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

International Tables for Crystallography (2006). Vol. A1, ch. 2.1, pp. 45-48   | 1 | 2 |

Section 2.1.3. I Maximal translationengleiche subgroups (t-subgroups)

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:

2.1.3. I Maximal translationengleiche subgroups (t-subgroups)

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In this block, all maximal t-subgroups [{\cal H}] of the plane groups and the space groups [{\cal G}] are listed individually. Maximal t-subgroups are always non-isomorphic.

For the sequence of the subgroups, see Section[link]. There are no lattice relations for t-subgroups because the lattice is retained. Therefore, the sequence is determined only by the rising value of the index and by the decreasing space-group number. 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].

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


  • 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]. Basis transformation and origin shift

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Each t-subgroup [{\cal H} \,\lt\, {\cal G}] is defined by its representatives, listed under `sequence' by numbers each of which designates an element of [{\cal G}]. These elements form the general position of [{\cal H}]. They are taken from the general position of [{\cal G}] and, therefore, are referred to the coordinate system of [{\cal G}]. In the general position of [{\cal H}], however, its elements are referred to the coordinate system of [{\cal H}]. In order to allow the transfer of the data from the coordinate system of [{\cal G}] to that of [{\cal H}], the tools for this transformation are provided in the columns `matrix' and `shift' of the subgroup tables. The designation of the quantities is that of IT A Part 5[link] and is repeated here for convenience.

In the following, columns and rows are designated by boldface italic lower-case letters. Point coordinates [{\bi x},{\bi x}'], translation parts [{\bi w}, {\bi w}'] of the symmetry operations and shifts [{\bi p}, {\bi q}=-{\bi P}^{-1}{\bi p}] are represented by columns. The sets of basis vectors [({\bf a},\,{\bf b},\,{\bf c})=({\bi a})^{\rm T}] and [({\bf a}',\,{\bf b}',\,{\bf c}')=({\bi a}')^{\rm T}] are represented by rows [indicated by [(\ldots)^{\rm T}], which means `transposed']. The quantities with unprimed symbols are referred to the coordinate system of [{\cal G}], those with primes are referred to the coordinate system of [{\cal H}].

The following columns will be used ([{\bi w}'] is analogous to w): [{\bi w} =  \left(\matrix { w_{1}\cr w_{2} \cr w_{3} } \right); \, \, {\bi x} = \left(\matrix { x \cr y \cr z } \right); \,\, {\bi x}\,'  =\left(\matrix { x' \cr y' \cr z' } \right); \,\, {\bi p} =  \left(\matrix { p_{1}\cr p_{2} \cr p_{3} } \right); \,\, {\bi q} = \left(\matrix { q_{1}\cr q_{2} \cr q_{3} } \right).]

The [(3\times3)] matrices W and [{\bi W}\,'] of the symmetry operations, as well as the matrix P for a change of basis and its inverse [{\bi Q}={\bi P}^{-1}], are designated by boldface italic upper-case letters ([{\bi W}\,'] is analogous to W): [{\bi W} = \left(\matrix {W_{11} &W_{12} &W_{13} \cr W_{21} & W_{22} & W_{23} \cr W_{31} & W_{32} & W_{33} } \right)\semi {\bi P} = \left(\matrix{P_{11} & P_{12} & P_{13} \cr P_{21} & P_{22} & P_{23} \cr P_{31} & P_{32} & P_{33} } \right)\semi {\bi Q} = \left(\matrix { Q_{11} & Q_{12} & Q_{13} \cr Q_{21} & Q_{22} & Q_{23} \cr Q_{31} & Q_{32} & Q_{33} } \right).]Let [{\bf a}, {\bf b}, {\bf c} = ({\bi a})^{\rm T}] be the row of basis vectors of [{\cal G}] and [{\bf a}', {\bf b}', {\bf c}'\,=\, ({\bi a}')^{\rm T}] the basis of [{\cal H}], then the basis [({\bi a}')^{\rm T}] is expressed in the basis [({\bi a})^{\rm T}] by the system of equations 2[\eqalignno{ {\bf a}' &=P_{11}{\bf a} +P_{21}{\bf b} + P_{31}{\bf c}& \cr {\bf b}' &=P_{12}{\bf a} +P_{22}{\bf b} + P_{32}{\bf c} &(\cr {\bf c}' &=P_{13}{\bf a} +P_{23}{\bf b} +P_{33}{\bf c}&}]or [({\bf a}', {\bf b}', {\bf c}') ^{\rm T} = ({\bf a}, {\bf b}, {\bf c}) ^{\rm T} \left(\matrix { P_{11} & P_{12} & P_{13} \cr P_{21} & P_{22} & P_{23} \cr P_{31} & P_{32} & P_{33} } \right). \eqno (]

