in the axes 4, [\overline{4}] and 6 and rotation axes 3 included in the axes [\overline{3}], 6 and [\overline{6}].

  • (iii) Additional symmetry elements occurring in space groups with centred unit cells, cf. Sections 1.4.2.4[link] and 1.5.4.1[link] . These types of operation can be deduced from the product of the centring translation (I, g) with a symmetry operation (W, w). The new symmetry operation [({\bi W},\; {\bi g} + {\bi w})] again has W as rotation part but a different glide/screw part if the component of g parallel to the symmetry element corresponding to W is not a lattice vector; cf. Section 1.5.4.1[link] .

    Example

    Space group [C2/c\ (15)] has a twofold axis along b with screw part [{\bi w}_{g} = \pmatrix{0\cr 0\cr 0}]. The translational part of the centring operation is [{\bi g} = \let\normalbaselines\relax\openup2pt\pmatrix{\textstyle{1\over 2}\cr \textstyle{1\over 2}\cr 0}].

    An additional axis parallel to b thus has a translation part [{\bi g} + {\bi w}_{g} = \let\normalbaselines\relax\openup2pt\pmatrix{\textstyle{1\over 2}\cr \textstyle{1\over 2}\cr 0}]. The component [\pmatrix{0\cr \textstyle{1\over 2}\cr 0}] indicates a screw axis [2_{1}] in the b direction, whereas the component [\pmatrix{\textstyle{1\over 2}\cr 0\cr 0}] indicates the location of this axis in [\textstyle{1\over 4},y,0]. Similarly, it can be shown that glide plane c combined with the centring gives a glide plane n.

    In the same way, in rhombohedral and cubic space groups, a rotation axis 3 is accompanied by screw axes [3_{1}] and [3_{2}].

    In space groups with centred unit cells, the location parts of different symmetry elements may coincide. In [I\overline{4}2m], for example, the mirror plane m contains simultaneously a non-symbolized glide plane n. The same applies to all mirror planes in Fmmm.

  • (iv) Symmetry elements with diagonal orientation always occur with different types of glide/screw parts simultaneously. In space group [P\overline{4}2m] (111) the translation vector along a can be decomposed as[{\bi w} = \pmatrix{1\cr 0\cr 0} = \let\normalbaselines\relax\openup2pt\pmatrix{\textstyle{1\over 2}\cr \textstyle{1\over 2}\cr 0} + \pmatrix{\textstyle{1\over 2}\cr -\textstyle{1\over 2}\cr 0} = {\bi w}_{g} + {\bi w}_{l}.]The diagonal mirror plane with normal along [[1\overline{1}0]] passing through the origin is accompanied by a parallel glide plane with glide part [\let\normalbaselines\relax\openup2pt\pmatrix{\textstyle{1\over 2}\cr \textstyle{1\over 2}\cr 0}] passing through [\textstyle{1\over 4},-\textstyle{1\over 4},0]. The same arguments lead to the occurrence of screw axes [2_{1}], [3_{1}] and [3_{2}] connected with diagonal rotation axes 2 or 3.

  • (v) For some investigations connected with klassengleiche subgroups (for subgroups of space groups, cf. Section 1.7.1[link] ), it is convenient to introduce an extended Hermann–Mauguin symbol that comprises all symmetry elements indicated in (iii)[link] and (iv)[link]. The basic concept may be found in papers by Hermann (1929[link]) and in IT (1952[link]). These concepts have been applied by Bertaut (1976[link]) and Zimmermann (1976[link]); cf. Section 1.5.4.1[link] .

Example

The full symbol of space group Imma (74) is[I{2_{1} \over m} {2_{1} \over m} {2_{1} \over a}.]

The I-centring operation introduces additional rotation axes and glide planes for all three sets of lattice symmetry directions. The extended Hermann–Mauguin symbol is[I{2,2_{1} \over m,n} {2,2_{1} \over m,n} {2,2_{1} \over a,b} \quad \hbox{or} \quad I\openup2pt\matrix{\displaystyle{2_{1} \over m} &\displaystyle{2_{1} \over m} &\displaystyle{2_{1} \over a}\cr \displaystyle{2 \over n} &\displaystyle{2 \over n} &\displaystyle{2 \over b}\cr}.] This symbol shows immediately the eight subgroups with a P lattice corresponding to point group mmm:[\displaylines{Pmma \sim Pmmb,\quad Pnma \sim Pmnb,\quad Pmna \sim Pnmb\quad \hbox{and}\cr \quad Pnna \sim Pnnb.\hfill}]

3.3.3.4. Standardization rules for short symbols

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The symbols of Bravais lattices and glide planes depend on the choice of basis vectors. As shown in the preceding section, additional translation vectors in centred unit cells produce new symmetry operations with the same rotation but different glide/screw parts. Moreover, it was shown that for diagonal orientations symmetry operations may be represented by different symbols. Thus, different short symbols for the same space group can be derived even if the rules for the selection of the generators and indicators are obeyed.

For the unique designation of a space-group type, a standardization of the short symbol is necessary. Rules for standardization were given first by Hermann (1931[link]) and later in a slightly modified form in IT (1952[link]).

These rules, which are generally followed in the present tables, are given below. Because of the historical development of the symbols (cf. Section 3.3.4[link]), some of the present symbols do not obey the rules, whereas others depending on the crystal class need additional rules for them to be uniquely determined. These exceptions and additions are not explicitly mentioned, but may be discovered from Table 3.3.3.1[link] in which the short symbols are listed for all space groups. A table for all settings may be found in Section 1.5.4[link] .

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Standard space-group symbols

