Tables for
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

International Tables for Crystallography (2011). Vol. A1, ch. 2.1, pp. 76-77   | 1 | 2 |

Section A description in close analogy with IT A

Hans Wondratscheka* and Mois I. Aroyob

aInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany, and bDepartamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain
Correspondence e-mail: A description in close analogy with IT A

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The listing is similar to that of IT A and presents on one line the following information for each subgroup [{\cal H}]: [[i]\ \hbox{HMS1 (No., HMS2) \hskip2em sequence \hskip2em matrix \hskip2em shift}]Conjugate subgroups are listed together and are connected by a left brace.

The symbols have the following meaning:

[i] index of [{\cal H}] in [{\cal G}];
HMS1 HM symbol of [{\cal H}] referred to the coordinate system and setting of [{\cal G}]. This symbol may be nonconventional;
No. space-group No. of [{\cal H}];
HMS2 conventional HM symbol of [{\cal H}] if HMS1 is not a conventional HM symbol;
sequence sequence of numbers; the numbers refer to those coordinate triplets of the general position of [{\cal G}] that are retained in [{\cal H}], cf. Remarks; for general position cf. Section[link];
matrix matrix part of the transformation to the conventional setting corresponding to HMS2, cf. Section[link];
shift column part of the transformation to the conventional setting corresponding to HMS2, cf. Section[link].


  • In the sequence column for space groups with centred lattices, the abbreviation `(numbers)[+]' means that the coordinate triplets specified by `numbers' are to be taken plus those obtained by adding each of the centring translations, see the comments following Examples[link] and[link].

  • The symbol HMS2 is omitted if HMS1 is a conventional HM symbol.

  • The following deviations from the listing of IT A are introduced in these tables:

    • No.: the space-group No. of [{\cal H}] is added.

    • HMS2: In order to specify the setting clearly, the full HM symbol is listed for monoclinic subgroups, not the standard (short) HM symbol as in IT A.

    • matrix, shift: These entries contain information on the trans­formation of [{\cal H}] from the setting of [{\cal G}] to the standard setting of [{\cal H}]. They are explained in Section[link].

In general, the numbers in the list `Sequence' of [\cal H] follow the order of the numbers in the group [\cal G], i.e. they rise monotonically. Sometimes this sequence is modified because those entries which have the same additional translations are joined together, see, e.g. the maximal k-subgroups of [Fm\overline3 m] with `Loss of centring translations'. In addition, in a class of conjugate subgroups, the mono­tonically rising order may be obeyed only for the first member of the conjugacy class. The order of the other members is then determined by the conjugation of the first member. (In IT A the monotonically rising order of the numbers is kept in all conjugate subgroups.)


[{\cal G}=Pm\overline 3m], No. 221, tetragonal t-subgroups

I Maximal translationengleiche subgroups[\Biggl\{\matrix{[3]\,\,P4/m12/m\,\,(123, P4/mmm)\quad1\semi2\semi3\semi4\semi13\semi14\semi15\semi16\semi\ldots\hfill\cr[3]\,\,P4/m12/m\,\,(123, P4/mmm)\quad1\semi4\semi2\semi3\semi18\semi19\semi17\semi20\semi\ldots\cr[3]\,\,P4/m12/m\,\,(123, P4/mmm)\quad1\semi3\semi4\semi2\semi22\semi24\semi23\semi21\semi\ldots}]Comments:

If [{\bi W}_1,{\bi W}_2, {\bi W}_3,\ldots] is the order of the first sequence, then the second sequence follows the order [{\bi C}^{-1}{\bi W}_1{\bi C}], [{\bi C}^{-1}{\bi W}_2{\bi C}], [{\bi C}^{-1}{\bi W}_3{\bi C},\ldots]. Here the C means a threefold rotation and is the conjugating element; for the second subgroup [{\bi C}=(9)\ y,z,x] of the general position of [Pm\overline 3m]; for the third subgroup [{\bi C}=(5)\ z,x,y]. In this example the columns w of the symmetry operations (and thus of the conjugating elements) are the zero columns o and could be omitted.

The description of the subgroups can be explained by the following four examples.


[{\cal G}=C1m1], No. 8, UNIQUE AXIS b

I Maximal translationengleiche subgroups

[[2] \,\, C1\,\, (1, \,\, P1) \quad 1+]


  • HMS1: [C1] is not a conventional HM symbol. Therefore, the conventional symbol [P1] is added as HMS2 after the space-group number 1 of [{\cal H}].