In matrix notation, this is [({\bi a}')^{\rm T} = ({\bi a})^{\rm T} \, {\bi P}. \eqno (]

The column p of coordinates of the origin [O'] of [{\cal H}] is referred to the coordinate system of [{\cal G}] and is called the origin shift. The matrix–column pair (P, p) describes the transformation from the coordinate system of [{\cal G}] to that of [{\cal H}], for details, cf. IT A, Part 5[link] . Therefore, P and p are chosen in the subgroup tables in the columns `matrix' and `shift', cf. Section[link]. The column `matrix' is empty if there is no change of basis, i.e. if P is the unit matrix I. The column `shift' is empty if there is no origin shift, i.e. if p is the column o consisting of zeroes only.

A change of the coordinate system, described by the matrix–column pair (P, p), changes the point coordinates from the column x to the column [{\bi x}\,']. The formulae for this change do not contain the pair (P, p) itself, but the related pair [({\bi Q},\, {\bi q})=({\bi P}^{-1},\, -{\bi P}^{-1}\,{\bi p})]: [ {\bi x}\,'={\bi Q}{\bi x}+{\bi q}={\bi P}^{-1}{\bi x}- {\bi P}^{-1}{\bi p}={\bi P}^{-1}({\bi x}-{\bi p}). \eqno (]

Not only the point coordinates but also the matrix–column pairs for the symmetry operations are changed by a change of the coordinate system. A symmetry operation [\sf W] is described in the coordinate system of [{\cal G}] by the system of equations [\eqalignno{ \tilde{x}&=W_{11}\,x+W_{12}\,y+W_{13}\,z+w_1 &\cr \tilde{y}&=W_{21}\,x+W_{22}\,y+W_{23}\,z+w_2 & (\cr \tilde{z}&=W_{31}\,x+W_{32}\,y+W_{33}\,z+w_3, &}] [{\tilde {\bi x}}={\bi W}{\bi x}+{\bi w}=({\bi W,w}){\bi x}, \eqno (]i.e. by the matrix–column pair (W, w). The symmetry operation [\sf W] will be described in the coordinate system of the subgroup [{\cal H}] by the equation [\tilde{\bi{x}}\,'=\bi{W}\,'\,\bi{x}\,'+\bi{w}\,'= (\bi{W}\,',\,\bi{w}\,')\,\bi{x}\,', \eqno (]and thus by the pair [(\bi{W}\,',\,\bi{w}\,')]. This pair can be calculated from the pair [(\bi{W},\ \bi{w})] by solving the equations [ \bi{W}\,'=\bi{Q}\,\bi{W}\,\bi{P}=\bi{P}^{-1}\bi{W}\bi{P} \eqno (]and [ \bi{w}\,'=\bi{q}+ \bi{Q}\bi{w}+\bi{Q}\bi{W}\bi{p}=\bi{P}^{-1}(\bi{w}+ \bi{W}\bi{p}-\bi{p})=\bi{P}^{-1}(\bi{w}+(\bi{W}-\bi{I}) \bi{p}). \eqno (]


Consider the data listed for the t-subgroups of [Pmn2_1], No. 31:[\matrix{{\rm Index} & {\rm HM\,\, \& \,\,No.} &{\rm sequence} &{\rm matrix} &{\rm shift} \cr [2] &P1n1\,\, (7, P1c1)\hfill &1\semi3 &{\bf c},{\bf b}, -{\bf a} -{\bf c}\hfill &\cr [2] &Pm11\,\,(6, P1m1)\hfill &1\semi4 &{\bf c}, {\bf a}, {\bf b}\hfill&\cr [2] &P112_{1}\,\,(4)\hfill &1\semi2 & &1/4, 0, 0\cr}]This means that the matrices and origin shifts are

  • (1) [{\bi P}_{1}= \left(\matrix { 0 & 0 & \overline{1} \cr 0 & 1 & 0 \cr 1 & 0 & \overline{1} } \right)\semi {\bi P}_{2} =\left(\matrix { 0 & 1 & 0 \cr 0 & 0 & 1 \cr 1 & 0 & 0 } \right)\semi {\bi P}_{3} = \left(\matrix { 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 } \right)]

  • (2) [{\bi p}_{1} = \left(\matrix { 0 \cr 0 \cr 0 } \right), \,\,{\bi p}_{2} = \left(\matrix {0 \cr 0 \cr 0 }\right), \,\,{\bi p}_{3} = \left(\matrix { {{1}\over{4}} \cr 0 \cr 0 } \right).]