No.Schoenflies symbolShubnikov symbolSymbols of International TablesComments
1935 EditionPresent Edition
ShortFullShortFull
1 [C_{1}^{1}] [(a/b/c)\cdot 1] P1 P1 P1 P1  
2 [C_{i}^{1}] [(a/b/c)\cdot \overline{2}] [P\overline{1}] [P\overline{1}] [P\overline{1}] [P\overline{1}] [(a/b/c)\cdot \overline{1}] (Sh–K)
3 [C_{2}^{1}] [(b\!:\!(c/a))\!:\!2] P2 P2 P2 P121  
    [(c\!:\!(a/b))\!:\!2]       P112  
4 [C_{2}^{2}] [(b\!:\!(c/a))\!:\!2_{1}] [P2_{1}] [P2_{1}] [P2_{1}] [P12_{1}1]  
    [(c\!:\!(a/b))\!:\!2_{1}]       [P112_{1}]  
5 [C_{2}^{3}] [\left(\displaystyle{a + b \over 2}\bigg/b\!:\!(c/a)\right)\!:\!2] C2 C2 C2 C121 B2, B112 (IT, 1952[link])
    [\left(\displaystyle{b + c \over 2}\bigg/c\!:\!(b/a)\right)\!:\!2]       A112 [\left(\displaystyle{a + c \over 2}\bigg/c\!:\!(a/b)\right)\!:\!2] (Sh–K)
6 [C_{2}^{1}] [(b\!:\!(c/a))\cdot m] Pm Pm Pm P1m1  
    [(c\!:\!(a/b))\cdot m]       P11m  
7 [C_{s}^{2}] [(b\!:\!(c/a))\cdot \tilde{c}] Pc Pc Pc P1c1 Pb, P11b (IT, 1952[link])
    [(c\!:\!(b/a))\cdot \tilde{a}]       P11a [(c\!:\!(a/b))\cdot \tilde{b}] (Sh–K)
8 [C_{s}^{3}] [\left(\displaystyle{a + b \over 2}\bigg/b\!:\!(c/a)\right)\cdot m] Cm Cm Cm C1m1 Bm, B11m (IT, 1952[link])
    [\left(\displaystyle{b + c \over 2}\bigg/c\!:\!(b/a)\right)\cdot m]       A11m [\left(\displaystyle{a + c \over 2}\bigg/c\!:\!(a/b)\right)\cdot m] (Sh–K)
9 [C_{s}^{4}] [\left(\displaystyle{a + b \over 2}\bigg/b\!:\!(c/a)\right)\cdot \tilde{c}] Cc Cc Cc C1c1 Bb, B11b (IT, 1952[link])
    [\left(\displaystyle{b + c \over 2}\bigg/c\!:\!(b/a)\right)\cdot \tilde{a}]       A11a [\left(\displaystyle{a + c \over 2}\bigg/c\!:\!(a/b)\right)\cdot \tilde{b}] (Sh–K)
10 [C_{2h}^{1}] [(b\!:\!(c/a))\cdot m\!:\!2] [P2/m] [P2/m] [P2/m] [P1\ {2/m}1]  
    [(c\!:\!(a/b))\cdot m\!:\!2]       [P11\ 2/m]  
11 [C_{2h}^{2}] [(b\!:\!(c/a))\cdot m\!:\!2_{1}] [P2_{1}/m] [P2_{1}/m] [P2_{1}/m] [P1\ 2_{1}/m\ 1]  
    [(c\!:\!(a/b))\cdot m\!:\!2_{1}]       [P11\ 2_{1}/m]  
12 [C_{2h}^{3}] [\left(\displaystyle{a + b \over 2}\bigg/b\!:\!(c/a)\right)\cdot m\!:\!2] [C2/m] [C2/m] [C2/m] [C1\ 2/m\ 1] [B2/m, B11\ 2/m] (IT, 1952[link])
    [\left(\displaystyle{b + c \over 2}\bigg/c\!:\!(b/a)\right)\cdot m\!:\!2]       [A11\ 2/m] [\left(\displaystyle{a + c \over 2}\bigg/c\!:\!(a/b)\right)\cdot m\!:\!2] (Sh–K)
13 [C_{2h}^{4}] [(b\!:\!(c/a))\cdot \tilde{c}\!:\!2] [P2/c] [P2/c] [P2/c] [P1\ 2/c\ 1] [P2/b, P11\ 2/b] (IT, 1952[link])
    [(c\!:\!(a/b))\cdot \tilde{a}\!:\!2]       [P11\ 2/a] [(c\!:\!(a/b))\cdot \tilde{b}\!:\!2] (Sh–K)
14 [C_{2h}^{5}] [(b\!:\!(c/a))\cdot \tilde{c}\!:\!2_{1}] [P2_{1}/c] [P2_{1}/c] [P2_{1}/c] [P1\ 2_{1}/c\ 1] [P2_{1}/b,P112_{1}/b] (IT, 1952[link])
    [(c\!:\!(a/b))\cdot \tilde{a}\!:\!2_{1}]       [P11\ 2_{1}/a] [(c\!:\!(a/b))\cdot b\!:\!2_{1}] (Sh–K)
15 [C_{2h}^{6}] [\left(\displaystyle{a + b \over 2}\bigg/b\!:\!(c/a)\right)\cdot \tilde{c}\!:\!2] [C2/c] [C2/c] [C2/c] [C1\ 2/c\ 1] [B2/b, B11\ 2/b] (IT, 1952[link])
    [\left(\displaystyle{b + c \over 2}\bigg/c\!:\!(b/a)\right)\cdot \tilde{a}\!:\!2]       A11 2a [\left(\displaystyle{a + c \over 2}\bigg/c\!:\!(a/b)\right)\cdot \tilde{b}\!:\!2] (Sh–K)
16 [D_{2}^{1}] [(c\!:\!(a\!:\!b))\!:\!2\!:\!2] P222 P222 P222 P222  
17 [D_{2}^{2}] [(c\!:\!(a\!:\!b))\!:\!2_{1}\!:\!2] [P222_{1}] [P222_{1}] [P222_{1}] [P222_{1}]  
18 [D_{2}^{3}] [\def\circcol{\mathop{\bigcirc\hskip -5pt{\raise.05pt\hbox{$\!:\!$}} \ }} (c\!:\!(a\!:\!b))\!:\!2\circcol\,2_{1}] [P2_{1}2_{1}2] [P2_{1}2_{1}2] [P2_{1}2_{1}2] [P2_{1}2_{1}2]  
19 [D_{2}^{4}] [\def\circcol{\mathop{\bigcirc\hskip -5pt{\raise.05pt\hbox{$\!:\!$}} \ }} (c\!:\!(a\!:\!b))\!:\!2_{1}\circcol\, 2_{1}] [P2_{1}2_{1}2_{1}] [P2_{1}2_{1}2_{1}] [P2_{1}2_{1}2_{1}] [P2_{1}2_{1}2_{1}]  
20 [D_{2}^{5}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\!:\!2_{1}\!:\!2] [C222_{1}] [C222_{1}] [C222_{1}] [C222_{1}]  
21 [D_{2}^{6}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\!:\!2\!:\!2] C222 C222 C222 C222  
22 [D_{2}^{7}] [\displaylines{\left(\displaystyle{a + c \over 2}\bigg/{b + c \over 2}\bigg/{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\hfill\cr \quad :\!2\!:\!2\hfill}] F222 F222 F222 F222  
23 [D_{2}^{8}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!2\!:\!2] I222 I222 I222 I222  
24 [D_{2}^{9}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!2\!:\!2_{1}] [I2_{1}2_{1}2_{1}] [I2_{1}2_{1}2_{1}] [I2_{1}2_{1}2_{1}] [I2_{1}2_{1}2_{1}]  
25 [C_{2v}^{1}] [(c\!:\!(a\!:\!b))\!:\!m\cdot 2] Pmm Pmm2 Pmm2 Pmm2  
26 [C_{2v}^{2}] [(c\!:\!(a\!:\!b))\!:\!\tilde{c}\cdot 2_{1}] Pmc [Pmc2_{1}] [Pmc2_{1}] [Pmc2_{1}]  
27 [C_{2v}^{3}] [(c\!:\!(a\!:\!b))\!:\!\tilde{c}\cdot 2] Pcc Pcc2 Pcc2 Pcc2  
28 [C_{2v}^{4}] [(c\!:\!(a\!:\!b))\!:\!\tilde{a}\cdot 2] Pma Pma2 Pma2 Pma2  
29 [C_{2v}^{5}] [(c\!:\!(a\!:\!b))\!:\!\tilde{a}\cdot 2_{1}] Pca [Pca2_{1}] [Pca2_{1}] [Pca2_{1}]  
30 [C_{2v}^{6}] [(c\!:\!(a\!:\!b))\!:\!\tilde{c} \bigodot 2] Pnc Pnc2 Pnc2 Pnc2 [(c\!:\!(a\!:\!b))\!:\!\widetilde{ac}\cdot 2] (Sh–K)
31 [C_{2v}^{7}] [(c\!:\!(a\!:\!b))\!:\!\widetilde{ac}\cdot 2_{1}] Pmn [Pmn2_{1}] [Pmn2_{1}] [Pmn2_{1}]  
32 [C_{2v}^{8}] [(c\!:\!(a\!:\!b))\!:\!\tilde{a}\bigodot 2] Pba Pba2 Pba2 Pba2  
33 [C_{2v}^{9}] [(c\!:\!(a\!:\!b))\!:\!\tilde{a}\bigodot 2_{1}] Pna [Pna2_{1}] [Pna2_{1}] [Pna2_{1}]  
34 [C_{2v}^{10}] [(c\!:\!(a\!:\!b))\!:\!\widetilde{ac}\bigodot 2] Pnn Pnn2 Pnn2 Pnn2  
35 [C_{2v}^{11}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\!:\!m\cdot 2] Cmm Cmm2 Cmm2 Cmm2  
36 [C_{2v}^{12}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\!:\!\tilde{c}\cdot 2_{1}] Cmc [Cmc2_{1}] [Cmc2_{1}] [Cmc2_{1}]  
37 [C_{2v}^{13}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\!:\!\tilde{c}\cdot 2] Ccc Ccc2 Ccc2 Ccc2  
