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

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. A. ch. 13.1, pp. 836-842
https://doi.org/10.1107/97809553602060000528

Chapter 13.1. Isomorphic subgroups

Y. Billieta‡ and E. F. Bertautb§

a Département de Chimie, Faculté des Sciences et Techniques, Université de Bretagne Occidentale, Brest, France, and bLaboratoire de Cristallographie, CNRS, Grenoble, France

In this chapter, the (infinitely many) isomorphic subgroups of a space group are defined. The matrices relating a plane group or space group [{\cal G}] and its isomorphic subgroup [{\cal H}] are tabulated for each crystal system. Conditions on the coefficients of the matrices are also given, as are examples of the use of the matrices.

Keywords: isomorphic subgroups; space groups; plane groups; symmetry operations; crystal systems.

13.1.1. Definitions

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A subgroup [{\cal H}] of a space group [{\cal G}] is an isomorphic subgroup if [{\cal H}] is of the same or the enantiomorphic space-group type as [{\cal G}]. Thus, isomorphic space groups are a special subset of klassengleiche subgroups. The maximal isomorphic subgroups of lowest index are listed under IIc in the space-group tables of this volume (Part 7[link] ) (cf. Section 2.2.15[link] ). Isomorphic subgroups can easily be recognized because the standard space-group symbols of [{\cal G}] and [{\cal H}] are the same [isosymbolic subgroups (Billiet, 1973[link])] or the symbol of [{\cal H}] is enantiomorphic to that of [{\cal G}]. Every space group has an infinite number of maximal isomorphic subgroups, whereas the number of maximal non-isomorphic subgroups is finite (cf. Section 8.3.3[link] ). For this reason, isomorphic subgroups are discussed in more detail in the present section.

If a, b, c are the basis vectors defining the conventional unit cell of [{\cal G}] and [{\bf a}',{\bf b}',{\bf c}'] the basis vectors corresponding to [{\cal H}] the relation [({\bf a}',{\bf b}',{\bf c}') = ({\bf a},{\bf b},{\bf c}){\bi S} \eqno(13.1.1.1)] holds, where (a, b, c) and [({\bf a}',{\bf b}',{\bf c}')] are row matrices and S is a [(3 \times 3)] matrix. The coefficients [S_{ij}] of S are integers.1

The index of [{\cal H}] in [{\cal G}] is equal to [|\det ({\bi S})|]1, which is the ratio of the volumes [[{\bf a}'{\bf b}'{\bf c}']] and [abc] of the two cells. [\det ({\bi S})] is positive if the bases of the two cells have the same handedness and negative if they have opposite handedness.

If O and O′ are the origins of the coordinate systems (O, a, b, c) and [(O',{\bf a}',{\bf b}',{\bf c}')], used for the description of [{\cal G}] and [{\cal H}], the column matrix of the coordinates of O′ referred to the system (O, a, b, c) will be denoted by s. Thus, the coordinate system [(O',{\bf a}',{\bf b}',{\bf c}')] will be specified completely by the square matrix S and the column matrix s, symbolized by [\specialfonts{\bbsf S}: ({\bi S}, {\bi s})].

An example of the application of equation (13.1.1.1)[link] is given at the end of this chapter.

13.1.1.1. The mathematical expression of equivalence

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Let [\specialfonts{\bbsf W} = ({\bi W}, {\bi w})] be the operator of a given symmetry operation of [{\cal H}] referred to (O, a, b, c) and [\specialfonts{\bbsf W}' = ({\bi W}', {\bi w}')] the operator of the same operation referred to [(O',{\bf a}',{\bf b}',{\bf c})]. Then the following relation applies [\specialfonts{\bbsf S}{\bbsf W}' = {\bbsf W}{\bbsf S}\quad\rm \hbox{or}\quad {\bbsf W}' = {\bbsf S} ^{\rm -1}{\bbsf W}{\bbsf S} \eqno(13.1.1.2)] (cf. Bertaut & Billiet, 1979[link]). The latter expression is more conventional, the former is easier to manipulate. Identifying the rotational (matrix) and translational (column) parts of [\specialfonts{\bbsf W}], one obtains the following two conditions: [\eqalign{{\bi S}{\bi W}' &= {\bi W}{\bi S},\cr {\bi s} + {\bi S}{\bi w}' = {\bi w} + {\bi W}{\bi s} &= \hat{{\bi w}} + {\bi t}_{\cal G} + {\bi W}{\bi s}} \eqno(13.1.1.2{a})] or [{\bi S}{\bi w}' - \hat{{\bi w}} + ({\bi I} - {\bi W}){\bi s} = {\bi t}_{\cal G}. \eqno(13.1.1.2b)]

Here we have split w into a fractional part [\hat{{\bi w}}] (smaller than any lattice translation) and [{\bi t}_{\cal G}] which describes a lattice translation in [{\cal G}].

The general expression of the matrix S is [{\bi S} = \pmatrix{S_{11} &S_{12} &S_{13}\cr S_{21} &S_{22} &S_{23}\cr S_{31} &S_{32} &S_{33}\cr}. \eqno(13.1.1.3)] This general form, without any restrictions on the coefficients, applies only to the triclinic space groups P1 and [P\bar{1}]; P1 has only isomorphic subgroups (cf. Billiet, 1979[link]; Billiet & Rolley Le Coz, 1980[link]). For other space groups, restrictions have to be imposed on the coefficients [S_{ij}].

13.1.2. Isomorphic subgroups

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For convenience, we consider first those crystal systems that possess a unique direction (the privileged axis being taken parallel to c). We also include here the monoclinic system (unique axis either c or b).

13.1.2.1. Monoclinic, tetragonal, trigonal, hexagonal systems

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If W is the matrix corresponding to a rotation about the c axis, [{\bi W}' = {\bi W}] holds if the positive direction is the same for c and c′.2 In consequence, W must commute with S [cf. equation (13.1.1.2a)[link]]. This condition imposes relations on the coefficients [S_{ij}] of the matrix so that S and [\det ({\bi S})] take the following forms:

  • Monoclinic system [{\bi M}_{c} = \pmatrix{S_{11}\hfill &S_{12}\hfill &0\hfill\cr S_{21}\hfill &S_{22}\hfill &0\hfill\cr 0\hfill &0\hfill &S_{33}\hfill\cr},\quad \det ({\bi M}_{c}) = S_{33}(S_{11}S_{22} - S_{12}S_{21});] or if b instead of c is used [{\bi M}_{b} = \pmatrix{S_{11}\hfill &0\hfill &S_{13}\hfill\cr 0\hfill &S_{22}\hfill &0\hfill\cr S_{31}\hfill &0\hfill &S_{33}\hfill\cr},\quad \det ({\bi M}_{b}) = S_{22}(S_{11}S_{33} - S_{13}S_{31}).]

  • Tetragonal system [{\bi T}_{1} = \pmatrix{S_{11}\hfill &-S_{21}\hfill &0\hfill\cr S_{21}\hfill &\phantom{-}S_{11}\hfill &0\hfill\cr 0\hfill &\phantom{-}0\hfill &S_{33}\hfill\cr},\quad \det ({\bi T}_{1}) = S_{33}(S_{11}^{2} + S_{21}^{2}).]

  • Hexagonal and trigonal systems [\eqalign{{\bi H}_{1} &= \pmatrix{S_{11}\hfill &-S_{21}\hfill &0\hfill\cr S_{21}\hfill &\phantom{-}S_{11} - S_{21}\hfill &0\hfill\cr 0\hfill &\phantom{-}0\hfill &S_{33}\hfill\cr},\hfill\cr \det ({\bi H}_{1}) &= S_{33}(S_{11}^{2} + S_{21}^{2} - S_{11}S_{21}).}]For rhombohedral space groups, the matrix [{\bi H}_{1}] applies only when hexagonal axes are used. If rhombohedral axes are used, the matrix S has the form [\eqalign{{\bi R}_{1} &= \pmatrix{S_{0} &S_{2} &S_{1}\cr S_{1} &S_{0} &S_{2}\cr S_{2} &S_{1} &S_{0}\cr},\cr \det ({\bi R}_{1}) &= S_{0}^{3} + S_{1}^{3} + S_{2}^{3} - 3S_{0}S_{1}S_{2}\cr &= (S_{0} + S_{1} + S_{2})\cr &\quad \times (S_{0}^{2} + S_{1}^{2} + S_{2}^{2} - S_{0}S_{1} - S_{1}S_{2} - S_{2}S_{0}).}]

13.1.2.1.1. Additional restrictions

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If mirror or glide planes parallel to and/or twofold rotation or screw axes perpendicular to the principal rotation axis exist, further conditions are imposed upon the coefficients [S_{ij}] and these are indicated below (cf. Bertaut & Billiet, 1979[link]).

  • Monoclinic system

    The matrices [{\bi M}_{c}] and [{\bi M}_{b}] apply without any further restrictions on the coefficients.

  • Tetragonal system

    The matrix [{\bi T}_{1}] is valid for all space groups belonging to the crystal classes 4, [\bar{4}] and [4/m].

    For all other space groups, restrictions apply to the coefficients [S_{21}] according to the following two rules which are consequences of equation (13.1.1.2a)[link]:

    • (i) If the last two letters of the Hermann–Mauguin symbol are different, [S_{21} = 0]; the corresponding matrix is called [{\bi T}_{2}].