  • (3) The first subgroup is monoclinic, the symmetry direction is the b axis, which is standard. However, the glide direction [{{1}\over{2}}({\bf a}+{\bf c})] is nonconventional. Therefore, the basis of [{\cal G}] is transformed to a basis of the subgroup [{\cal H}] such that the b axis is retained, the glide direction becomes the [c'] axis and the [a'] axis is chosen such that the basis is a right-handed one, the angle [\beta'\ge90^{\circ}] and the transformation matrix P is simple. This is done by the chosen matrix [{\bi P}_1]. The origin shift is the o column.

    With equations ([link] and ([link], one obtains for the glide reflection [x,\overline{y},z-{{1}\over{2}}], which is [x,\overline{y},z+{{1}\over{2}}] after standardization by [0\le w_j \,\lt\, 1].

  • (4) For the second monoclinic subgroup, the symmetry direction is the (nonconventional) a axis. The rules of Section[link] require a change to the setting `unique axis b'. A cyclic permutation of the basis vectors is the simplest way to achieve this. The reflection [\overline{x}, y, z] is now described by [x, \overline{y}, z]. Again there is no origin shift.

  • (5) The third monoclinic subgroup is in the conventional setting `unique axis c', but the origin must be shifted onto the [2_1] screw axis. This is achieved by applying equation ([link] with [{\bi p}_3], which changes [\overline{x}+{{1}\over{2}},\overline{y},z+{{1}\over{2}}] of [Pmn2_1] to [\overline{x},\overline{y},z+{{1}\over{2}}] of [P112_1].


Evaluation of the t-subgroup data of the space group [P3_112], No. 151, started in Example[link]. The evaluation is now continued with the columns `sequence', `matrix' and `shift'. They are used for the transformation of the elements of [{\cal H}] to their conventional form. Only the monoclinic t-subgroups are of interest here because the trigonal subgroup is already in the standard setting.

One takes from the tables of subgroups in Chapter 2.3[link] [\displaylines{\,\,\,\,{\rm Index} \,\,\,\quad{\rm HM\,\, \& \,\, No.}\quad {\rm sequence} \quad \quad{\rm matrix} \quad\quad\quad\quad\,\,\, {\rm shift}\hfill\cr \Biggl\{\matrix{[3] &P112\,\,(5,\,C121) &\,\,1\semi6 &\quad{\bf b},-2{\bf a}- {\bf b},{\bf c} \hfill & \cr [3] &P112\,\,(5,\,C121) & \,\,1\semi4 &\quad -{\bf a}-{\bf b}, {\bf a}-{\bf b}, {\bf c}\hfill & 0,0,1/3 \cr [3] &P112\,\,(5,\,C121) & \,\,1\semi5 &\quad {\bf a},{\bf a} + 2{\bf b},{\bf c}\hfill & 0,0,2/3\cr}}]Designating the three matrices by [{\bi P}_6], [{\bi P}_4], [{\bi P}_5], one obtains [{\bi P}_{6} = \left(\matrix { 0 & -2 & 0 \cr 1 & -1 & 0 \cr 0 & 0 & 1 } \right), \,\, {\bi P}_{4} = \left(\matrix {-1 & 1 & 0 \cr -1 & -1 & 0 \cr 0 & 0 & 1 } \right), \,\, {\bi P}_{5} = \left(\matrix {1 & 1 & 0 \cr 0 & 2 & 0 \cr 0 & 0 & 1 } \right)]with the corresponding inverse matrices [{\bi Q}_{6} = \left(\matrix { -{{1}\over{2}} & 1 & 0 \cr -{{1}\over{2}} & 0 & 0 \cr 0 & 0 & 1 } \right),\,\, {\bi Q}_{4} = \left(\matrix { -{{1}\over{2}} & -{{1}\over{2}} & 0 \cr {{1}\over{2}} & -{{1}\over{2}} & 0 \cr 0 & 0 & 1 } \right),\,\, {\bi Q}_{5} = \left(\matrix { 1 & -{{1}\over{2}} & 0 \cr 0 & {{1}\over{2}} & 0 \cr 0 & 0 & 1 } \right)]and the origin shifts [{\bi p}_{6} = \left(\matrix { 0 \cr 0 \cr 0 } \right), \, \, {\bi p}_{4} = \left(\matrix { 0 \cr 0 \cr {{1}\over{3}} } \right),\, \, {\bi p}_{5} = \left(\matrix { 0 \cr 0 \cr {{2}\over{3}}} \right).]For the three new bases this means [\eqalign{ &{\bf a}_{6}' = {\bf b},\,\, {\bf b}_{6}'=-2{\bf a}-{\bf b}, \,\,{\bf c}_6'={\bf c} \cr &{\bf a}_4'=-{\bf a}-{\bf b},\,\, {\bf b}_4'={\bf a}-{\bf b}, \,\,{\bf c}_4'={\bf c}\,\, {\rm and} \cr& {\bf a}_5'={\bf a}, \,\,{\bf b}_5'={\bf a}+2{\bf b}, \,\,{\bf c}_5'={\bf c}. }]All these bases span ortho-hexagonal cells with twice the volume of the original hexagonal cell because for the matrices [\det({\bi P}_{i})=2] holds.