38 [C_{2v}^{14}] [\left(\displaystyle{b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!m\cdot 2] Amm Amm2 Amm2 Amm2  
39 [C_{2v}^{15}] [\left(\displaystyle{b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!m\cdot 2_{1}] Abm Abm2 Aem2 Aem2 [\cases{\!\!\left(\displaystyle{b + c \over 2}\!\big/\!c\!:\!(a\!:\!b)\right)\!:\!\tilde{c}\cdot 2\cr\quad (\rm{Sh\!-\!K})\cr \hbox{Use former symbol}\cr Abm2\ \hbox{for generation}\cr}]
40 [C_{2v}^{16}] [\left(\displaystyle{b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!\tilde{a}\cdot 2] Ama Ama2 Ama2 Ama2  
41 [C_{2v}^{17}] [\left(\displaystyle{b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!\tilde{a}\cdot 2_{1}] Aba Aba2 Aea2 Aea2 [\cases{\!\!\left(\displaystyle{b + c \over 2}\!\big/c\!:\!(a\!:\!b)\right)\!:\!\widetilde{ac}\cdot 2 \cr\quad (\rm{Sh\!-\!K})\cr \hbox{Use former symbol}\cr Aba2\ \hbox{for generation}\cr}]
42 [C_{2v}^{18}] [\displaylines{\left(\displaystyle{a + c \over 2}\bigg/{b + c \over 2}\bigg/{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\hfill\cr\quad:m\cdot 2\hfill}] Fmm Fmm2 Fmm2 Fmm2  
43 [C_{2v}^{19}] [\left(\displaystyle{a + c \over 2}\bigg/{b + c \over 2}\bigg/{a + b \over 2}\!:\!\tilde{c}\!:\!(a\!:\!b)\right)][:{1\over 2}\widetilde{ac}\bigodot 2] Fdd Fdd2 Fdd2 Fdd2  
44 [C_{2v}^{20}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!m\cdot 2] Imm Imm2 Imm2 Imm2  
45 [C_{2v}^{21}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!\tilde{c}\cdot 2] Iba Iba2 Iba2 Iba2 [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!\tilde{a}\cdot 2_{1}](Sh–K)
46 [C_{2v}^{22}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!\tilde{a}\cdot 2] Ima Ima2 Ima2 Ima2  
47 [D_{2h}^{1}] [(c\!:\!(a\!:\!b))\cdot m\!:\!2\cdot m] Pmmm [P2/m\ 2/m\ 2/m] Pmmm [P2/m\ 2/m\ 2/m]  
48 [D_{2h}^{2}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot \widetilde{ab}\!:\!2\circdot \widetilde{ac}] Pnnn [P2/n\ 2/n\ 2/n] Pnnn [P2/n\ 2/n\ 2/n]  
49 [D_{2h}^{3}] [(c\!:\!(a\!:\!b))\cdot m\!:\!2\cdot \tilde{c}] Pccm [P2/c\ 2/c\ 2/m] Pccm [P2/c\ 2/c\ 2/m]  
50 [D_{2h}^{4}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot \widetilde{ab}\!:\!2\circdot \tilde{a}] Pban [P2/b\ 2/a\ 2/n] Pban [P2/b\ 2/a\ 2/n]  
51 [D_{2h}^{5}] [(c\!:\!(a\!:\!b))\cdot \tilde{a}\!:\!2\cdot m] Pmma [P2_{1}/m\ 2/m\ 2/a] Pmma [P2_{1}/m\ 2/m\ 2/a]  
52 [D_{2h}^{6}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot \tilde{a}\!:\!2\circdot \widetilde{ac}] Pnna [P2/n\ 2_{1}/n\ 2/a] Pnna [P2/n\ 2_{1}/n\ 2/a]  
53 [D_{2h}^{7}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot \tilde{a}\!:\!2_{1}\cdot \widetilde{ac}] Pmna [P2/m\ 2/n\ 2_{1}/a] Pmna [P2/m\ 2/n\ 2_{1}/a]  
54 [D_{2h}^{8}] [(c\!:\!(a\!:\!b))\cdot \tilde{a}\!:\!2\cdot \tilde{c}] Pcca [P2_{1}/c\ 2/c\ 2/a] Pcca [P2_{1}/c\ 2/c\ 2/a]  
55 [D_{2h}^{9}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot m\!:\!2\circdot \tilde{a}] Pbam [P2_{1}/b\ 2_{1}/a\ 2/m] Pbam [P2_{1}/b\ 2_{1}/a\ 2/m]  
56 [D_{2h}^{10}] [(c\!:\!(a\!:\!b))\cdot \widetilde{ab}\!:\!2\cdot \tilde{c}] Pccn [P2_{1}/c\ 2_{1}/c\ 2/n] Pccn [P2_{1}/c\ 2_{1}/c\ 2/n]  
57 [D_{2h}^{11}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot m\!:\!2_{1}\circdot \tilde{c}] Pbcm [P2/b\ 2_{1}/c\ 2_{1}/m] Pbcm [P2/b\ 2_{1}/c\ 2_{1}/m]  
58 [D_{2h}^{12}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot m\!:\!2\circdot \widetilde{ac}] Pnnm [P2_{1}/n\ 2_{1}/n\ 2/m] Pnnm [P2_{1}/n\ 2_{1}/n\ 2/m]  
59 [D_{2h}^{13}] [(c\!:\!(a\!:\!b))\cdot \widetilde{ab}\!:\!2\cdot m] Pmmn [P2_{1}/m\ 2_{1}/m\ 2/n] Pmmn [P2_{1}/m\ 2_{1}/m\ 2/n]  
60 [D_{2h}^{14}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot \widetilde{ab}\!:\!2_{1}\circdot \tilde{c}] Pbcn [P2_{1}/b\ 2/c\ 2_{1}/n] Pbcn [P2_{1}/b\ 2/c\ 2_{1}/n]  
61 [D_{2h}^{15}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot \tilde{a}\!:\!2_{1}\circdot \tilde{c}] Pbca [P2_{1}/b\ 2_{1}/c\ 2_{1}/a] Pbca [P2_{1}/b\ 2_{1}/c\ 2_{1}/a]  
62 [D_{2h}^{16}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot \tilde{a}\!:\!2_{1}\circdot m] Pnma [P2_{1}/n\ 2_{1}/m\ 2_{1}/a] Pnma [P2_{1}/n\ 2_{1}/m\ 2_{1}/a]  
63 [D_{2h}^{17}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\cdot m\!:\!2_{1}\cdot \tilde{c}] Cmcm [C2/m\ 2/c\ 2_{1}/m] Cmcm [C2/m\ 2/c\ 2_{1}/m]  
64 [D_{2h}^{18}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\cdot \tilde{a}\!:\!2_{1}\cdot \tilde{c}] Cmca [C2/m\ 2/c\ 2_{1}/a] Cmce [C2/m\ 2/c\ 2_{1}/e] Use former symbol Cmca for generation
65 [D_{2h}^{19}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\cdot m\!:\!2\cdot m] Cmmm [C2/m\ 2/m\ 2/m] Cmmm [C2/m\ 2/m\ 2/m]  
66 [D_{2h}^{20}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\cdot m\!:\!2\cdot \tilde{c}] Cccm [C2/c\ 2/c\ 2/m] Cccm [C2/c\ 2/c\ 2/m]  
67 [D_{2h}^{21}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\cdot \tilde{a}\!:\!2\cdot m] Cmma [C2/m\ 2/m\ 2/a] Cmme [C2/m\ 2/m\ 2/e] Use former symbol Cmma for generation
68 [D_{2h}^{22}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\cdot \tilde{a}\!:\!2\cdot \tilde{c}] Ccca [C2/c\ 2/c\ 2/a] Ccce [C2/c\ 2/c\ 2/e] Use former symbol Ccca for generation
69 [D_{2h}^{23}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\hfill\cr \quad\cdot \,m\!:\!2\cdot m\hfill\cr}] Fmmm [F2/m\ 2/m\ 2/m] Fmmm [F2/m\ 2/m\ 2/m]  
70 [D_{2h}^{24}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} \displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\hfill\cr \quad\cdot\, {\textstyle{1 \over 2}}\widetilde{ab}\!:\!2\circdot {\textstyle{1 \over 2}}\widetilde{ac}\hfill\cr}] Fddd [F2/d\ 2/d\ 2/d] Fddd [F2/d\ 2/d\ 2/d]  