      Example: [P4_{2}/mmc]

      [{\bi T}_{2} = \pmatrix{S_{11}\hfill &0\hfill &0\hfill\cr 0\hfill &S_{11}\hfill &0\hfill\cr 0\hfill &0\hfill &S_{33}\hfill\cr},\quad \det ({\bi T}_{2}) = S_{33}S_{11}^{2}.]

    • (ii) If the last two letters are the same (except for the three cases mentioned below), two matrices have to be applied, the matrix [{\bi T}_{2}] introduced above and the matrix [{\bi T}_{1}] with [S_{21} = S_{11}]; the corresponding matrix is called [{\bi T}_{3}]. [{\bi T}_{3} = \pmatrix{S_{11}\hfill &-S_{11}\hfill &0\hfill\cr S_{11}\hfill &\phantom{-}S_{11}\hfill &0\hfill\cr 0\hfill &\phantom{-}0\hfill &S_{33}\hfill\cr},\quad \det ({\bi T}_{3}) = 2S_{33}S_{11}^{2}.]

      The following space groups have matrices [{\bi T}_{2}] and [{\bi T}_{3}]: P422, P4mm, [P4/mmm], [P4_{1}22], [P4_{3}22], [P4_{2}22], P4cc, [P4/mcc], I422, I4mm and [I4/mmm]. The three exceptions to the rule mentioned above are the space groups [P4/nmm], [P4/ncc] and [I4_{1}22], which allow only [{\bi T}_{2}].

  • Hexagonal and trigonal systems

    The matrix [{\bi H}_{1}] is valid for all space groups belonging to the crystal classes 6, [\bar{6}], [6/m], 3 and [\bar{3}].

    For all other space groups for which the last two letters of the Hermann–Mauguin symbol are different, [S_{22} = S_{11}], and the matrix is called [{\bi H}_{2}]. Examples are [P6_{3}/mcm], P312 and [P\bar{6}2m]. [{\bi H}_{2} = \pmatrix{S_{11}\hfill &0\hfill &0\hfill\cr 0\hfill &S_{11}\hfill &0\hfill\cr 0\hfill &0\hfill &S_{33}\hfill\cr},\quad \det({\bi H}_{2}) = S_{33}S_{11}^{2}.]

    If the last two letters of the Hermann–Mauguin symbol are the same, two matrices have to be applied, the matrix [{\bi H}_{2}] introduced above and the matrix [{\bi H}_{1}] with [S_{11} = 2S'_{11}] and [S_{21} = S'_{11}]; this matrix is called [{\bi H}_{3}], [{\bi H}_{3} = \pmatrix{2S'_{11}\hfill &-S'_{11}\hfill &0\hfill\cr \phantom{2}S'_{11}\hfill &\phantom{-}S'_{11}\hfill &0\hfill\cr \phantom{2}0\hfill &\phantom{-}0\hfill &S_{33}\hfill\cr},\quad \det({\bi H}_{3}) = 3S_{33}{S}_{11}'^{2}.] Examples are P622, [P6/mmm] and P6cc.

  • Rhombohedral space groups

    For R3 and [R\bar{3}], one has the matrix [{\bi H}_{1}] for hexagonal axes and [{\bi R}_{1}] for rhombohedral axes. For all other rhombohedral space groups, one has [{\bi H}_{2}] (hexagonal axes) and the matrix [{\bi R}_{1}] with [S_{1} = S_{2}] (rhombohedral axes). This last matrix is called [{\bi R}_{2}]. Example: R32. [{\bi R}_{2} = \pmatrix{S_{0}\hfill &S_{1}\hfill &S_{1}\hfill\cr S_{1}\hfill &S_{0}\hfill &S_{1}\hfill\cr S_{1}\hfill &S_{1}\hfill &S_{0}\hfill\cr},\quad \det({\bi R}_{2}) = (S_{0} + 2S_{1})(S_{0} - S_{1})^{2}.]

13.1.2.2. Cubic and orthorhombic systems

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  • Cubic system

    For cubic space groups, equation (13.1.1.2a[link]) leads to the matrix C: [{\bi C} = \pmatrix{S\hfill &0\hfill &0\hfill\cr 0\hfill &S\hfill &0\hfill\cr 0\hfill &0\hfill &S\hfill\cr},\quad \det({\bi C}) = S^{3}.]

  • Orthorhombic system

    There are six choices of matrices [{\bi O}_{i}\ (i = 1,2,3,4,5,6)] corresponding to the identical orientation [({\bi O}_{1})], to cyclic permutations of the three axes ([{\bi O}_{2}] and [{\bi O}_{3}]) and to the interchange of two axes ([{\bi O}_{4}, {\bi O}_{5}] and [{\bi O}_{6}]), i.e. to the six orthorhombic `settings'. [\eqalignno{{\bi O}_{1}\hfill &= \pmatrix{S_{11}\hfill &0\hfill &0 \hfill\cr 0\hfill &S_{22}\hfill &0\hfill\cr 0\hfill &0\hfill &S_{33}\cr};\quad {\bi O}_{2} = \pmatrix{0\hfill &S_{12}\hfill &0\hfill\cr 0\hfill &0\hfill &S_{23}\cr S_{31}\hfill &0\hfill &0\hfill\cr};\cr {\bi O}_{3} &= \pmatrix{0\hfill &0\hfill &S_{13}\cr S_{21}\hfill &0\hfill &0\hfill\cr 0\hfill &S_{32}\hfill &0\hfill\cr};\quad {\bi O}_{4} = \pmatrix{S_{11}\hfill &0\hfill &0\hfill\cr 0\hfill &0\hfill &S_{23}\cr 0\hfill &S_{32}\hfill &0\hfill\cr};\cr {\bi O}_{5}\hfill &= \pmatrix{0\hfill &0\hfill &S_{13}\cr 0\hfill &S_{22}\hfill &0\hfill\cr S_{31}\hfill &0\hfill &0\hfill\cr};\quad {\bi O}_{6} = \pmatrix{0\hfill &S_{12}\hfill &0\hfill\cr S_{21}\hfill &0\hfill &0\hfill\cr 0\hfill &0\hfill &S_{33}\cr}.}]

    The determinant is always equal to the product of the three non-zero coefficients, [\det({\bi O}_{i}) = \pm S_{1j}S_{2k}S_{3l}].

    The following general rule exists: only those matrices [{\bi O}_{i}] are permissible for which, if the non-zero coefficients are replaced by 1, the corresponding transformation of the axes conserves the Hermann–Mauguin symbol.

    Examples

    • (1) When the three letters of the Hermann–Mauguin symbol are the same, as in P222, Pmmm, Pnnn etc., the Hermann–Mauguin symbol does not change and all six matrices are valid.

    • (2) When the z axis plays a privileged role and when the x and y axes are equivalent, only [{\bi O}_{1}] and [{\bi O}_{6}] apply. Examples are [P222_{1}], Pbam and Ccca. In Pmma, the x and y axes are not equivalent because the interchange leads to Pmmb (the non-equivalence of the x and y axes can also be recognized by inspection of the full symbol [P2_{1}/m\ 2/m\ 2/a]).

    • (3) Matrix [{\bi O}_{1}] always applies.

13.1.2.3. Triclinic system

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As stated above, P1 has only isomorphic subgroups and the general nature of the matrix S [equation (13.1.1.3)][link] requires the use of special techniques (cf. Chapter 13.2[link] , Derivative lattices); they apply also to [P\bar{1}].

13.1.2.4. Parity conditions

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In equation (13.1.1.2b)[link], there occurs the choice of the origin by means of s, the nature of the lattice by means of [{\bi t}_{\cal G}] and the nature of the symmetry operations by means of the column matrices w and w′. The three factors, origin, lattice type, screw and glide components, impose parity conditions on the coefficients of the matrix S. Only a few examples are given here.

When (W, w) and [({\bi W}', {\bi w}')] are operators of lattice translations of [{\cal H}], say [({\bi I}, {\bi t}_{\cal G})] and [({\bi I}, {\bi t}_{\cal H})], equation (13.1.1.2b)[link] reduces to [{\bi S}\cdot {\bi t}_{\cal H} = {\bi t}_{\cal G}. \eqno(13.1.1.4)]

Example

In a tetragonal I lattice, [{\bi t}_{\cal G}] is either an integral or a fractional translation. If [{\bi t}_{\cal H}] is [({1 \over 2},{1 \over 2},{1 \over 2})] and the matrix S is replaced by [{\bi T}_{1}], one obtains [{{\bi S} \openup3pt\pmatrix{{1 \over 2}\cr {1 \over 2}\cr {1 \over 2}\cr} = \pmatrix{{1 \over 2}(S_{11} - S_{21})\cr {1 \over 2}(S_{11} + S_{21})\cr {1 \over 2}S_{33}\hfill\cr} = \pmatrix{n_{1}\cr n_{2}\cr n_{3}\cr} \hbox{ or } \pmatrix{n_{1} + {1 \over 2}\cr n_{2} + {1 \over 2}\cr n_{3} + {1 \over 2}\cr}} \eqno(13.1.1.4a)] with [n_{1}, n_{2}, n_{3}] integers.

From this it follows that either [S_{11}] and [S_{21}] have the same parity and [S_{33}] is even, or else [S_{11}] and [S_{21}] have opposite parities and [S_{33}] is odd.

