Please note

The information given in this document reflects the conventions used in SgInfo Version 1.01 as released in 1996. There are subtle differences compared to International Tables for Crystallography Volume B (ITVB), 2001, pp. 112-119. The unusual lattice symbol S in Table 1 below was dropped in ITVB, and some Hall symbols in Table 6 below were replaced with equivalent alternatives.

See also: ITVB 2001 Table A1.4.2.7 Hall symbols


Concise Space-Group Symbols

Sydney R. Hall
Ralf W. Grosse-Kunstleve


Hall Symbols

The explicit-origin space group notation proposed by Hall (1981) [1], [2] is based on the minimum number of symmetry operations, in the form of Seitz matrices, needed to uniquely define a space group. The concise unambiguous nature of this notation makes it well suited to handling symmetry in computing and database applications.

The notation has the general form:

             L [NAT]1 ... [NAT]p V

where L is the symbol specifying the lattice translational symmetry (see Table 1), NAT identifies the 4x4 Seitz matrix of a symmetry element in the minimum set which defines the space-group symmetry (see Tables 2 3, 4, and 5), and p is the number of elements in the set. V is a translation vector which shifts the origin of the generator matrices by fractions of the unit cell lengths ab and c.

The matrix symbol NAT is composed of three parts:

Table 6 lists space group notation in several formats. The computer-entry representation of the Hall symbols is listed in column 3. The computer-entry format is the general notation expressed as case insensitive ASCII characters, with the overline (bar) symbol replaced by a minus sign. Column 1 of Table 6 contains the space-group number with an appended code which identifies the non-standard settings. Column 2 contains the full Hermann-Mauguin symbols in computer-entry format with appended codes which identify the origin and cell choice when there are alternatives.

The computer-entry format of the Hall notation contains the rotation-order symbol N as positive integers 1, 2, 3, 4, or 6 for proper rotations and a negative integers -1, -2, -3, -4 or -6 for improper rotations. The T translation symbols 1, 2, 3, 4, 5, a, b, c, n, u, v, w, d are described in Table 2. These translations apply additively (e.g. ad signifies a (3/4,1/4,1/4)) translation).
The A axis symbols x, y, z denote rotations about the axes a, b, c, respectively (see Table 3). The axis symbols " and ' signal rotations about the body-diagonal vectors a+b (or alternatively b+c or c+a) and a-b (or alternatively b-c or c-a) (see Table 4). The axis symbol * always refers to a 3-fold rotation along a+b+c (see Table 5).
The origin-shift translation vector V has the construction (va vb vc), where va, vb and vc denote the shifts in 12ths parallel to the cell edges a, b and c, respectively. va/12, vb/12 and vc/12 are the coordinates of the unshifted origin in the shifted basis system. The shifted Seitz matrices Sn' are derived from the unshifted matrices Sn with the transformation

                  (1 0 0 va/12)        (1 0 0 -va/12)
            Sn' = (0 1 0 vb/12) * Sn * (0 1 0 -vb/12)
                  (0 0 1 vc/12)        (0 0 1 -vc/12)
                  (0 0 0   1  )        (0 0 0    1  )

Default axes

For most Hall symbols the rotation axes applicable to each N are implied and an explicit axis symbol A is not needed. The rules for default axis directions are:

  1. the first rotation has an axis direction of c
  2. the second rotation (if N is 2) has an axis direction of
  3. the third rotation (N is always 3) has an axis direction of

Example matrices

Here are several simple examples of how NAT symbols expand to Seitz matrices.

  1. The notation -2xc represents an improper 2-fold rotation along a and a c/2 translation:
                           (-1  0  0   0 )
                    -2xc = ( 0  1  0   0 )
                           ( 0  0  1  1/2)
                           ( 0  0  0   1 )
    
  2. The notation 3* represents a 3-fold rotation along a+b+c:
                           ( 0  0  1   0 )
                      3* = ( 1  0  0   0 )
                           ( 0  1  0   0 )
                           ( 0  0  0   1 )
    