71 [D_{2h}^{25}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\cdot m\!:\!2\cdot m] Immm [I2/m\ 2/m\ 2/m] Immm I2/m 2/m 2/m  
72 [D_{2h}^{26}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\cdot m\!:\!2\cdot \tilde{c}] Ibam [I2/b\ 2/a\ 2/m] Ibam [I2/b\ 2/a\ 2/m]  
73 [D_{2h}^{27}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\cdot \tilde{a}\!:\!2\cdot \tilde{c}] Ibca [I2_{1}/b\ 2_{1}/c\ 2_{1}/a] Ibca [I2_{1}/b\ 2_{1}/c\ 2_{1}/a] [I2/b\ 2/c\ 2/a] (IT, 1952[link])
74 [D_{2h}^{28}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\cdot \tilde{a}\!:\!2\cdot m] Imma [I2_{1}/m\ 2_{1}/m\ 2_{1}/a] Imma [I2_{1}/m\ 2_{1}/m\ 2_{1}/a] [I2/m\ 2/m\ 2/a] (IT, 1952[link])
75 [C_{4}^{1}] [(c\!:\!(a\!:\!a))\!:\!4] P4 P4 P4 P4  
76 [C_{4}^{2}] [(c\!:\!(a\!:\!a))\!:\!4_{1}] [P4_{1}] [P4_{1}] [P4_{1}] [P4_{1}]  
77 [C_{4}^{3}] [(c\!:\!(a\!:\!a))\!:\!4_{2}] [P4_{2}] [P4_{2}] [P4_{2}] [P4_{2}]  
78 [C_{4}^{4}] [(c\!:\!(a\!:\!a))\!:\!4_{3}] [P4_{3}] [P4_{3}] [P4_{3}] [P4_{3}]  
79 [C_{4}^{5}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!4] I4 I4 I4 I4  
80 [C_{4}^{6}] [\left(\displaystyle{a - b - c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!4_{1}] [I4_{1}] [I4_{1}] [I4_{1}] [I4_{1}]  
81 [S_{4}^{1}] [(c\!:\!(a\!:\!a))\!:\!\tilde{4}] [P\overline{4}] [P\overline{4}] [P\overline{4}] [P\overline{4}]  
82 [S_{4}^{2}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!\tilde{4}] [I\overline{4}] [I\overline{4}] [I\overline{4}] [I\overline{4}]  
83 [C_{4h}^{1}] [(c\!:\!(a\!:\!a))\cdot m\!:\!4] [P4/m] [P4/m] [P4/m] [P4/m]  
84 [C_{4h}^{2}] [(c\!:\!(a\!:\!a))\cdot m\!:\!4_{2}] [P4_{2}/m] [P4_{2}/m] [P4_{2}/m] [P4_{2}/m]  
85 [C_{4h}^{3}] [(c\!:\!(a\!:\!a))\cdot \widetilde{ab}\!:\!4] [P4/n] [P4/n] [P4/n] [P4/n]  
86 [C_{4h}^{4}] [(c\!:\!(a\!:\!a))\cdot \widetilde{ab}\!:\!4_{2}] [P4_{2}/n] [P4_{2}/n] [P4_{2}/n] [P4_{2}/n]  
87 [C_{4h}^{5}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\cdot m\!:\!4] [I4/m] [I4/m] [I4/m] [I4/m]  
88 [C_{4h}^{6}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\cdot \tilde{a}\!:\!4_{1}] [I4_{1}/a] [I4_{1}/a] [I4_{1}/a] [I4_{1}/a]  
89 [D_{4}^{1}] (c:(a:a)):4:2 P42 P422 P422 P422  
90 [D_{4}^{2}] [\def\circcol{\mathop{\bigcirc\hskip -5pt{\raise.05pt\hbox{$\!:\!$}} \ }} (c\!:\!(a\!:\!a))\!:\!4 \circcol\, 2_{1}] [P42_{1}] [P42_{1}2] [P42_{1}2] [P42_{1}2]  
91 [D_{4}^{3}] [(c\!:\!(a\!:\!a))\!:\!4_{1}\!:\!2] [P4_{1}2] [P4_{1}22] [P4_{1}22] [P4_{1}22]  
92 [D_{4}^{4}] [\def\circcol{\mathop{\bigcirc\hskip -5pt{\raise.05pt\hbox{$\!:\!$}} \ }} (c\!:\!(a\!:\!a))\!:\!4_{1}\circcol\, 2_{1}] [P4_{1}2_{1}] [P4_{1}2_{1}2] [P4_{1}2_{1}2] [P4_{1}2_{1}2]  
93 [D_{4}^{5}] [(c\!:\!(a\!:\!a))\!:\!4_{2}\!:\!2] [P4_{2}2] [P4_{2}22] [P4_{2}22] [P4_{2}22]  
94 [D_{4}^{6}] [\def\circcol{\mathop{\bigcirc\hskip -5pt{\raise.05pt\hbox{$\!:\!$}} \ }} (c\!:\!(a\!:\!a))\!:\!4_{2}\circcol\, 2_{1}] [P4_{2}2_{1}] [P4_{2}2_{1}2] [P4_{2}2_{1}2] [P4_{2}2_{1}2]  
95 [D_{4}^{7}] [(c\!:\!(a\!:\!a))\!:\!4_{3}\!:\!2] [P4_{3}2] [P4_{3}22] [P4_{3}22] [P4_{3}22]  
96 [D_{4}^{8}] [\def\circcol{\mathop{\bigcirc\hskip -5pt{\raise.05pt\hbox{$\!:\!$}} \ }} (c\!:\!(a\!:\!a))\!:\!4_{3}\circcol\, 2_{1}] [P4_{3}2_{1}] [P4_{3}2_{1}2] [P4_{3}2_{1}2] [P4_{3}2_{1}2]  
97 [D_{4}^{9}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!4\!:\!2] I42 I422 I422 I422  
98 [D_{4}^{10}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!4_{1}\!:\!2] [I4_{1}2] [I4_{1}22] [I4_{1}22] [I4_{1}22]  
99 [C_{4v}^{1}] [(c\!:\!(a\!:\!a))\!:\!4\cdot m] P4mm P4mm P4mm P4mm  
100 [C_{4v}^{2}] [ (c\!:\!(a\!:\!a))\!:\!4\bigodot \tilde{a}] P4bm P4bm P4bm P4bm  
101 [C_{4v}^{3}] [(c\!:\!(a\!:\!a))\!:\!4_{2}\cdot \tilde{c}] P4cm [P4_{2}cm] [P4_{2}cm] [P4_{2}cm]  
102 [C_{4v}^{4}] [ (c\!:\!(a\!:\!a))\!:\!4_{2}\bigodot \widetilde{ac}] P4nm [P4_{2}nm] [P4_{2}nm] [P4_{2}nm]  
103 [C_{4v}^{5}] [(c\!:\!(a\!:\!a))\!:\!4\cdot \tilde{c}] P4cc P4cc P4cc P4cc  
104 [C_{4v}^{6}] [ (c\!:\!(a\!:\!a))\!:\!4\bigodot \widetilde{ac}] P4nc P4nc P4nc P4nc  
105 [C_{4v}^{7}] [(c\!:\!(a\!:\!a))\!:\!4_{2}\cdot m] P4mc [P4_{2}mc] [P4_{2}mc] [P4_{2}mc]  
106 [C_{4v}^{8}] [(c\!:\!(a\!:\!a))\!:\!4_{2}\bigodot \tilde{a}] P4bc [P4_{2}bc] [P4_{2}bc] [P4_{2}bc]  
107 [C_{4v}^{9}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!4\cdot m] I4mm I4mm I4mm I4mm  
108 [C_{4v}^{10}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!4\cdot \tilde{c}] I4cm I4cm I4cm I4cm  
109 [C_{4v}^{11}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!4_{1}\bigodot m] I4md [I4_{1}md] [I4_{1}md] [I4_{1}md]  
110 [C_{4v}^{12}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!4_{1}\bigodot \tilde{c}] I4cd [I4_{1}cd] [I4_{1}cd] [I4_{1}cd] [\left(\displaystyle{a + b + c \over 2}\!\!\bigg/\!\!c\!:\!a\!:\!a\right)\!:\!4_{1}\cdot \tilde{a}](Sh–K)
111 [D_{2d}^{1}] [(c\!:\!(a\!:\!a))\!:\!\tilde{4}\!:\!2] [P\overline{4}2m] [P\overline{4}2m] [P\overline{4}2m] [P\overline{4}2m]  
112 [D_{2d}^{2}] [\def\circcol{\mathop{\bigcirc\hskip -5pt{\raise.05pt\hbox{$\!:\!$}} \ }} (c\!:\!(a\!:\!a))\!:\!\tilde{4}\circcol\, 2] [P\overline{4}2c] [P\overline{4}2c] [P\overline{4}2c] [P\overline{4}2c]  
113 [D_{2d}^{3}] [(c\!:\!(a\!:\!a))\!:\!\tilde{4}\cdot \widetilde{ab}] [P\overline{4}2_{1}m] [P\overline{4}2_{1}m] [P\overline{4}2_{1}m] [P\overline{4}2_{1}m]  
114 [D_{2d}^{4}] [(c\!:\!(a\!:\!a))\!:\!\tilde{4}\cdot \widetilde{abc}] [P\overline{4}2_{1}c] [P\overline{4}2_{1}c] [P\overline{4}2_{1}c] [P\overline{4}2_{1}c]  
115 [D_{2d}^{5}] [(c\!:\!(a\!:\!a))\!:\!\tilde{4}\cdot m] [C\overline{4}2m] [C\overline{4}2m] [P\overline{4}m2] [P\overline{4}m2]  
116 [D_{2d}^{6}] [(c\!:\!(a\!:\!a))\!:\!\tilde{4}\cdot \tilde{c}] [C\overline{4}2c] [C\overline{4}2c] [P\overline{4}c2] [P\overline{4}c2]  