In the general position of [{\cal G}=P3_{1}12], No.151, one finds[(1)\,\, x,y,z; \quad (4)\,\, \overline{y},\overline{x}, \overline{z}+{\textstyle{{2}\over{3}}};\quad (5) \,\,\overline{x}+y,y, \overline{z}+{\textstyle{{1}\over{3}}};\quad (6) \,\,x,x-y,\overline{z}.]These entries represent the matrix–column pairs [({\bi W},\,{\bi w})]: [\displaylines{(1)\, \left(\matrix {1 &0 &0 \cr 0 &1 &0 \cr 0 &0 &1 } \right), \left(\matrix { 0 \cr 0 \cr 0 } \right)\semi \quad (4)\, \left(\matrix { 0&{\overline 1}&0 \cr {\overline 1}&0&0 \cr 0&0&\overline{1}} \right), \left(\matrix { 0 \cr 0 \cr {{2}\over{3}} } \right)\semi\cr\cr (5)\, \left(\matrix {\overline{1} & 1 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & \overline{1}} \right), \left(\matrix { 0 \cr 0 \cr {{1}\over{3}} } \right) \semi \quad (6)\, \left(\matrix { 1&0&0 \cr 1&\overline{1}&0 \cr 0&0&\overline{1} } \right), \left(\matrix { 0 \cr 0 \cr 0 } \right).}]Application of equations ([link] on the matrices [{\bi W}_k] and ([link] on the columns [{\bi w}_k] of the matrix–column pairs results in [{\bi W}\,'_{4} = {\bi W}\,'_{5} = {\bi W}\,'_{6} = \left(\matrix { \overline{1} & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & \overline{1} } \right)\semi \,\, {\bi w}'_{4} = {\bi w}'_{6} = {\bi o}\semi \, {\bi w}'_{5} = \left(\matrix { 0 \cr 0 \cr \overline{1} } \right).]All translation vectors of [{\cal G}] are retained in the subgroups but the volume of the cells is doubled. Therefore, there must be centring-translation vectors in the new cells. For example, the application of equation ([link] with [({\bi P}_{6},\, {\bi p}_{6})] to the translation of [{\cal G}] with the vector [-{\bf a}], i.e. [{\bi w} = -(1,0,0)], results in the column [{\bi w}' = ({{1}\over{2}}, {{1}\over{2}}, 0)], i.e. the centring translation [{{1}\over{2}}({\bf a}'+{\bf b}')] of the subgroup. Either by calculation or, more easily, from a small sketch one sees that the vectors [-{\bf b}] for [{\bi P}_4], [{\bf a}+{\bf b}] for [{\bi P}_5] (and [-{\bf a}] for [{\bi P}_6]) correspond to the cell-centring translation vectors of the subgroup cells.


This example reveals that the conjugation of conjugate subgroups does not necessarily imply the conjugation of the representatives of these subgroups in the general positions of IT A. The three monoclinic subgroups [C121] in this example are conjugate in the group [{\cal G}] by the [3_1] screw rotation. Conjugation of the representative (4) by the [3_1] screw rotation of [{\cal G}] results in the representative (5) with the column [{\bi w}_{5} = 0,0,{{4}\over{3}}], which is not exactly the representative (5) but one of its translationally equivalent elements of [{\cal G}] retained in [{\cal H}].

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