  3. The notation 4vw represents a 4-fold rotation along c (implied) and translation of b/4 and c/4:
                           ( 0 -1  0   0 )
                     4vw = ( 1  0  0  1/4)
                           ( 0  0  1  1/4)
                           ( 0  0  0   1 )
    
  4. The notation 61 2 (0 0 -1) represents a 61 screw along c and a 2-fold rotation along a-b. The translation component 5/6 of the second matrix is the result of the origin shift of -c/12:
                           ( 1 -1  0   0 )  ( 0 -1  0   0 )
           61 2 (0 0 -1) = ( 1  0  0   0 )  (-1  0  0   0 )
                           ( 0  0  1  1/6)  ( 0  0 -1  5/6)
                           ( 0  0  0   1 )  ( 0  0  0   1 )
    

Table 1. Lattice Symbol L

The lattice symbol L specifies one or more Seitz matrices which are needed to generate the space-group symmetry elements. For noncentrosymmetric lattices the rotation matrices are for 1 (see Table 3). For centrosymmetric lattices the lattice symbols are preceded by a minus sign `-', rotations are 1 and -1, and the total number of generator matrices implied by each symbol is twice the number of implied lattice translations.


Non-centrosymmetric
symbol
Number of lattice
translations
Implied lattice
translation(s)
P 1 (0,0,0)
A 2 (0,0,0), (0,1/2,1/2)
B 2 (0,0,0), (1/2,0,1/2)
C 2 (0,0,0), (1/2,1/2,0)
I 2 (0,0,0), (1/2,1/2,1/2)
R 3 (0,0,0), (2/3,1/3,1/3), (1/3,2/3,2/3)
S 3 (0,0,0), (1/3,1/3,2/3), (2/3,2/3,1/3)
T 3 (0,0,0), (1/3,2/3,1/3), (2/3,1/3,2/3)
F 4 (0,0,0), (0,1/2,1/2), (1/2,0,1/2), (1/2,1/2,0)

The unusual lattice symbols S and T are necessary to allow for obverse and reverse settings for all of 3x, 3y, and 3z, respectively. Table 1.1. summarizes the relationsships.
Table 1.1. Lattice symbol
Unique axis R S T
3z obverse - reverse
3y reverse obverse -
3x - reverse obverse


Table 2. Translation symbol T

The symbol T specifies the translation elements of a Seitz matrix. Alphabetical symbols (column 1 below) specify translations along a fixed direction. Numerical symbols (column 3 below) specify translations as a fraction of the rotation order N, and in the direction of the implied or explicitly defined axis.


Translation
symbol
Translation
vector
Subscript
symbol
Fractional
translation
a 1/2,0,0 1 in 31 1/3
b 0,1/2,0 2 in 32 2/3
c 0,0,1/2 1 in 41 1/4
n 1/2,1/2,1/2 3 in 43 3/4
u 1/4,0,0 1 in 61 1/6
v 0,1/4,0 2 in 62 1/3
w 0,0,1/4 4 in 64 2/3
d 1/4,1/4,1/4 5 in 65 5/6

Table 3. Rotation matrices for principal axes

The 3x3 matrices for proper rotations along the three principal unit-cell directions. The matrices for improper rotations (-1, -2, -3, -4 and -6) are identical except that the signs are reversed.


Rotation Order: 1            2            3            4            6
    Symbol
Axis  A
          (  1  0  0)  (  1  0  0)  (  1  0  0)  (  1  0  0)  (  1  0  0)
  a   x   (  0  1  0)  (  0 -1  0)  (  0  0 -1)  (  0  0 -1)  (  0  1 -1)
          (  0  0  1)  (  0  0 -1)  (  0  1 -1)  (  0  1  0)  (  0  1  0)

          (  1  0  0)  ( -1  0  0)  ( -1  0  1)  (  0  0  1)  (  0  0  1)
  b   y   (  0  1  0)  (  0  1  0)  (  0  1  0)  (  0  1  0)  (  0  1  0)
          (  0  0  1)  (  0  0 -1)  ( -1  0  0)  ( -1  0  0)  ( -1  0  1)

          (  1  0  0)  ( -1  0  0)  (  0 -1  0)  (  0 -1  0)  (  1 -1  0)
  c   z   (  0  1  0)  (  0 -1  0)  (  1 -1  0)  (  1  0  0)  (  1  0  0)
          (  0  0  1)  (  0  0  1)  (  0  0  1)  (  0  0  1)  (  0  0  1)

Table 4. Rotation matrices for face-diagonal axes

The symbols for face-diagonal 2-fold rotations are 2' and 2". The face-diagonal axis direction is determined by the axis of the preceding rotation Nx, Ny or Nz. Note that the single quote symbol ' is the default and may be omitted.