117 [D_{2d}^{7}] [(c\!:\!(a\!:\!a))\!:\!\tilde{4}\bigodot \tilde{a}] [C\overline{4}2b] [C\overline{4}2b] [P\overline{4}b2] [P\overline{4}b2]  
118 [D_{2d}^{8}] [(c\!:\!(a\!:\!a))\!:\!\tilde{4}\cdot \widetilde{ac}] [C\overline{4}2n] [C\overline{4}2n] [P\overline{4}n2] [P\overline{4}n2]  
119 [D_{2d}^{9}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!\tilde{4}\cdot m] [F\overline{4}2m] [F\overline{4}2m] [I\overline{4}m2] [I\overline{4}m2]  
120 [D_{2d}^{10}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!\tilde{4}\cdot \tilde{c}] [F\overline{4}2c] [F\overline{4}2c] [I\overline{4}c2] [I\overline{4}c2]  
121 [D_{2d}^{11}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!\tilde{4}\!:\!2] [I\overline{4}2m] [I\overline{4}2m] [I\overline{4}2m] [I\overline{4}2m]  
122 [D_{2d}^{12}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!\tilde{4}\bigodot {1 \over 2}\widetilde{abc}] [I\overline{4}2d] [I\overline{4}2d] [I\overline{4}2d] [I\overline{4}2d]  
123 [D_{4h}^{1}] [(c\!:\!(a\!:\!a))\cdot m\!:\!4\cdot m] [P4/mmm] [P4/m\ 2/m\ 2/m] [P4/mmm] [P4/m\ 2/m\ 2/m]  
124 [D_{4h}^{2}] [(c\!:\!(a\!:\!a))\cdot m\!:\!4\cdot \tilde{c}] [P4/mcc] [P4/m\ 2/c\ 2/c] [P4/mcc] [P4/m\ 2/c\ 2/c]  
125 [D_{4h}^{3}] [(c\!:\!(a\!:\!a))\cdot \widetilde{ab}\!:\!4\bigodot \tilde{a}] [P4/nbm] [P4/n\ 2/b\ 2/m] [P4/nbm] [P4/n\ 2/b\ 2/m] [\def\bigodot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!a\!:\!a)\cdot \widetilde{ab}\!:\!4\bigodot \tilde{b}] (Sh–K)
126 [D_{4h}^{4}] [(c\!:\!(a\!:\!a))\cdot \widetilde{ab}\!:\!4\bigodot \widetilde{ac}] [P4/nnc] [P4/n\ 2/n\ 2/c] [P4/nnc] [P4/n\ 2/n\ 2/c]  
127 [D_{4h}^{5}] [(c\!:\!(a\!:\!a))\cdot m\!:\!4\bigodot \tilde{a}] [P4/mbm] [P4/m\ 2_{1}/b\ 2/m] [P4/mbm] [P4/m\ 2_{1}/b\ 2/m] [\def\bigodot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!a\!:\!a)\cdot m\!:\!4\bigodot \tilde{b}] (Sh–K)
128 [D_{4h}^{6}] [(c\!:\!(a\!:\!a))\cdot m\!:\!4\bigodot \widetilde{ac}] [P4/mnc] [P4/m\ 2_{1}/n\ 2/c] [P4/mnc] [P4/m\ 2_{1}/n\ 2/c]  
129 [D_{4h}^{7}] [(c\!:\!(a\!:\!a))\cdot \widetilde{ab}\!:\!4\cdot m] [P4/nmm] [P4/n\ 2_{1}/m\ 2/m] [P4/nmm] [P4/n\ 2_{1}/m\ 2/m]  
130 [D_{4h}^{8}] [(c\!:\!(a\!:\!a)\cdot \widetilde{ab}\!:\!4\cdot \tilde{c}] [P4/ncc] [P4/n\ 2/c\ 2/c] [P4/ncc] [P4/n\ 2/c\ 2/c]  
131 [D_{4h}^{9}] [(c\!:\!(a\!:\!a))\cdot m\!:\!4_{2}\cdot m] [P4/mmc] [P4_{2}/m\ 2/m\ 2/c] [P4_{2}/mmc] [P4_{2}/m\ 2/m\ 2/c]  
132 [D_{4h}^{10}] [(c\!:\!(a\!:\!a))\cdot m\!:\!4_{2}\cdot \tilde{c}] [P4/mcm] [P4_{2}/m\ 2/c\ 2/m] [P4_{2}/mcm] [P4_{2}/m\ 2/c\ 2/m]  
133 [D_{4h}^{11}] [(c\!:\!(a\!:\!a))\cdot \widetilde{ab}\!:\!4_{2}\bigodot \tilde{a}] [P4/nbc] [P4_{2}/n\ 2/b\ 2/c] [P4_{2}/nbc] [P4_{2}/n\ 2/b\ 2/c] [(c\!:\!a\!:\!a)\cdot \widetilde{ab}\!:\!4_{2}\bigodot \tilde{b}] (Sh–K)
134 [D_{4h}^{12}] [(c\!:\!(a\!:\!a))\cdot \widetilde{ab}\!:\!4_{2}\bigodot \widetilde{ac}] [P4/nnm] [P4_{2}/n\ 2/n\ 2/m] [P4_{2}/nnm] [P4_{2}/n\ 2/n\ 2/m]  
135 [D_{4h}^{13}] [(c\!:\!(a\!:\!a))\cdot n\!:\!4_{2}\bigodot \tilde{a}] [P4/mbc] [P4_{2}/m\ 2_{1}/b\ 2/c] [P4_{2}/mbc] [P4_{2}/m\ 2_{1}/b\ 2/c] [(c\!:\!a\!:\!a)\cdot m\!:\!4_{2}\bigodot \tilde{b}] (Sh–K)
136 [D_{4h}^{14}] [(c\!:\!(a\!:\!a))\cdot m\!:\!4_{2}\bigodot \widetilde{ac}] [P4/mnm] [P4_{2}/m\ 2_{1}/n\ 2/m] [P4_{2}/mnm] [P4_{2}/m\ 2_{1}/n\ 2/m]  
137 [D_{4h}^{15}] [(c\!:\!(a\!:\!a))\cdot \widetilde{ab}\!:\!4_{2}\cdot m] [P4/nmc] [P4_{2}/n\ 2_{1}/m\ 2/c] [P4_{2}/nmc] [P4_{2}/n\ 2_{1}/m\ 2/c]  
138 [D_{4h}^{16}] [(c\!:\!(a\!:\!a))\cdot ab\!:\!4_{2}\cdot \tilde{c}] [P4/ncm] [P4_{2}/n\ 2_{1}/c\ 2/m] [P4_{2}/ncm] [P4_{2}/n\ 2_{1}/c\ 2/m]  
139 [D_{4h}^{17}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!c\!:\!(a\!:\!a)\right)\cdot m\!:\!4\cdot m] [I4/mmm] [I4/m\ 2/m\ 2/m] [I4/mmm] [I4/m\ 2/m\ 2/m]  
140 [D_{4h}^{18}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!c\!:\!(a\!:\!a)\right)\cdot m\!:\!4\cdot \tilde{c}] [I4/mcm] [I4/m\ 2/c\ 2/m] [I4/mcm] [I4/m\ 2/c\ 2/m]  
141 [D_{4h}^{19}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!c\!:\!(a\!:\!a)\right)\cdot \tilde{a}\!:\!4_{1}\bigodot m] [I4/amd] [I4_{1}/a\ 2/m\ 2/d] [I4_{1}/amd] [I4_{1}/a\ 2/m\ 2/d]  
142 [D_{4h}^{20}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!c\!:\!(a\!:\!a)\right)\cdot \tilde{a}\!:\!4_{1}\bigodot \tilde{c}] [I4/acd] [I4_{1}/a\ 2/c\ 2/d] [I4_{1}/acd] [I4_{1}/a\ 2/c\ 2/d]  
143 [C_{3}^{1}] [(c\!:\!(a/a))\!:\!3] C3 C3 P3 P3  
144 [C_{3}^{2}] [(c\!:\!(a/a))\!:\!3_{1}] [C3_{1}] [C3_{1}] [P3_{1}] [P3_{1}]  
145 [C_{3}^{3}] [(c\!:\!(a/a))\!:\!3_{2}] [C3_{2}] [C3_{2}] [P3_{2}] [P3_{2}]  
146 [C_{3}^{4}] [\displaylines{\left(\displaystyle{2a + b + c \over 3}\!\!\bigg/\!\!{a + 2b + 2c \over 3}\!\!\bigg/\!\!c\!:\!(a/a)\right)\hfill\cr\quad:3\hfill}] R3 R3 R3 R3 Hexagonal setting (Sh–K)
    [(a/a/a)/3]         Rhombohedral setting (Sh–K)
147 [C_{3i}^{1}] [(c\!:\!(a/a))\!:\!\tilde{6}] [C\overline{3}] [C\overline{3}] [P\overline{3}] [P\overline{3}]  
148 [C_{3i}^{2}] [\displaylines{\left(\displaystyle{2a + b + c \over 3}\!\bigg/\!{a + 2b + 2c \over 3}\!\bigg/\!c\!:\!(a/a)\right)\hfill\cr\quad :\tilde{6}\hfill}] [R\overline{3}] [R\overline{3}] [R\overline{3}] [R\overline{3}] Hexagonal setting (Sh–K)
    [(a/a/a)/\tilde{6}]         Rhombohedral setting (Sh–K)
149 [D_{3}^{1}] [(c\!:\!(a/a))\!:\!2\!:\!3] H32 H321 P312 P312  
150 [D_{3}^{2}] [(c\!:\!(a/a))\!:\!2\!:\!3] C32 C321 P321 P321  
151 [D_{3}^{3}] [(c\!:\!(a/a))\!:\!2\!:\!3_{1}] [H3_{1}2] [H3_{1}21] [P3_{1}12] [P3_{1}12]  
152 [D_{3}^{4}] [(c\!:\!(a/a))\!:\!2\!:\!3_{1}] [C3_{1}2] [C3_{1}21] [P3_{1}21] [P3_{1}21]  
153 [D_{3}^{5}] [(c\!:\!(a/a))\!:\!2\!:\!3_{2}] [H3_{2}2] [H3_{2}21] [P3_{2}12] [P3_{2}12]  