If (W, w) and [({\bi W}', {\bi w}')] represent the same operation [4_{1}] in the two P lattices of [{\cal G}] and [{\cal H}], with [\hat{{\bi w}} = {\bi w}' = (00{1 \over 4})], for [{\bi s} = {\bf 0}], equation (13.1.1.2b)[link] reduces to [{\textstyle{1 \over 4}} (S_{33} - 1) = n \quad \hbox{or}\quad S_{33} = 4n + 1,\quad n\ \hbox{integer}.]

There are similar conditions for glide and other screw operations.

The location part s, i.e. the relative positions of the origins O and O′ of group and subgroup, is another important problem. This problem can be approached in two ways. First, only standard settings and origins (as defined in the space-group tables of this volume) of the group [{\cal G}] and its subgroup [{\cal H}] are considered. In this case, the origin relation between [{\cal G}] and [{\cal H}] must be indicated by the appropriate column s. For instance, in [P2_{1}2_{1}2_{1}] the matrices [{\bi O}_{4}], [{\bi O}_{5}], [{\bi O}_{6}] are only possible for [s \neq o], say [s = (-{1 \over 4},\! -{1 \over 4},\! -{1 \over 4})]. Second, one can describe [{\cal H}] based on the same origin as [{\cal G}], i.e. O and O′ coincide. In this case, nonstandard descriptions of [{\cal H}] frequently result and one has to indicate the location of the symmetry elements of [{\cal H}] with respect to the origin of [{\cal G}].

Unfortunately, it was not possible to incorporate in the present tables the implications that the choice of the origin has on the coefficients [S_{ij}] (cf. Bertaut, 1956[link]; Billiet, 1978[link]; Bertaut & Billiet, 1979[link]).

The explicit forms of the matrices S for each space group and each plane group are given in Tables 13.1.2.1[link] and 13.1.2.2[link] without origin indications.

Table 13.1.2.1| top | pdf |
Isomorphic subgroups of the plane groups

OBLIQUE SYSTEM

[{\bi S} = \pmatrix{S_{11}\hfill &S_{12}\cr S_{21}\hfill &S_{22}\cr}]
[\hbox{Conditions: } S_{11} \gt 0,S_{22} \gt 0,S_{11}S_{22} \gt 1,S_{21} = 0], [-S_{11}/2 \lt S_{12} \leq S_{11}/2]

RECTANGULAR SYSTEM

[{\bi O} = \pmatrix{S_{11}\hfill &0\hfill\cr 0\hfill &S_{22}\cr}]
[\hbox{Conditions: } S_{11} \gt 0,S_{22} \gt 0,S_{11}S_{22} \gt 1]
 [{\bi O}^{a}][{\bi O}^{b}][{\bi O}^{c}][{\bi O}^{d}][{\bi O}^{e}]
[S_{11}][n_{1}][n_{1}][2n_{1} + 1][2n_{1}][2n_{1} + 1]
[S_{22}][n_{2}][2n_{2} + 1][2n_{2} + 1][2n_{2}][n_{2}]

SQUARE SYSTEM

[{\bi T}_{1} = \pmatrix{S_{11}\hfill &-S_{21}\cr S_{21}\hfill &{\phantom -}S_{11}\cr}]
[\hbox{Conditions: } S_{11} \gt 0,S_{21} \geq 0,S_{11}^{2} + S_{21}^{2} \gt 1]
[{\bi T}_{2} = \pmatrix{S_{11}\hfill &0\hfill\cr 0\hfill &S_{11}\cr}]
[\hbox{Conditions: } {\bi T}_{2}^{a}:S_{11} \gt 1;\ {\bi T}_{2}^{b}:S_{11} = 2n_{1} + 1 \gt 1]
[{\bi T}_{3} = \pmatrix{S_{11}\hfill &-S_{11}\cr S_{11}\hfill &{\phantom -}S_{11}\cr}]
[\hbox{Condition: } S_{11} \gt 0]

HEXAGONAL SYSTEM

[{\bi H}_{1} = \pmatrix{S_{11}\hfill &-S_{21}\hfill\cr S_{21}\hfill &S_{11} - S_{21}\hfill\cr}]
[\hbox{Conditions: } S_{11} \gt 0,0 \leq S_{21} \lt S_{11}], [(S_{11}^{2} + S_{21}^{2} - S_{11}S_{21}) \gt 1]
[{\bi H}_{2} = \pmatrix{S_{11}\hfill &0\hfill\cr 0\hfill &S_{11}\cr}]
[\hbox{Condition: } S_{11} \gt 1]
[{\bi H}_{3} = \pmatrix{2S_{11}\hfill &-S_{11}\cr S_{11}\hfill &S_{11}\cr}]
[\hbox{Condition: } S_{11} \gt 0]

Table of plane subgroups

[\eqalign{&\hbox{No. }1\ p1: {\bi S};\ \hbox{No. }2\ p2: {\bi S};\ \hbox{No. }3\ pm: {\bi O}^{a};\ \hbox{No. }4\ pg: {\bi O}^{b};\cr &\hbox{No. }5\ cm: {\bi O}^{c}, {\bi O}^{d};\ \hbox{No. }6\ p2mm:{\bi O}^{a};\ \hbox{No. }7\ p2mg: {\bi O}^{e};\ \hbox{No. }8\ p2gg: {\bi O}^{c};\cr &\hbox{No. }9 \ c2mm: {\bi O}^{c}, {\bi O}^{d};\ \hbox{No. }10\ p4: {\bi T}_{1};\ \hbox{No. }11\ p4mm: {\bi T}_{2}^{a}, {\bi T}_{3};\cr &\hbox{No. }12\ p4gm: {\bi T}_{2}^{b};\ \hbox{No. }13\ p3: {\bi H}_{1};\ \hbox{No. }14\ p3m1: {\bi H}_{2};\ \hbox{No. }15\ p31m: {\bi H}_{2};\cr &\hbox{No. }16\ p6: {\bi H}_{1};\ \hbox{No. }17\ p6mm: {\bi H}_{2}, {\bi H}_{3}.}]

Table 13.1.2.2| top | pdf |
Isomorphic subgroups of the space groups

TRICLINIC SYSTEM

[{\bi S} = \pmatrix{S_{11}\hfill &S_{12}\hfill &S_{13}\cr S_{21}\hfill &S_{22}\hfill &S_{23}\cr S_{31}\hfill &S_{32}\hfill &S_{33}\cr}]
[\eqalign{\hbox{Conditions:}\ &S_{11} \gt 0, S_{22} \gt 0, S_{33} \gt 0,S_{11}S_{22}S_{33} \gt 1,\cr &S_{21} = S_{31} = S_{32} = 0, -\! S_{11}/2 \lt S_{12} \leq S_{11}/2,\cr &-S_{11}/2 \lt S_{13} \leq S_{11}/2, -\! S_{22}/2 \lt S_{23} \leq S_{22}/2}]