  • sequence: `1[+]' means [x,y,z]; [x\, +\, {{1}\over{2}},y\, +\,{{1}\over{2}},z].


[{\cal G}=Fdd2], No. 43

I Maximal translationengleiche subgroups


[[2]\,\, F112\,\, (5,\, A112)\quad (1;\,2)+]


  • HMS1: [F112] is not a conventional HM symbol; therefore, the conventional symbol [A112] is added to the space-group No. 5 as HMS2. The setting unique axis c is inherited from [{\cal G}].

  • sequence: (1, 2)[+] means:[\quad\matrix{x,y,z\semi &x,y+{{1}\over{2}}, z+{{1}\over{2}}\semi &x+{{1}\over{2}},y,z+{{1}\over{2}}\semi &x+{{1}\over{2}}, y+{{1}\over{2}}, z\semi \cr {\bar x},{\bar y},z\semi &{\bar x},{\bar y}+{{1}\over{2}},z+{{1}\over{2}}\semi &{\bar x}+{{1}\over{2}}, {\bar y},z+{{1}\over{2}}\semi &{\bar x}+{{1}\over{2}}, {\bar y}+{{1}\over{2}}, z\semi}]


[{\cal G}=P\,4_{2}/nmc=P4_{2}/n\, 2_{1}/m\, 2/c], No. 137, ORIGIN CHOICE 2

I Maximal translationengleiche subgroups

[[2]\,\,P2/n\ 2_{1}/m\ 1] (59, [Pmmn]) 1; 2; 5; 6; 9; 10; 13; 14


  • HMS1: The sequence of the directions in the HM symbol for a tetragonal space group is c, a, [{\bf a-b}]. From the parts [4_{2}/n], [2_{1}/m] and [2/c] of the full HM symbol of [{\cal G}], only [2/n], [2_{1}/m] and 1 remain in [{\cal H}]. Therefore, HMS1 is [P2/n\,2_{1}/m\,1], and the con­ven­tional symbol [Pmmn] is added as HMS2.

  • No.: The space-group number of [{\cal H}] is 59. The setting origin choice 2 of [{\cal H}] is inherited from [{\cal G}].

  • sequence: The coordinate triplets of [{\cal G}] retained in [{\cal H}] are: (1) [x,y,z]; (2) [\overline{x}+{{1}\over{2}},\overline{y}+{{1}\over{2}},z]; (5) [\overline{x},y+{{1}\over{2}},\overline{z}]; (6) [x+{{1}\over{2}},\overline{y}, \overline{z}]; (9) [\overline{x}, \overline{y}, \overline{z}]; etc.


[{\cal G}=P3_{1} 12], No. 151

I Maximal translationengleiche subgroups[\ \matrix{\,\ \ [2] \, P3_{1}11\,\, (144,\,P3_{1})\,\, 1\semi 2\semi 3 \hfill\cr \Biggl\{\matrix{[3]\, P112 \,\, (5, C121)\hfill & {1\semi 6\hfill} & {\bf b},-2{\bf a}- {\bf b}, {\bf c} &\cr [3]\, P112 \,\, (5, C121)\hfill &{1\semi 4\hfill} & -{\bf a} -{ \bf b}, {\bf a} -{\bf b}, {\bf c} & 0,0,1/3 \cr [3]\, P112 \,\, (5, C121)\hfill &{1\semi 5\hfill} &{\bf a}, {\bf a} + 2 {\bf b}, {\bf c} \hfil &0,0, 2/3 \hfil\cr}}]Comments:

  • brace: The brace on the left-hand side connects the three conjugate monoclinic subgroups.

  • HMS1: [P112] is not the conventional HM symbol for unique axis c but the constituent `2' of the nonconventional HM symbol refers to the directions [-2{\bf a}-{\bf b}], [{\bf a}-{\bf b}] and [{\bf a} + 2{\bf b}], in the hexagonal basis. According to the rules of Section[link], the standard setting is unique axis b, as expressed by the HM symbol [C121].

  • HMS2: Note that the conventional monoclinic cell is centred.

  • matrix, shift: The entries in the columns `matrix' and `shift' are explained in the following Section[link] and evaluated in Example[link].

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