154 [D_{3}^{6}] [(c\!:\!(a/a))\!:\!2\!:\!3_{2}] [C3_{2}2] [C3_{2}21] [P3_{2}21] [P3_{2}21]  
155 [D_{3}^{7}] [\displaylines{\left(\displaystyle{2a + b + c \over 3}\!\bigg/\!{a + 2b + 2c \over 3}\!\bigg/\!c\!:\!(a/a)\right)\hfill\cr \quad\cdot 2\!:\!3\hfill\cr}] R32 R32 R32 R32 Hexagonal setting (Sh–K)
    [(a/a/a)/3\!:\!2]         Rhombohedral setting (Sh–K)
156 [C_{3v}^{1}] [(c\!:\!(a/a))\!:\!m\!:\!3] C3m C3m1 P3m1 P3m1  
157 [C_{3v}^{2}] [(a\!:\!c\!:\!a)\!:\!m\!:\!3] H3m H3m1 P31m P31m [(c\!:\!(a/a))\cdot m\cdot 3] (Sh–K) with special comment
158 [C_{3v}^{3}] [(c\!:\!(a/a))\!:\!\tilde{c}\!:\!3] C3c C3c1 P3c1 P3c1  
159 [C_{3v}^{4}] [(a\!:\!c\!:\!a)\!:\!\tilde{c}\!:\!3] H3c H3c1 P31c P31c [(c\!:\!(a/a))\cdot \tilde{c}\cdot 3] (Sh–K) with special comment
160 [C_{3v}^{5}] [\displaylines{\left(\displaystyle{2a + b + c \over 3}\!\bigg/\!{a + 2b + 2c \over 3}\!\bigg/\!c\!:\!(a/a)\right)\hfill\cr \quad\cdot\, m\cdot 3\hfill\cr}] R3m R3m R3m R3m Hexagonal setting (Sh–K)
    [(a/a/a)/3\cdot m]         Rhombohedral setting (Sh–K)
161 [C_{3v}^{6}] [\displaylines{\left(\displaystyle{2a + b + c \over 3}\!\bigg/\!{a + 2b + 2c \over 3}\!\bigg/\!c\!:\!(a/a)\right) \hfill\cr\quad \cdot \,\tilde{c}\cdot 3\hfill\cr}] R3c R3c R3c R3c Hexagonal setting (Sh–K)
    [(a/a/a)/3\cdot \widetilde{abc}]         Rhombohedral setting (Sh–K)
162 [D_{3d}^{1}] [(a\!:\!c\!:\!a)\cdot m\cdot \tilde{6}] [H\overline{3}m] [H\overline{3}\ 2/m\ 1] [P\overline{3}1m] [P\overline{3}1\ 2/m] [(c\!:\!(a/a))\cdot m\cdot \tilde{6}] (Sh–K) with special comment
163 [D_{3d}^{2}] [(a\!:\!c\!:\!a)\cdot \tilde{c}\cdot \tilde{6}] [H\overline{3}c] [H\overline{3}\ 2/c\ 1] [P\overline{3}1c] [P\overline{3}1\ 2/c] [(c\!:\!(a/a)\cdot \tilde{c}\cdot \tilde{6}] (Sh–K) with special comment
164 [D_{3d}^{3}] [(c\!:\!(a/a))\!:\!m\cdot \tilde{6}] [C\overline{3}m] [C\overline{3}\ 2/m\ 1] [P\overline{3}m1] [P\overline{3}\ 2/m\ 1]  
165 [D_{3d}^{4}] [(c\!:\!(a/a))\!:\!\tilde{c}\cdot \tilde{6}] [C\overline{3}c] [C\overline{3}\ 2/c\ 1] [P\overline{3}c1] [P\overline{3}\ 2/c\ 1]  
166 [D_{3d}^{5}] [\displaylines{\left(\displaystyle{2a + b + c \over 3}\!\bigg/\!{a + 2b + 2c \over 3}\!\bigg/\!c\!:\!(a/a)\right) \hfill\cr \quad:m\cdot \tilde{6}\hfill\cr}] [R\overline{3}m] [R\overline{3}\ 2/m] [R\overline{3}m] [R\overline{3}\ 2/m] Hexagonal setting (Sh–K)
    [(a/a/a)/\tilde{6}\cdot m]         Rhombohedral setting (Sh–K)
167 [D_{3d}^{6}] [\displaylines{\left(\displaystyle{2a + b + c \over 3}\!\bigg/\!{a + 2b + 2c \over 3}\!\bigg/\!c\!:\!(a/a)\right)\hfill\cr \quad:\tilde{c}\cdot \tilde{6}\hfill\cr}] [R\overline{3}c] [R\overline{3}\ 2/c] [R\overline{3}c] [R\overline{3}\ 2/c] Hexagonal setting (Sh–K)
    [(a/a/a)/\tilde{6}\cdot \widetilde{abc}]         Rhombohedral setting (Sh–K)
168 [C_{6}^{1}] [(c\!:\!(a/a))\!:\!6] C6 C6 P6 P6  
169 [C_{6}^{2}] [(c\!:\!(a/a))\!:\!6_{1}] [C6_{1}] [C6_{1}] [P6_{1}] [P6_{1}]  
170 [C_{6}^{3}] [(c\!:\!(a/a))\!:\!6_{5}] [C6_{5}] [C6_{5}] [P6_{5}] [P6_{5}]  
171 [C_{6}^{4}] [(c\!:\!(a/a))\!:\!6_{2}] [C6_{2}] [C6_{2}] [P6_{2}] [P6_{2}]  
172 [C_{6}^{5}] [(c\!:\!(a/a))\!:\!6_{4}] [C6_{4}] [C6_{4}] [P6_{4}] [P6_{4}]  
173 [C_{6}^{6}] [(c\!:\!(a/a))\!:\!6_{3}] [C6_{3}] [C6_{3}] [P6_{3}] [P6_{3}]  
174 [C_{3h}^{1}] [(c\!:\!(a/a))\!:\!3\!:\!m] [C\overline{6}] [C\overline{6}] [P\overline{6}] [P\overline{6}]  
175 [C_{6h}^{1}] [(c\!:\!(a/a))\cdot m\!:\!6] [C6/m] [C6/m] [P6/m] [P6/m]  
176 [C_{6h}^{2}] [(c\!:\!(a/a))\cdot m\!:\!6_{3}] [C6_{3}/m] [C6_{3}/m] [P6_{3}/m] [P6_{3}/m]  
177 [D_{6}^{1}] [(c\!:\!(a/a))\cdot 2\!:\!6] C62 C622 P622 P622  
178 [D_{6}^{2}] [(c\!:\!(a/a))\cdot 2\!:\!6_{1}] [C6_{1}2] [C6_{1}22] [P6_{1}22] [P6_{1}22]  
179 [D_{6}^{3}] [(c\!:\!(a/a))\cdot 2\!:\!6_{5}] [C6_{5}2] [C6_{5}22] [P6_{5}22] [P6_{5}22]  
180 [D_{6}^{4}] [(c\!:\!(a/a))\cdot 2\!:\!6_{2}] [C6_{2}2] [C6_{2}22] [P6_{2}22] [P6_{2}22]  
181 [D_{6}^{5}] [(c\!:\!(a/a))\cdot 2\!:\!6_{4}] [C6_{4}2] [C6_{4}22] [P6_{4}22] [P6_{4}22]  
182 [D_{6}^{6}] [(c\!:\!(a/a))\cdot 2\!:\!6_{3}] [C6_{3}2] [C6_{3}22] [P6_{3}22] [P6_{3}22]  
183 [C_{6v}^{1}] [(c\!:\!(a/a))\!:\!m\cdot 6] C6mm C6mm P6mm P6mm  
184 [C_{6v}^{2}] [(c\!:\!(a/a))\!:\!\tilde{c}\cdot 6] C6cc C6cc P6cc P6cc  
185 [C_{6v}^{3}] [(c\!:\!(a/a))\!:\!\tilde{c}\cdot 6_{3}] C6cm [C6_{3}cm] [P6_{3}cm] [P6_{3}cm]  
186 [C_{6v}^{4}] [(c\!:\!(a/a))\!:\!m\cdot 6_{3}] C6mc [C6_{3}mc] [P6_{3}mc] [P6_{3}mc]  
187 [D_{3h}^{1}] [(c\!:\!(a/a))\!:\!m\cdot 3\!:\!m] [C\overline{6}m2] [C\overline{6}m2] [P\overline{6}m2] [P\overline{6}m2]  
188 [D_{3h}^{2}] [(c\!:\!(a/a))\!:\!\tilde{c}\cdot 3\!:\!m] [C\overline{6}c2] [C\overline{6}c2] [P\overline{6}c2] [P\overline{6}c2]  
189 [D_{3h}^{3}] [(c\!:\!(a/a))\cdot m\!:\!3\cdot m] [H\overline{6}m2] [H\overline{6}m2] [P\overline{6}2m] [P\overline{6}2m]  
190 [D_{3h}^{4}] [(c\!:\!(a/a))\cdot m\!:\!3\cdot \tilde{c}] [H\overline{6}c2] [H\overline{6}c2] [P\overline{6}2c] [P\overline{6}2c]  
191 [D_{6h}^{1}] [(c\!:\!(a/a))\cdot m\!:\!6\cdot m] [C6/mmm] [C6/m\ 2/m\ 2/m] [P6/mmm] [P6/m\ 2/m\ 2/m]  
192 [D_{6h}^{2}] [(c\!:\!(a/a))\cdot m\!:\!6\cdot \tilde{c}] [C6/mcc] [C6/m\ 2/c\ 2/c] [P6/mcc] [P6/m\ 2/c\ 2/c]  
193 [D_{6h}^{3}] [(c\!:\!(a/a))\cdot m\!:\!6_{3}\cdot \tilde{c}] [C6/mcm] [C6_{3}/m\ 2/c\ 2/m] [P6_{3}/mcm] [P6_{3}/m\ 2/c\ 2/m]  
194 [D_{6h}^{4}] [(c\!:\!(a/a))\cdot m\!:\!6_{3}\cdot m] [C6/mmc] [C6_{3}/m\ 2/m\ 2/c] [P6_{3}/mmc] [P6_{3}/m\ 2/m\ 2/c]  
195 [T^{1}] [(a\!:\!(a/a))\!:\!2/3] P23 P23 P23 P23  
196 [T^{2}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right)\hfill\cr\quad:2/3\hfill}] F23 F23 F23 F23  
197 [T^{3}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\!:\!2/3] I23 I23 I23 I23  