MONOCLINIC SYSTEM

Unique axis c
[{\bi M}_{c} = \pmatrix{S_{11}\hfill &S_{12}\hfill &0\hfill\cr S_{21}\hfill &S_{22}\hfill &0\hfill\cr 0\hfill &0\hfill &S_{33}\cr}]
[\hbox{Conditions: } S_{33} \gt 0,(S_{11}S_{22} - S_{12}S_{21})S_{33} \gt 1]
 [S_{11}][S_{12}][S_{21}][S_{22}][S_{33}]Extra condition
[{\bi M}_{c}^{a}][n_{1} \gt 0]0[n_{3}][n_{4} \gt 0][n_{5}][- n_{4}/2 \lt n_{3} \leq n_{4}/2]
[{\bi M}_{c}^{b}][n_{1} \gt 0]0[n_{3}][n_{4} \gt 0][2n_{5} + 1][- n_{4}/2 \lt n_{3} \leq n_{4}/2]
[{\bi M}_{c}^{c}][n_{1} \gt 0][2n_{2}]0[2n_{4} + 1 \gt 0][2n_{5} + 1][- n_{1}/2 \lt n_{2} \leq n_{1}/2]
[{\bi M}_{c}^{d}][n_{1} \gt 0][2n_{2}]0[2n_{4} \gt 0][2n_{5}][- n_{1}/2 \lt n_{2} \leq n_{1}/2]
[{\bi M}_{c}^{e}][n_{1}][2n_{2} \gt 0][n_{3} \lt 0]0[2n_{5}][- n_{2} \lt n_{1} \leq n_{2}]
[{\bi M}_{c}^{f}][2n_{1} + 1 \gt 0]0[2n_{3}][n_{4} \gt 0][n_{5}][- n_{4}/2 \lt n_{3} \leq n_{4}/2]
[{\bi M}_{c}^{g}][2n_{1} + 1 \gt 0][2n_{2}]0[2n_{4} + 1 \gt 0][2n_{5} + 1][- (2n_{1} + 1)/2 \lt n_{2} \leq (2n_{1} + 1)/2]
[{\bi M}_{c}^{h}][2n_{1} + 1 \gt 0][2n_{2}]0[2n_{4} \gt 0][2n_{5}][- (2n_{1} + 1)/2 \lt n_{2} \leq (2n_{1} + 1)/2]
[{\bi M}_{c}^{i}][2n_{1} + 1][2n_{2} \gt 0][n_{3} \lt 0]0[2n_{5}][- (n_{2} + 1)/2 \lt n_{1} \leq (n_{2} - 1)/2]
[{\bi M}_{c}^{\;j}][2n_{1} + 1 \gt 0]0[2n_{3}][n_{4} \gt 0][2n_{5} + 1][- n_{4}/2 \lt n_{3} \leq n_{4}/2]
Unique axis b
[{\bi M}_{b}= \pmatrix{S_{11}\hfill &0\hfill &S_{13}\cr 0\hfill &S_{22}\hfill &0\hfill\cr S_{31}\hfill &0\hfill &S_{33}\cr}]
[\hbox{Conditions: } S_{22} \gt 0,(S_{11}S_{33} - S_{13}S_{31})S_{22} \gt 1]
 [S_{11}][S_{13}][S_{22}][S_{31}][S_{33}]Extra condition
[{\bi M}_{b}^{a}][n_{1} \gt 0][n_{2}][n_{3}]0[n_{5} \gt 0][- n_{1}/2 \lt n_{2} \leq n_{1}/2]
[{\bi M}_{b}^{b}][n_{1} \gt 0][n_{2}][2n_{3} + 1]0[n_{5} \gt 0][- n_{1}/2 \lt n_{2} \leq n_{1}/2]
[{\bi M}_{b}^{c}][2n_{1} + 1 \gt 0]0[2n_{3} + 1][2n_{4}][n_{5} \gt 0][- n_{5}/2 \lt n_{4} \leq n_{5}/2]
[{\bi M}_{b}^{d}][2n_{1} \gt 0]0[2n_{3}][2n_{4}][n_{5} \gt 0][- n_{5}/2 \lt n_{4} \leq n_{5}/2]
[{\bi M}_{b}^{e}]0[n_{2} \lt 0][2n_{3}][2n_{4} \gt 0][n_{5}][- n_{4} \lt n_{5} \leq n_{4}]
[{\bi M}_{b}^{f}][n_{1}][2n_{2}][n_{3}]0[2n_{5} + 1 \gt 0][- n_{1}/2 \lt n_{2} \leq n_{1}/2]
[{\bi M}_{b}^{g}][2n_{1} + 1 \gt 0]0[2n_{3} + 1][2n_{4}][2n_{5} + 1 \gt 0][- (2n_{5} + 1)/2 \lt n_{4} \leq (2n_{5} + 1)/2]
[{\bi M}_{b}^{h}][2n_{1} \gt 0]0[2n_{3}][2n_{4}][2n_{5} + 1 \gt 0][- (2n_{5} + 1)/2 \lt n_{4} \leq (2n_{5} + 1)/2]
[{\bi M}_{b}^{i}]0[n_{2} \lt 0][2n_{3}][2n_{4} \gt 0][2n_{5} + 1][- (n_{4} + 1)/2 \lt n_{5} \leq (n_{4} - 1)/2]
[{\bi M}_{b}^{\;j}][n_{1} \gt 0][2n_{2}][2n_{3} + 1]0[2n_{5} + 1 \gt 0][- n_{1}/2 \lt n_{2} \leq n_{1}/2]

ORTHORHOMBIC SYSTEM

[{\bi O}_{1} = \pmatrix{S_{11}\hfill &0\hfill &0\hfill\cr 0\hfill &S_{22}\hfill &0\hfill\cr 0\hfill &0\hfill &S_{33}\cr}]
[\hbox{Conditions: } S_{11} \gt 0, S_{22} \gt 0, S_{33} \gt 0, S_{11}S_{22}S_{33} \gt 1]
 [S_{11}][S_{22}][S_{33}]
[{\bi O}_{1}^{a}][n_{1}][n_{2}][n_{3}]
[{\bi O}_{1}^{b}][n_{1}][n_{2}][2n_{3} + 1]
[{\bi O}_{1}^{c}][2n_{1} + 1][2n_{2} + 1][n_{3}]
[{\bi O}_{1}^{d}][2n_{1} + 1][2n_{2} + 1][2n_{3} + 1]
[{\bi O}_{1}^{e}][2n_{1}][2n_{2}][2n_{3} + 1]
[{\bi O}_{1}^{f}][2n_{1}][2n_{2}][n_{3}]
[{\bi O}_{1}^{g}][2n_{1}][2n_{2}][2n_{3}]
[{\bi O}_{1}^{h}][2n_{1} + 1][n_{2}][n_{3}]
[{\bi O}_{1}^{i}][2n_{1} + 1][n_{2}][2n_{3} + 1]
[{\bi O}_{1}^{\;j}][n_{1}][2n_{2} + 1][2n_{3} + 1]
[{\bi O}_{1}^{k}][n_{1}][2n_{2}][2n_{3}]
[{\bi O}_{1}^{l}][2n_{1} + 1][2n_{2}][2n_{3}]
[{\bi O}_{4} = \pmatrix{S_{11}\hfill &0\hfill &0\hfill\cr 0\hfill &0\hfill &S_{23}\cr 0\hfill &S_{32}\hfill &0\hfill\cr}]
[\hbox{Conditions: } S_{11} = 2n_{1} \gt 0, S_{23} = - n_{2} \lt 0, S_{32} = 2n_{3} \gt 0]
[{\bi O}_{5} = \pmatrix{0\hfill &0\hfill &S_{13}\cr 0\hfill &S_{22}\hfill &0\hfill\cr S_{31}\hfill &0\hfill &0\hfill\cr}]
[\hbox{Conditions: } S_{13} = - n_{1} \lt 0, S_{22} = 2n_{2} \gt 0, S_{31} = 2n_{3} \gt 0]
[{\bi O}_{6} = \pmatrix{0\hfill &S_{12}\hfill &0\hfill\cr S_{21}\hfill &0\hfill &0\hfill\cr 0\hfill &0\hfill &S_{33}\cr}]
[\hbox{Conditions: } S_{12} = 2n_{1} \gt 0, S_{21} = - n_{2} \lt 0, S_{33} = 2n_{3} \gt 0]

TETRAGONAL SYSTEM

[{\bi T}_{1} = \pmatrix{S_{11}\hfill &-S_{21}\hfill &0\hfill\cr S_{21}\hfill &S_{11}\hfill &0\hfill\cr 0\hfill &0\hfill &S_{33}\hfill\cr}]
[\hbox{Conditions: } S_{11} \gt 0, S_{21} \geq 0, S_{33} \gt 0, (S_{11}^{2} + S_{21}^{2})S_{33} \gt 1]
 [S_{11}][S_{21}][S_{33}]
[{\bi T}_{1}^{a}][n_{1}][n_{2}][n_{3}]
[{\bi T}_{1}^{b}][n_{1}][n_{2}][4n_{3} + 1]
[{\bi T}_{1}^{c}][n_{1}][n_{2}][4n_{3} + 3]
[{\bi T}_{1}^{d}][n_{1}][n_{2}][2n_{3} + 1]
[{\bi T}_{1}^{e}][2n_{1} + 1][2n_{2}][2n_{3} + 1]
[{\bi T}_{1}^{f}][2n_{1}][2n_{2} + 1][2n_{3} + 1]
[{\bi T}_{1}^{g}][2n_{1} + 1][2n_{2} + 1][2n_{3}]
[{\bi T}_{1}^{h}][2n_{1}][2n_{2}][2n_{3}]
[{\bi T}_{1}^{i}][2n_{1} + 1][2n_{2}][n_{3}]
[{\bi T}_{1}^{\;j}][2n_{1}][2n_{2} + 1][n_{3}]
[{\bi T}_{2} = \pmatrix{S_{11}\hfill &0\hfill &0\hfill\cr 0\hfill &S_{11}\hfill &0\hfill\cr 0\hfill &0\hfill &S_{33}\cr}]
[\hbox{Conditions: } S_{11} \gt 0,S_{33} \gt 0,S_{11} S_{33} \gt 1]
 [S_{11}] [S_{33}]
[{\bi T}_{2}^{a}][n_{1}] [n_{2}]
[{\bi T}_{2}^{b}][2n_{1} + 1] [n_{2}]
[{\bi T}_{2}^{c}][n_{1}] [4n_{2} + 1]
[{\bi T}_{2}^{d}][n_{1}] [4n_{2} + 3]
[{\bi T}_{2}^{e}][2n_{1} + 1] [4n_{2} + 1]
[{\bi T}_{2}^{f}][2n_{1} + 1] [4n_{2} + 3]
[{\bi T}_{2}^{g}][n_{1}] [2n_{2} + 1]
[{\bi T}_{2}^{h}][2n_{1} + 1] [2n_{2} + 1]
[{\bi T}_{2}^{i}][2n_{1}] [2n_{2}]
[{\bi T}_{3} = \pmatrix{S_{11}\hfill &-S_{11}\hfill &0\hfill\cr S_{11}\hfill &S_{11}\hfill &0\hfill\cr 0\hfill &0\hfill &S_{33}\cr}]
[\hbox{Conditions}: S_{11} \gt 0,S_{33} \gt 0]
[{\bi T}_{3}^{a}:S_{33} = n_{1};\ {\bi T}_{3}^{b}:S_{33} = 4n_{1} + 1;\ {\bi T}_{3}^{c}:S_{33} = 4n_{1} + 1];
[{\bi T}_{3}^{d}:S_{33} = 2n_{1} + 1;\ {\bi T}_{3}^{e}:S_{33} = 2n_{1}]

TRIGONAL AND HEXAGONAL SYSTEMS

Hexagonal axes
[{\bi H}_{1} = \pmatrix{S_{11}\hfill &-S_{21}\hfill &0\hfill\cr S_{21}\hfill &S_{11} - S_{21}\hfill &0\hfill\cr 0\hfill &0\hfill &S_{33}\cr}]
[\eqalign{\hbox{Conditions:}\ &S_{11} \gt 0, 0 \leq S_{21} \lt S_{11}, S_{33} \gt 0, (S_{11}^{2} + S_{21}^{2} - S_{11} S_{21}) S_{33} \gt 1\cr &{\bi H}_{1}^{a}:S_{33} = n_{1};\ {\bi H}_{1}^{b}:S_{33} = 3n_{1} + 1;\ {\bi H}_{1}^{c}:S_{33} = 3n_{1} + 2;\cr&{\bi H}_{1}^{d}:S_{33} = 6n_{1} + 1;\ {\bi H}_{1}^{e}:S_{33} = 6n_{1} + 5;\ {\bi H}_{1}^{f}:S_{33} = 2n_{3} + 1\cr}]
[\hbox{Conditions: } S_{11} \gt 0, S_{21} \geq 0, S_{33} \gt 0, (S_{11}^{2} + S_{21}^{2} - S_{11} S_{21}) S_{33} \gt 1]
 [S_{11}][S_{21}][S_{33}]
[{\bi H}_{1}^{g}][3n_{1} + 1][3n_{2}][3n_{3} + 1]
[{\bi H}_{1}^{h}][3n_{1} + 2][3n_{2} + 2][3n_{3} + 1]
[{\bi H}_{1}^{i}][3n_{1}][3n_{2} + 1][3n_{3} + 1]
[{\bi H}_{1}^{\;j}][3n_{1} + 1][3n_{2} + 1][3n_{3} + 2]
[{\bi H}_{1}^{k}][3n_{1} + 2][3n_{2}][3n_{3} + 2]
[{\bi H}_{1}^{l}][3n_{1}][3n_{2} + 2][3n_{3} + 2]
[{\bi H}_{1}^{m}][3n_{1} + 1][3n_{2} + 2][3n_{3}]
[{\bi H}_{1}^{n}][3n_{1} + 2][3n_{2} + 1][3n_{3}]
[{\bi H}_{1}^{o}][3n_{1}][3n_{2}][3n_{3}]
[{\bi H}_{2} = \pmatrix{S_{11}\hfill &0\hfill &0\hfill\cr 0\hfill &S_{11}\hfill &0\hfill\cr 0\hfill &0\hfill &S_{33}\cr}]
[\hbox{Conditions:}\ S_{11} \gt 0, S_{33} \gt 0, S_{11} S_{33} \gt 1]
 [S_{11}] [S_{33}]
[{\bi H}_{2}^{a}][n_{1}] [n_{2}]
[{\bi H}_{2}^{b}][n_{1}] [3n_{2} + 1]
[{\bi H}_{2}^{c}][n_{1}] [3n_{2} + 2]
[{\bi H}_{2}^{d}][3n_{1} + 1] [3n_{2} + 1]
[{\bi H}_{2}^{e}][3n_{1} + 2] [3n_{2} + 2]
[{\bi H}_{2}^{f}][3n_{1}] [3n_{2}]
[{\bi H}_{2}^{g}][n_{1}] [2n_{2} + 1]
[{\bi H}_{2}^{h}][3n_{1} + 1] [6n_{2} + 1]
[{\bi H}_{2}^{i}][3n_{1} + 2] [6n_{2} + 5]
[{\bi H}_{2}^{\;j}][3n_{1}] [6n_{2} + 3]
[{\bi H}_{2}^{k}][n_{1}] [6n_{2} + 1]
[{\bi H}_{2}^{l}][n_{1}] [6n_{2} + 5]
[{\bi H}_{3} = \pmatrix{2S_{11}\hfill &-S_{11}\hfill &0\hfill\cr S_{11}\hfill &S_{11}\hfill &0\hfill\cr 0\hfill &0\hfill &S_{33}\cr}]
[\eqalign{\hbox{Conditions:}\ &S_{11} \gt 0, S_{33} \gt 0\cr &{\bi H}_{3}^{a}:S_{33} = n_{1};\ {\bi H}_{3}^{b}:S_{33} = 6n_{1} + 1;{\bi H}_{3}^{c}:S_{33} = 6n_{1} + 5;\cr&{\bi H}_{3}^{d}:S_{33} = 3n_{1} + 1;{\bi H}_{3}^{e}:S_{33} = 3n_{1} + 2;\ {\bi H}_{3}^{f}:S_{33} = 2n_{1} + 1\cr &}]
Rhombohedral axes
[{\bi R}_{1} = \pmatrix{S_{0}\hfill &S_{2}\hfill &S_{1}\cr S_{1}\hfill &S_{0}\hfill &S_{2}\cr S_{2}\hfill &S_{1}\hfill &S_{0}\cr}]
[\eqalign{\hbox{Conditions:}\ &S_{0} \gt 0, S_{1} \leq S_{0}, - (S_{0} + S_{1}) \lt S_{2} \lt S_{0},\cr &(S_{0} + S_{1} + S_{2})(S_{0}^{2} + S_{1}^{2} + S_{2}^{2} - S_{0}S_{1} - S_{1}S_{2} - S_{2}S_{0}) \gt 1}]
[{\bi R}_{2} = \pmatrix{S_{0}\hfill &S_{1}\hfill &S_{1}\cr S_{1}\hfill &S_{0}\hfill &S_{1}\cr S_{1}\hfill &S_{1}\hfill &S_{0}\cr}]
[\eqalign{\hbox{Conditions:}\ &{\bi R}_{2}^{a}:S_{0} \gt 0, - S_{0}/2 \lt S_{1} \lt S_{0};\cr &{\bi R}_{b}^{2}:S_{0} \gt 0, - S_{0}/2 \lt S_{1} \lt S_{0}, S_{0} + 2S_{1} = 2n + 1}]

CUBIC SYSTEM

[{\bi C} = \pmatrix{S_{11}\hfill &0\hfill &0\hfill\cr 0\hfill &S_{11}\hfill &0\hfill\cr 0\hfill &0\hfill &S_{11}\cr}]
[\eqalign{\hbox{Condition:}\ &S_{11} \gt 1\cr &{\bi C}^{a}:S_{11} = n_{1};\ {\bi C}^{b}:S_{11} = 2n_{1} + 1;\cr&{\bi C}^{c}:S_{11} = 4n_{1} + 1;\ {\bi C}^{d}:S_{11} = 4n_{1} + 3}]