198 [T^{4}] [(a\!:\!(a\!:\!a))\!:\!2_{1}//3] [P2_{1}3] [P2_{1}3] [P2_{1}3] [P2_{1}3]  
199 [T^{5}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\!:\!2_{1}//3] [I2_{1}3] [I2_{1}3] [I2_{1}3] [I2_{1}3]  
200 [T_{h}^{1}] [(a\!:\!(a\!:\!a))\cdot m/\tilde{6}] Pm3 [P2/m\ \overline{3}] [Pm\overline{3}] [P2/m\ \overline{3}] Pm3 (IT, 1952[link])
201 [T_{h}^{2}] [(a\!:\!(a\!:\!a))\cdot \widetilde{ab}/\tilde{6}] Pn3 [P2/n\ \overline{3}] [Pn\overline{3}] [P2/n\ \overline{3}] Pn3 (IT, 1952[link])
202 [T_{h}^{3}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right)\hfill\cr\quad\cdot\, m/\tilde{6}\hfill}] Fm3 [F2/m\ \overline{3}] [Fm\overline{3}] [F2/m\ \overline{3}] Fm3 (IT, 1952[link])
203 [T_{h}^{4}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right) \hfill\cr\quad\cdot \,{\textstyle{1 \over 2}}ab/\tilde{6}\hfill}] Fd3 [F2/d\ \overline{3}] [Fd\overline{3}] [F2/d\ \overline{3}] Fd3 (IT, 1952[link])
204 [T_{h}^{5}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\cdot m/\tilde{6}] Im3 [I2/m\ \overline{3}] [Im\overline{3}] [I2/m\ \overline{3}] Im3 (IT, 1952[link])
205 [T_{h}^{6}] [(a\!:\!(a\!:\!a))\cdot \tilde{a}/\tilde{6}] Pa3 [P2_{1}/a\ \overline{3}] [Pa\overline{3}] [P2_{1}/a\ \overline{3}] Pa3 (IT, 1952[link])
206 [T_{h}^{7}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\cdot \tilde{a}/\tilde{6}] Ia3 [I2_{1}/a\ \overline{3}] [Ia\overline{3}] [I2_{1}/a\ \overline{3}] Ia3 (IT, 1952[link])
207 [O^{1}] [(a\!:\!(a\!:\!a))\!:\!4/3] P43 P432 P432 P432  
208 [O^{2}] [(a\!:\!(a\!:\!a))\!:\!4_{2}//3] [P4_{2}3] [P4_{2}32] [P4_{2}32] [P4_{2}32]  
209 [O^{3}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right)\hfill\cr\quad :4/3\hfill}] F43 F432 F432 F432  
210 [O^{4}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right)\hfill\cr\quad :4_{1}//3\hfill}] [F4_{1}3] [F4_{1}32] [F4_{1}32] [F4_{1}32]  
211 [O^{5}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\!:\!4/3] I43 I432 I432 I432  
212 [O^{6}] [(a\!:\!(a\!:\!a))\!:\!4_{3}//3] [P4_{3}3] [P4_{3}32] [P4_{3}32] [P4_{3}32]  
213 [O^{7}] [(a\!:\!(a\!:\!a))\!:\!4_{1}//3] [P4_{1}3] [P4_{1}32] [P4_{1}32] [P4_{1}32]  
214 [O^{8}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\!:\!4_{1}//3] [I4_{1}3] [I4_{1}32] [I4_{1}32] [I4_{1}32]  
215 [T_{d}^{1}] [(a\!:\!(a\!:\!a))\!:\!\tilde{4}/3] [P\overline{4}3m] [P\overline{4}3m] [P\overline{4}3m] [P\overline{4}3m]  
216 [T_{d}^{2}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right)\hfill\cr\quad :\tilde{4}/3\hfill}] [F\overline{4}3m] [F\overline{4}3m] [F\overline{4}3m] [F\overline{4}3m]  
217 [T_{d}^{3}] [\displaylines{\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\hfill\cr\quad :\tilde{4}/3\hfill}] [I\overline{4}3m] [I\overline{4}3m] [I\overline{4}3m] [I\overline{4}3m]  
218 [T_{d}^{4}] [(a\!:\!(a\!:\!a))\!:\!\tilde{4}//3] [P\overline{4}3n] [P\overline{4}3n] [P\overline{4}3n] [P\overline{4}3n]  
219 [T_{d}^{5}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right)\hfill\cr\quad :\tilde{4}//3\hfill}] [F\overline{4}3c] [F\overline{4}3c] [F\overline{4}3c] [F\overline{4}3c]  
220 [T_{d}^{6}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\!:\!\tilde{4}//3] [I\overline{4}3d] [I\overline{4}3d] [I\overline{4}3d] [I\overline{4}3d]  
221 [O_{h}^{1}] [(a\!:\!(a\!:\!a))\!:\!4/\tilde{6}\cdot m] Pm3m [P4/m\ \overline{3}\ 2/m] [Pm\overline{3}m] [P4/m\ \overline{3}\ 2/m] Pm3m (IT, 1952[link])
222 [O_{h}^{2}] [(a\!:\!(a\!:\!a))\!:\!4/\tilde{6}\cdot \widetilde{abc}] Pn3n [P4/n\ \overline{3}\ 2/n] [Pn\overline{3}n] [P4/n\ \overline{3}\ 2/n] Pn3n (IT, 1952[link])
223 [O_{h}^{3}] [(a\!:\!(a\!:\!a))\!:\!4_{2}//\tilde{6}\cdot \widetilde{abc}] Pm3n [P4_{2}/m\ \overline{3}\ 2/n] [Pm\overline{3}n] [P4_{2}/m\ \overline{3}\ 2/n] Pm3n (IT, 1952[link])
224 [O_{h}^{4}] [(a\!:\!(a\!:\!a))\!:\!4_{2}//\tilde{6}\cdot m] Pn3m [P4_{2}/n\ \overline{3}\ 2/m] [Pn\overline{3}m] [P4_{2}/n\ \overline{3}\ 2/m] Pn3m (IT ,1952[link])
225 [O_{h}^{5}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right) \hfill\cr\quad :4/\tilde{6}\cdot m\hfill}] Fm3m [F4/m\ \overline{3}\ 2/m] [Fm\overline{3}m] [F4/m\ \overline{3}\ 2/m] Fm3m (IT, 1952[link])
226 [O_{h}^{6}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right)\hfill\cr\quad :4/\tilde{6}\cdot \tilde{c}\hfill}] Fm3c [F4/m\ \overline{3}\ 2/c] [Fm\overline{3}c] [F4/m\ \overline{3}\ 2/c] Fm3c (IT, 1952[link])
227 [O_{h}^{7}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a(a\!:\!a)\right)\hfill\cr\quad :\!4_{1}//\tilde{6}\cdot m\hfill\cr}] Fd3m [F4_{1}/d\ \overline{3}\ 2/m] [Fd\overline{3}m] [F4_{1}/d\ \overline{3}\ 2/m] Fd3m (IT, 1952[link])
228 [O_{h}^{8}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right)\hfill\cr \quad:\!4_{1}//\tilde{6}\cdot \tilde{c}\hfill\cr}] Fd3c [F4_{1}/d\ \overline{3}\ 2/c] [Fd\overline{3}c] [F4_{1}/d\ \overline{3}\ 2/c] Fd3c (IT, 1952[link])
229 [O_{h}^{9}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\!:\!4/\tilde{6}\cdot m] Im3m [I4/m\ \overline{3}\ 2/m] [Im\overline{3}m] [I4/m\ \overline{3}\ 2/m] Im3m (IT, 1952[link])
230 [O_{h}^{10}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\!:\!4_{1}//\tilde{6}\cdot {1 \over 2}\widetilde{abc}] Ia3d [I4_{1}/a\ \overline{3}\ 2/d] [Ia\overline{3}d] [I4_{1}/a\ \overline{3}\ 2/d] Ia3d (IT, 1952[link])
Abbreviations used in the column Comments: IT, 1952[link]: International Tables for X-ray Crystallography, Vol. I (1952[link]); Sh–K; Shubnikov & Koptsik (1972[link]). Note that this table contains only one notation for the b-unique setting and one notation for the c-unique setting in the monoclinic case, always referring to cell choice 1 of the space-group tables.