Table of space subgroups

No. 1 P1 : S; No. 2 [P\bar{1}:{\bi S }]; No. 3 [P112:{\bi M}_{c}^{a};\ P121:{\bi M}_{a}^{b}];
No. 4 [P112_{1}:{\bi M}_{c}^{b};\ P12_{1}1:{\bi M}_{b}^{b}]; No. 5 [A112:{\bi M}_{c}^{c},{\bi M}_{c}^{d},{\bi M}_{c}^{e}];
[C121:{\bi M}_{b}^{c},{\bi M}_{b}^{d},{\bi M}_{b}^{e}]; No. 6 [P11m:{\bi M}_{c}^{a};\ P1m1:{\bi M}_{b}^{a}]; No. 7 [P11a:{\bi M}_{c}^{f}];
[P1c1:{\bi M}_{b}^{f}]; No. 8 [A11m:{\bi M}_{c}^{c},{\bi M}_{c}^{d},{\bi M}_{c}^{e};\ C1m1:{\bi M}_{b}^{c},{\bi M}_{b}^{d},{\bi M}_{b}^{e}];
No. 9 [A11a:{\bi M}_{c}^{g},{\bi M}_{c}^{h},{\bi M}_{c}^{i};\ C1c1:{\bi M}_{b}^{g},{\bi M}_{b}^{h},{\bi M}_{b}^{i}]; No. 10 [P112/m:{\bi M}_{c}^{a}];
[P12/m1:{\bi M}_{b}^{a}]; No. 11 [P112_{1}/m:{\bi M}_{c}^{b};\ P12_{1}/m1:{\bi M}_{b}^{b}];
No. 12 [A112/m:{\bi M}_{c}^{c},{\bi M}_{c}^{d},{\bi M}_{c}^{e};\ C12/m1:{\bi M}_{b}^{c},{\bi M}_{b}^{d},{\bi M}_{b}^{e}];
No. 13 [P112/a:{\bi M}_{c}^{f};\ P12/c1:{\bi M}_{b}^{f}]; No. 14 [P112_{1}/a:{\bi M}_{c}^{\;j};\ P12_{1}/c1:{\bi M}_{b}^{\;j}];
No. 15 [A112/a:{\bi M}_{c}^{g},{\bi M}_{c}^{h},{\bi M}_{c}^{i};\ C12/c1;\ {\bi M}_{b}^{g},{\bi M}_{b}^{h},{\bi M}_{b}^{i}]; No. 16 [P222:{\bi O}_{1}^{a}];
No. 17 [P222_{1}:{\bi O}_{1}^{b}]; No. 18 [P2_{1}2_{1}2:{\bi O}_{1}^{c}]; No. 19 [P2_{1}2_{1}2_{1}:{\bi O}_{1}^{d}];
No. 20 [C222_{1}:{\bi O}_{1}^{d},{\bi O}_{1}^{e}]; No. 21 [C222:{\bi O}_{1}^{c},{\bi O}_{1}^{f},{\bi O}_{4},{\bi O}_{5}];
No. 22 [F222:{\bi O}_{1}^{d},{\bi O}_{1}^{g}]; No. 23 [I222:{\bi O}_{1}^{d},{\bi O}_{1}^{g}]; No. 24 [I2_{1}2_{1}2_{1}:{\bi O}_{1}^{d}];
No. 25 [Pmm2:{\bi O}_{1}^{a}]; No. 26 [Pmc2_{1};\ {\bi O}_{1}^{b}]; No. 27 [Pcc2:{\bi O}_{1}^{b}];
No. 28 [Pma2:{\bi O}_{1}^{h}]; No. 29 [Pca2_{1}:{\bi O}_{1}^{i}]; No. 30 [Pnc2:{\bi O}_{1}^{\;j}];
No. 31 [Pmn2_{1}:{\bi O}_{1}^{i}]; No. 32 [Pba2:{\bi O}_{1}^{c}]; No. 33 [Pna2:{\bi O}_{1}^{d}];
No. 34 [Pnn2:{\bi O}_{1}^{d}]; No. 35 [Cmm2:{\bi O}_{1}^{c},{\bi O}_{1}^{f}]; No. 36 [Cmc2_{1}:{\bi O}_{1}^{d}];
No. 37 [Ccc2:{\bi O}_{1}^{d},{\bi O}_{1}^{e}]; No. 38 [Amm2:{\bi O}_{1}^{\;j},{\bi O}_{1}^{k},{\bi O}_{6}]; No. 39 [Abm2:{\bi O}_{1}^{\;j}];
No. 40 [Ama2:{\bi O}_{1}^{d},{\bi O}_{1}^{l}]; No. 41 [Aba2:{\bi O}_{1}^{d}]; No. 42 [Fmm2:{\bi O}_{1}^{d},{\bi O}_{1}^{g}];
No. 43 [Fdd2:{\bi O}_{1}^{d}]; No. 44 [Imm2:{\bi O}_{1}^{d},{\bi O}_{1}^{g}]; No. 45 [Iba2:{\bi O}_{1}^{d}];
No. 46 [Ima2:{\bi O}_{1}^{d}]; No. 47 [Pmmm:{\bi O}_{1}^{a}]; No. 48 [Pnnn:{\bi O}_{1}^{d}];
No. 49 [Pccm:{\bi O}_{1}^{b}]; No. 50 [Pban:{\bi O}_{1}^{c}]; No. 51 [Pmma:{\bi O}_{1}^{h}]
No. 52 [Pnna:{\bi O}_{1}^{d}]; No. 53 [Pmna:{\bi O}_{1}^{i}]; No. 54 [Pcca:{\bi O}_{1}^{i}];
No. 55 [Pbam:{\bi O}_{1}^{c}]; No. 56 [Pccn:{\bi O}_{1}^{d}]; No. 57 [Pbcm:{\bi O}_{1}^{\;j}];
No. 58 [Pnnm:{\bi O}_{1}^{d}]; No. 59 [Pmmn:{\bi O}_{1}^{c}]; No. 60 [Pbcn:{\bi O}_{1}^{d}];
No. 61 [Pbca:{\bi O}_{1}^{d}]; No. 62 [Pnma:{\bi O}_{1}^{d}]; No. 63 [Cmcm:{\bi O}_{1}^{d},{\bi O}_{1}^{e}];
No. 64 [Cmca:{\bi O}_{1}^{d}]; No. 65 [Cmmm:{\bi O}_{1}^{c},{\bi O}_{1}^{f},{\bi O}_{4},{\bi O}_{5}];
No. 66 [Cccm:{\bi O}_{1}^{d},{\bi O}_{1}^{e}]; No. 67 [Cmma:{\bi O}_{1}^{c}]; No. 68 [Ccca:{\bi O}_{1}^{d}];
No. 69 [Fmmm:{\bi O}_{1}^{d},{\bi O}_{1}^{g}]; No. 70 [Fddd:{\bi O}_{1}^{d}]; No. 71 [Immm:{\bi O}_{1}^{d},{\bi O}_{1}^{g}];
No. 72 [Ibam:{\bi O}_{1}^{d}]; No. 73 [Ibca:{\bi O}_{1}^{d}]; No. 74 [Imma:{\bi O}_{1}^{d}];
No. 75 [P4:{\bi T}_{1}^{a}]; No. 76 [P4_{1}:{\bi T}_{1}^{b}\hbox{(subgroups}\ P4_{1}), {\bi T}_{1}^{c}(\hbox{subgroups } P4_{3})];
No. 77 [P4_{2}:{\bi T}_{1}^{d}]; No. 78 [P4_{3}:{\bi T}_{1}^{b}(\hbox{subgroups}\ P4_{3}), {\bi T}_{1}^{c}(\hbox{subgroups}\ P4_{1})];
No. 79 [I4:{\bi T}_{1}^{e},{\bi T}_{1}^{f},{\bi T}_{1}^{g},{\bi T}_{1}^{h}]; No. 80 [I4_{1}:{\bi T}_{1}^{e},{\bi T}_{1}^{f}]; No. 81 [P\bar{4}:{\bi T}_{1}^{a}];
No. 82 [I\bar{4}:{\bi T}_{1}^{e},{\bi T}_{1}^{f},{\bi T}_{1}^{g},{\bi T}_{1}^{h}]; No. 83 [P4/m:{\bi T}_{1}^{a}]; No. 84 [P4_{2}/m:{\bi T}_{1}^{d}];
No. 85 [P4/n:{\bi T}_{1}^{i},{\bi T}_{1}^{\;j}]; No. 86 [P4_{2}/n:{\bi T}_{1}^{e},{\bi T}_{1}^{f}];
No. 87 [I4/m:{\bi T}_{1}^{e},{\bi T}_{1}^{f},{\bi T}_{1}^{g},{\bi T}_{1}^{h}]; No. 88 [I4_{1}/a:{\bi T}_{1}^{e},{\bi T}_{1}^{f}];
No. 89 [P422:{\bi T}_{2}^{a},{\bi T}_{3}^{a}]; No. 90 [P42_{1}2:{\bi T}_{2}^{b}];
No. 91 [P4_{1}22:{\bi T}_{2}^{c}(\hbox{subgroups}\ P4_{1}22), {\bi T}_{2}^{d}(\hbox{subgroups}\ P4_{3}22)],
[{\bi T}_{3}^{b}(\hbox{subgroups}\ P4_{1}22), {\bi T}_{3}^{c}(\hbox{subgroups}\ P4_{3}22)];
No. 92 [P4_{1}2_{1}2:{\bi T}_{2}^{e}(\hbox{subgroups}\ P4_{1}2_{1}2), {\bi T}_{2}^{f}(\hbox{subgroups}\ P4_{3}2_{1}2)];
No. 93 [P4_{2}22:{\bi T}_{2}^{g},{\bi T}_{3}^{d}]; No. 94 [P4_{2}2_{1}2:{\bi T}_{2}^{h}];
No. 95 [P4_{3}22:{\bi T}_{2}^{c}(\hbox{subgroups}\ P4_{3}22),{\bi T}_{2}^{d}(\hbox{subgroups}\ P4_{1}22)],
[{\bi T}_{3}^{b}(\hbox{subgroups}\ P4_{3}22), {\bi T}_{3}^{c}(\hbox{subgroups}\ P4_{1}22)];
No. 96 [P4_{3}2_{1}2:{\bi T}_{2}^{e}(\hbox{subgroups}\ P4_{3}2_{1}2), {\bi T}_{2}^{f}(\hbox{subgroups}\ P4_{1}2_{1}2)];
No. 97 [I422:{\bi T}_{2}^{h},{\bi T}_{2}^{i},{\bi T}_{3}^{e}]; No. 98 [I4_{1}22:{\bi T}_{2}^{h}]; No. 99 [P4mm:{\bi T}_{2}^{a},{\bi T}_{3}^{a}];
No. 100 [P4bm:{\bi T}_{2}^{b}]; No. 101 [P4_{2}cm:{\bi T}_{2}^{g}]; No. 102 [P4_{2}nm:{\bi T}_{2}^{h}];
No. 103 [P4cc:{\bi T}_{2}^{g},{\bi T}_{3}^{d}]; No. 104 [P4nc:{\bi T}_{2}^{h}]; No. 105 [P4_{2}mc:{\bi T}_{2}^{g}];
No. 106 [P4_{2}bc:{\bi T}_{2}^{h}]; No. 107 [I4mm:{\bi T}_{2}^{h},{\bi T}_{2}^{i},{\bi T}_{3}^{e}]; No. 108 [I4cm:{\bi T}_{2}^{h}];
No. 109 [I4_{1}md:{\bi T}_{2}^{h}]; No. 110 [I4_{1}cd:{\bi T}_{2}^{h}]; No. 111 [P\bar{4}2m:{\bi T}_{2}^{a}];
No. 112 [P\bar{4}2c:{\bi T}_{2}^{g}]; No. 113 [P\bar{4}2_{1}m:{\bi T}_{2}^{b}]; No. 114 [P\bar{4}2_{1}c:{\bi T}_{2}^{h}];
No. 115 [P\bar{4}m2:{\bi T}_{2}^{a}]; No. 116 [P\bar{4}c2:{\bi T}_{2}^{g}]; No. 117 [P\bar{4}b2:{\bi T}_{2}^{b}];
No. 118 [P\bar{4}n2:{\bi T}_{2}^{h}]; No. 119 [I\bar{4}m2:{\bi T}_{2}^{h},{\bi T}_{2}^{i}]; No. 120 [I\bar{4}c2:{\bi T}_{2}^{h}];
No. 121 [I\bar{4}2m:{\bi T}_{2}^{h},{\bi T}_{2}^{i}]; No. 122 [I\bar{4}2d:{\bi T}_{2}^{h}]; No. 123 [P4/mmm:{\bi T}_{2}^{a},{\bi T}_{3}^{a}];
No. 124 [P4/mcc:{\bi T}_{2}^{g},{\bi T}_{3}^{d}]; No. 125 [P4/nbm:{\bi T}_{2}^{b}]; No. 126 [P4/nnc:{\bi T}_{2}^{h}];
No. 127 [P4/mbm:{\bi T}_{2}^{b}]; No. 128 [P4/mnc:{\bi T}_{2}^{h}]; No. 129 [P4/nmm:{\bi T}_{2}^{b}];
No. 130 [P4/ncc: {\bi T}_{2}^{h}]; No. 131 [P4_{2}/mmc: {\bi T}_{2}^{g}]; No. 132 [P4_{2}/mcm: {\bi T}_{2}^{g}];
No. 133 [P4_{2}/nbc: {\bi T}_{2}^{h}]; No. 134 [P4_{2}/nnm: {\bi T}_{2}^{h}]; No. 135 [P4_{2}/mbc: {\bi T}_{2}^{h}];
No. 136 [P4_{2}/mnm: {\bi T}_{2}^{h}]; No. 137 [P4_{2}/nmc: {\bi T}_{2}^{h}]; No. 138 [P4_{2}/ncm: {\bi T}_{2}^{h}];
No. 139 [I4/mmm: {\bi T}_{2}^{h}], [{\bi T}_{2}^{i}], [{\bi T}_{3}^{e}]; No. 140 [I4/mcm: {\bi T}_{2}^{h}];
No. 141 [I4_{1}/amd: {\bi T}_{2}^{h}]; No. 142 [I4_{1}/acd: {\bi T}_{2}^{h}]; No. 143 [P3: {\bi H}_{1}^{a}];
No. 144 [P3_{1}: {\bi H}_{1}^{b}(\hbox{subgroups}\ P3_{1})], [{\bi H}_{1}^{c}(\hbox{subgroups}\ P3_{2})];
No. 145 [P3_{2}: {\bi H}_{1}^{b}(\hbox{subgroups}\ P3_{2})], [{\bi H}_{1}^{c}(\hbox{subgroups}\ P3_{1})];
No. 146 R3 (hexagonal axes): [{\bi H}_{1}^{g}], [{\bi H}_{1}^{h}], [{\bi H}_{1}^{i}], [{\bi H}_{1}^{\;j}], [{\bi H}_{1}^{k}], [{\bi H}_{1}^{l}], [{\bi H}_{1}^{m}], [{\bi H}_{1}^{n}], [{\bi H}_{1}^{o}];
R3 (rhombohedral axes): [{\bi R}_{1}]; No. 147 [P\bar{3}: {\bi H}_{1}^{a}];
No. 148 [R\bar{3}] (hexagonal axes): [{\bi H}_{1}^{g}], [{\bi H}_{1}^{h}], [{\bi H}_{1}^{i}], [{\bi H}_{1}^{\;j}], [{\bi H}_{1}^{k}], [{\bi H}_{1}^{l}], [{\bi H}_{1}^{m}], [{\bi H}_{1}^{n}], [{\bi H}_{1}^{o}];
[R\bar{3}] (rhombohedral axes): [{\bi R}_{1}]; No. 149 [P312: {\bi H}_{2}^{a}]; No. 150 [P321: {\bi H}_{2}^{a}];
No. 151 [P3_{1}12: {\bi H}_{2}^{b}(\hbox{subgroups}\ P3_{1}12)], [{\bi H}_{2}^{c}(\hbox{subgroups}\ P3_{2}12)];
No. 152 [P3_{1}21: {\bi H}_{2}^{b}(\hbox{subgroups}\ P3_{1}21)], [{\bi H}_{2}^{c}(\hbox{subgroups}\ P3_{2}21)];
No. 153 [P3_{2}12: {\bi H}_{2}^{b}(\hbox{subgroups}\ P3_{2}12)], [{\bi H}_{2}^{c}(\hbox{subgroups}\ P3_{1}12)];
No. 154 [P3_{2}21: {\bi H}_{2}^{b}(\hbox{subgroups}\ P3_{2}21)], [{\bi H}_{2}^{c}(\hbox{subgroups}\ P3_{1}21)];
No. 155 R32 (hexagonal axes): [{\bi H}_{2}^{d}], [{\bi H}_{2}^{e}], [{\bi H}_{2}^{f}]; R32 (rhombohedral axes):
[{\bi R}_{2}^{a}]; No. 156 [P3m1: {\bi H}_{2}^{a}]; No. 157 [P31m: {\bi H}_{2}^{a}]; No. 158 [P3c1: {\bi H}_{2}^{g}];
No. 159 [P31c: {\bi H}_{2}^{g}]; No. 160 R3m (hexagonal axes): [{\bi H}_{2}^{d},{\bi H}_{2}^{e},{\bi H}_{2}^{f}];
R3m (rhombohedral axes): [{\bi R}_{2}^{a}]; No. 161 R3c (hexagonal axes):
[{\bi H}_{2}^{h},{\bi H}_{2}^{i},{\bi H}_{2}^{\;j}]; R3c (rhombohedral axes): [{\bi R}_{2}^{b}]; No. 162 [P\bar{3}1m: {\bi H}_{2}^{a}];
No. 163 [P\bar{3}1c: {\bi H}_{2}^{g}]; No. 164 [P\bar{3}m1: {\bi H}_{2}^{a}]; No. 165 [P\bar{3}c1: {\bi H}_{2}^{g}];
No. 166 [R\bar{3}m] (hexagonal axes): [{\bi H}_{2}^{d},{\bi H}_{2}^{e},{\bi H}_{2}^{f}]; [R\bar{3}m] (rhombohedral
axes): [{\bi R}_{2}^{a}]; No. 167 [R\bar{3}c] (hexagonal axes): [{\bi H}_{2}^{h},{\bi H}_{2}^{i},{\bi H}_{2}^{\;j}];
[R\bar{3}c] (rhombohedral axes): [{\bi R}_{2}^{b}]; No. 168 P6: [{\bi H}_{1}^{a}];
No. 169 [P6_{1}: {\bi H}_{1}^{d}(\hbox{subgroups}\ P6_{1})], [{\bi H}_{1}^{e}(\hbox{subgroups}\ P6_{5})];
No. 170 [P6_{5}:{\bi H}_{1}^{d}(\hbox{subgroups}\ P6_{5})], [{\bi H}_{1}^{e}(\hbox{subgroups}\ P6_{1})];
No. 171 [P6_{2}:{\bi H}_{1}^{b}(\hbox{subgroups}\ P6_{2})], [{\bi H}_{1}^{c}(\hbox{subgroups}\ P6_{4})];
No. 172 [P6_{4}:{\bi H}_{1}^{b}(\hbox{subgroups}\ P6_{4})], [{\bi H}_{1}^{c}(\hbox{subgroups}\ P6_{2})];
No. 173 [P6_{3}: {\bi H}_{1}^{f}]; No. 174 [P\bar{6}: {\bi H}_{1}^{a}]; No. 175 [P6/m: {\bi H}_{1}^{a}];
No. 176 [P6_{3}/m: {\bi H}_{1}^{f}]; No. 177 [P622: {\bi H}_{2}^{a}]; [{\bi H}_{3}^{a}];
No. 178 [P6_{1}22: {\bi H}_{2}^{k}(\hbox{subgroups}\ P6_{1}22)], [{\bi H}_{2}^{i}(\hbox{subgroups}\ P6_{5}22)],
[{\bi H}_{3}^{b}(\hbox{subgroups}\ P6_{1}22)], [{\bi H}_{3}^{c}(\hbox{subgroups}\ P6_{5}22)];
No. 179 [P6_{5}22:{\bi H}_{2}^{k}(\hbox{subgroups}\ P6_{5}22)], [{\bi H}_{2}^{l}(\hbox{subgroups}\ P6_{1}22)],
[{\bi H}_{3}^{b}(\hbox{subgroups}\ P6_{5}22)], [{\bi H}_{3}^{c}(\hbox{subgroups}\ P6_{1}22)];
No. 180 [P6_{2}22:{\bi H}_{2}^{b}(\hbox{subgroups}\ P6_{2}22)], [{\bi H}_{2}^{c}(\hbox{subgroups}\ P6_{4}22)],
[{\bi H}_{3}^{d}(\hbox{subgroups}\ P6_{2}22)], [{\bi H}_{3}^{e}(\hbox{subgroups}\ P6_{4}22)];
No. 181 [P6_{4}22: {\bi H}_{2}^{b}(\hbox{subgroups}\ P6_{4}22)], [{\bi H}_{2}^{c}(\hbox{subgroups}\ P6_{2}22)],
[{\bi H}_{3}^{d}(\hbox{subgroups}\ P6_{4}22)], [{\bi H}_{3}^{e}(\hbox{subgroups}\ P6_{2}22)]; No. 182 [P6_{3}22: {\bi H}_{2}^{g},{\bi H}_{3}^{f}];
No. 183 [P6mm: {\bi H}_{2}^{a},{\bi H}_{3}^{a}]; No. 184 [P6cc: {\bi H}_{2}^{g},{\bi H}_{3}^{f}]; No. 185 [P6_{3}cm: {\bi H}_{2}^{g}];
No. 186 [P6_{3}mc: {\bi H}_{2}^{g}]; No. 187 [P\bar{6}m2: {\bi H}_{2}^{a}]; No. 188 [P\bar{6}c2: {\bi H}_{2}^{g}];
No. 189 [P\bar{6}2m: {\bi H}_{2}^{a}]; No. 190 [P\bar{6}2c: {\bi H}_{2}^{g}]; No. 191 [P6/mmm:{\bi H}_{2}^{a},{\bi H}_{3}^{a}];
No. 192 [P6/mcc: {\bi H}_{2}^{g},{\bi H}_{3}^{f}]; No. 193 [P6_{3}/mcm: {\bi H}_{2}^{g}];
No. 194 [P6_{3}/mmc: {\bi H}_{2}^{g}]; No. 195 [P23:{\bi C}^{a}]; No. 196 [F23:{\bi C}^{a}];
No. 197 [I23:{\bi C}^{a}]; No. 198 [P2_{1}3: {\bi C}^{b}]; No. 199 [I2_{1}3: {\bi C}^{b}];
No. 200 [Pm\bar{3}: {\bi C}^{a}]; No. 201 [Pn\bar{3}: {\bi C}^{b}]; No. 202 [Fm\bar{3}: {\bi C}^{a}];
No. 203 [Fd\bar{3}: {\bi C}^{b}]; No. 204 [Im\bar{3}: {\bi C}^{a}]; No. 205 [Pa\bar{3}: {\bi C}^{b}];
No. 206 [Ia\bar{3}: {\bi C}^{b}]; No. 207 [P432: {\bi C}^{a}]; No. 208 [P4_{2}32: {\bi C}^{b}];
No. 209 [F432: {\bi C}^{a}]; No. 210 [F4_{1}32: {\bi C}^{b}]; No. 211 [I432: {\bi C}^{a}];
No. 212 [P4_{3}32: {\bi C}^{c}(\hbox{subgroups}\ P4_{3}32)], [{\bi C}^{d}(\hbox{subgroups}\ P4_{1}32)];
No. 213 [P4_{1}32:{\bi C}^{c}(\hbox{subgroups}\ P4_{1}32)], [{\bi C}^{d}(\hbox{subgroups}\ P4_{3}32)];
No. 214 [I4_{1}32: {\bi C}^{b}]; No. 215 [P\bar{4}3m: {\bi C}^{a}]; No. 216 [F\bar{4}3m: {\bi C}^{a}];
No. 217 [I\bar{4}3m: {\bi C}^{a}]: No. 218 [P\bar{4}3n: {\bi C}^{b}]; No. 219 [F\bar{4}3c: {\bi C}^{b}];
No. 220 [I\bar{4}3d: {\bi C}^{b}]; No. 221 [Pm\bar{3}m: {\bi C}^{a}]; No. 222 [Pn\bar{3}n: {\bi C}^{b}];
No. 223 [Pm\bar{3}n: {\bi C}^{b}]; No. 224 [Pn\bar{3}m: {\bi C}^{b}]; No. 225 [Fm\bar{3}m: {\bi C}^{a}];
No. 226 [Fm\bar{3}c: {\bi C}^{b}]; No. 227 [Fd\bar{3}m: {\bi C}^{b}]; No. 228 [Fd\bar{3}c: {\bi C}^{b}];
No. 229 [Im\bar{3}m: {\bi C}^{a}]; No. 230 [Ia\bar{3}d: {\bi C}^{b}].