Triclinic symbols are unique if the unit cell is primitive. For the standard setting of monoclinic space groups, the lattice symmetry direction is labelled b. From the three possible centrings A, I and C, the latter one is favoured. If glide components occur in the plane perpendicular to [010], the glide direction c is preferred. In the space groups corresponding to the orthorhombic group mm2, the unique direction of the twofold axis is chosen along c. Accordingly, the face centring C is employed for centrings perpendicular to the privileged direction. For space groups with possible A or B centring, the first one is preferred. For groups 222 and mmm, no privileged symmetry direction exists, so the different possibilities of one-face centring can be reduced to C centring by change of the setting. The choices of unit cell and centring type are fixed by the conventional basis in systems with higher symmetry.

When more than one kind of symmetry elements exist in one representative direction, in most cases the choice for the space-group symbol is made in order of decreasing priority: for reflections and glide reflections m, a, b, c, n, d; for proper rotations and screw rotations [6,\ 6_{1},\ 6_{2},\ 6_{3},\ 6_{4},\ 6_{5}]; [4,\ 4_{1},\ 4_{2},\ 4_{3}]; [3,\ 3_{1},\ 3_{2}]; [2,\ 2_{1}] [cf. IT (1952[link]), p. 55, and Section 1.4.1[link] ].

3.3.3.5. Systematic absences

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Hermann (1928a[link]) emphasized that the short symbols permit the derivation of systematic absences of X-ray reflections caused by the glide/screw parts of the symmetry operations. If [{\bi h} = (hkl)] describes the X-ray reflection and [({\bi W},{\bi w})] is the matrix representation of a symmetry operation, the matrix can be expanded as follows:[({\bi W},{\bi w}) = ({\bi W}, {\bi w}_{g} + {\bf w}_{l}) = ({\bi W},\openup3pt\left(\matrix{\hfill w_{g,1}\cr \hfill w_{g,2}\cr \hfill w_{g,3}\cr}\right) + {\bi w}_{l}).]The absence of a reflection is governed by the relation (i) [{\bi h} \cdot {\bi W}] = [{\bi h}] and the scalar product (ii) [{\bi h} \cdot {\bi w}_{g} = hw_{g,1} + kw_{g,2} + lw_{g,3}]. A reflection h is absent if condition (i) holds and the scalar product (ii) is not an integer. The calculation must be made for all generators and indicators of the short symbol. Systematic absences, introduced by the further symmetry operations generated, are obtained by the combination of the extinction rules derived for the generators and indicators.

Example: Space group [D_{4}^{10} = I4_{1}22\ (98)]

The generators of the space group are the integral translations and the centring translation [x+\textstyle{1\over 2},y+\textstyle{1\over 2} ,z+\textstyle{1\over 2}], the rotation 2 in direction [100]: [x,\overline{y},\overline{z}] and the rotation 2 in direction [\hbox{[}1\overline{1}0\hbox{]}]: [\overline{y},\overline{x},\overline{z} -\textstyle{1\over 4}]. The combination of the two generators gives the operation corresponding to the indicator, namely [\overline y, x, z+\textstyle{1\over 4}], which represents a fourfold screw rotation in the direction [001].

The integral translations imply no restriction because the scalar product is always an integer. For the centring, condition (i) with [{\bi W} = {\bi I}] holds for all reflections (integral condition), but the scalar product (ii) is an integer only for [h + k + l = 2n]. Thus, reflections hkl with [h + k + l \ne 2n] are absent. The screw rotation 4 has the screw part [{\bi w}_{g} = \pmatrix{0\cr 0\cr \textstyle{1\over 4}}]. Only 00l reflections obey condition (i) (serial extinction). An integral value for the scalar product (ii) requires [l = 4n]. The twofold axes in the directions [100] and [\hbox{[}1\overline{1}0\hbox{]}] do not imply further absences because [{\bi w}_{g} = {\bi o}].

Detailed discussion of the theoretical background of conditions for possible general reflections and their derivation is given in Chapter 1.6[link] .

3.3.3.6. Generalized symmetry

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The international symbols can be suitably modified to describe generalized symmetry, e.g. colour groups, which occur when the symmetry operations are combined with changes of physical properties. For the description of antisymmetry (or `black–white' symmetry), the symbols of the Bravais lattices are supplemented by additional letters for centrings accompanied by a change in colour. For symmetry operations that are not translations, a prime is added to the usual symbol if a change of colour takes place. A complete description of the symbols and a detailed list of references are given by Koptsik (1966[link]). The Shubnikov symbols have not been extended to colour symmetry.

An introduction to the structure, properties and symbols of magnetic subperiodic and magnetic space groups is given in Chapter 3.6[link] .

References

International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Revised editions: 1965, 1969 and 1977. Abbreviated as IT (1952).]
Bertaut, E. F. (1976). Study of principal subgroups and their general positions in C and I groups of class mmm – D2h. Acta Cryst. A32, 380–387.
Burzlaff, H. & Zimmermann, H. (1980). On the choice of origins in the description of space groups. Z. Kristallogr. 153, 151–179.
Burzlaff, H. & Zimmermann, H. (2002). On the treatment of settings of space groups and crystal structures by specialized short Hermann–Mauguin space-group symbols. Z. Kristallogr. 217, 135–138.
Hermann, C. (1928a). Zur systematischen Strukturtheorie I. Eine neue Raumgruppensymbolik. Z. Kristallogr. 68, 257–287.
Hermann, C. (1929). Zur systematischen Strukturtheorie IV. Untergruppen. Z. Kristallogr. 69, 533–555.
Hermann, C. (1931). Bemerkungen zu der vorstehenden Arbeit von Ch. Mauguin. Z. Kristallogr. 76, 559–561.
Koptsik, V. A. (1966). Shubnikov Groups. Moscow University Press. (In Russian.)
Zimmermann, H. (1976). Ableitung der Raumgruppen aus ihren klassengleichen Untergruppenbeziehungen. Z. Kristallogr. 143, 485–515.








































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