Example

Consider space groups [P4/mmm] and [P4/mcc] and envisage the matrix [{\bi T}_{3}] which has the determinant [2S_{33}S_{11}^{2}]. Its lowest value is 2 with [S_{11} = S_{33} = 1]. The matrix [{\bi T}_{3}] then implies the axis transformation [{\bf a}' = {\bf a} + {\bf b},\quad {\bf b}' = - {\bf a} + {\bf b},\quad {\bf c}' = {\bf c.}]

Envisage also the matrix [{\bi T}_{2}] which has the determinant [S_{11}^{2}S_{33}]. Its lowest value for a proper subgroup is 2 with [S_{11} = 1] and [S_{33} = 2]. The axis transformation is here [{\bf a}' = {\bf a},\quad {\bf b}' = {\bf b},\quad {\bf c}' = 2{\bf c.}]

This transformation is permitted for [P4/mmm] but not for [P4/mcc] where the parity rules require [S_{33} = 2n + 1]. Thus, the lowest value of [S_{11}^{2}S_{33}] for a subgroup of space group [P4/mcc] is 3 and the axis transformation is [{\bf a}' = {\bf a},\quad {\bf b}' = {\bf b},\quad {\bf c}' = 3{\bf c.}]

13.1.2.5. Plane groups

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There is no difficulty in reducing the preceding considerations to plane groups, where S is a [(2 \times 2)] matrix and s a [(2 \times 1)] matrix.

13.1.2.6. Tables of matrices for isomorphic subgroups

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The matrices and the restrictions on the coefficients are listed for the plane groups in Table 13.1.2.1[link] and for the space groups in Table 13.1.2.2[link].

For the triclinic and monoclinic systems, there is an infinite choice of matrices for each index, owing to the infinite number of equivalent unit cells. For the other space groups, several different (but finitely many) choices of matrix occur. In all cases, we have restricted this choice to one matrix for each group–subgroup relation so that each subgroup is listed exactly once (apart from origin choice).

Example

No. 178, [P6_{1}22], has the matrix [{\bi H}_{2}^{k}] for all isomorphic and isosymbolic subgroups [P6_{1}22], having the lattices [n_{11}{\bf a}], [n_{11}{\bf b}], [(6n_{33} + 1) {\bf c}\ (n_{11} \geq 1, n_{33} \geq 0)], whilst [{\bi H}_{3}^{b}] is used for all isomorphic and isosymbolic subgroups with the lattices [n_{11}(2{\bf a} + {\bf b})], [n_{11}( - {\bf a} + {\bf b})], [(6n_{33} + 1){\bf c}] [ (n_{11} \geq 1, n_{33} \geq 0)]. These two kinds of subgroups are obviously different, having different translation lattices. The same group, [P6_{1}22], has the matrix [{\bi H}_{2}^{l}] for the isomorphic and enantiomorphic subgroups [P6_{5}22] of lattices [n_{11}{\bf a}], [n_{11}{\bf b}], [(6n_{33} + 5){\bf c}] whilst [{\bi H}_{3}^{c}] is used for the isomorphic and enantiomorphic subgroups [P6_{5}22] of lattices [n_{11}(2{\bf a} + {\bf b})], [n_{11}( - {\bf a} + {\bf b})], [(6n_{33} + 5){\bf c}].

In the tables, each system is preceded by the appropriate general form of the matrix, which is also given in this chapter, followed by the more specialized matrices such as [{\bi T}_{2}, {\bi T}_{3}]. Under Conditions, we have listed the nonredundant inequalities and parity conditions3 that ensure the uniqueness of the matrix for each subgroup. Also, we have used rules in order to avoid repetition of equivalent unit cells. For instance, for trigonal and hexagonal groups (rhombohedral groups excepted), we have restricted [{\bf a}'] to lie between a and a + b excluding this last vector from the sector of 60° because there is a repetition after a 60° rotation of the unit cell.

References

First citation Bertaut, E. F. (1956). Structure de FeS stoechiométrique. Bull. Soc. Fr. Minéral. Cristallogr. 79, 276–292.Google Scholar
 First citation Bertaut, E. F. & Billiet, Y. (1979). On equivalent subgroups and supergroups of the space groups. Acta Cryst. A35, 733–745.Google Scholar
 First citation Billiet, Y. (1973). Les sous-groupes isosymboliques des groupes spatiaux. Bull. Soc. Fr. Minéral. Cristallogr. 96, 327–334.Google Scholar
 First citation Billiet, Y. (1978). Some remarks on the `family tree' of Bärnighausen. Acta Cryst. A34, 1023–1025.Google Scholar
 First citation Billiet, Y. (1979). Le groupe P1 et ses sous-groupes. I. Outillage mathématique: automorphisme et factorisation matricielle. Acta Cryst. A35, 485–496.Google Scholar
 First citation Billiet, Y. & Rolley Le Coz, M. (1980). Le groupe P1 et ses sous-groupes. II. Tables de sous-groupes. Acta Cryst. A36, 242–248.Google Scholar








































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