Preceding
  rotation:      Nx                     Ny                     Nz
Notation:   2'         2"          2'         2"          2'         2"
Axis:      b-c        b+c         a-c        a+c         a-b        a+b
       (-1  0  0) (-1  0  0)  ( 0  0 -1) ( 0  0  1)  ( 0 -1  0) ( 0  1  0)
       ( 0  0 -1) ( 0  0  1)  ( 0 -1  0) ( 0 -1  0)  (-1  0  0) ( 1  0  0)
       ( 0 -1  0) ( 0  1  0)  (-1  0  0) ( 1  0  0)  ( 0  0 -1) ( 0  0 -1)

Table 5. Rotation matrix for the body-diagonal axis

The symbol for the 3-fold rotation in the a+b+c direction is 3*. Note that for cubic space groups the body-diagonal axis is implied, and the asterisk * may be omitted.


        Axis      Notation
                            ( 0  0  1)
        a+b+c        3*     ( 1  0  0)
                            ( 0  1  0)

Table 6. Concise space-group symbols

The codes appended to space-group numbers listed in column 1 of Table 6 identify the relationship of the symmetry elements to the crystal cell. The appended codes are separated from the space-group number by a colon. When a code is omitted the first listed choice applies.

  Monoclinic                  code  =  <unique axis><cell choice>
       Unique axis choices(+          b  -b  c  -c  a  -a
       Cell choices(+                 1  2  3

  Orthorhombic                code  =  <origin choice><setting>
       Origin choices                 1  2
       Setting choices(+              abc  ba-c  cab  -cba  bca  a-cb

  Tetragonal, Cubic           code  =  <origin choice>
       Origin choices                 1  2

  Trigonal                    code  =  <cell choice>
       Cell choices                   H (hex)   R (rhomb)

  (+ cf. IT Vol. A 1983 Table 4.3.1


  Number   Hermann-Mauguin   Hall
  ------   ---------------   ----
     1        P 1            P 1
     2        P -1          -P 1
     3:b      P 1 2 1        P 2y
     3:c      P 1 1 2        P 2
     3:a      P 2 1 1        P 2x
     4:b      P 1 21 1       P 2yb
     4:c      P 1 1 21       P 2c
     4:a      P 21 1 1       P 2xa
     5:b1     C 1 2 1        C 2y
     5:b2     A 1 2 1        A 2y
     5:b3     I 1 2 1        I 2y
     5:c1     A 1 1 2        A 2
     5:c2     B 1 1 2        B 2
     5:c3     I 1 1 2        I 2
     5:a1     B 2 1 1        B 2x
     5:a2     C 2 1 1        C 2x
     5:a3     I 2 1 1        I 2x
     6:b      P 1 m 1        P -2y
     6:c      P 1 1 m        P -2
     6:a      P m 1 1        P -2x
     7:b1     P 1 c 1        P -2yc
     7:b2     P 1 n 1        P -2yac
     7:b3     P 1 a 1        P -2ya
     7:c1     P 1 1 a        P -2a
     7:c2     P 1 1 n        P -2ab
     7:c3     P 1 1 b        P -2b
     7:a1     P b 1 1        P -2xb
     7:a2     P n 1 1        P -2xbc
     7:a3     P c 1 1        P -2xc
     8:b1     C 1 m 1        C -2y
     8:b2     A 1 m 1        A -2y
     8:b3     I 1 m 1        I -2y
     8:c1     A 1 1 m        A -2
     8:c2     B 1 1 m        B -2
     8:c3     I 1 1 m        I -2
     8:a1     B m 1 1        B -2x
     8:a2     C m 1 1        C -2x
     8:a3     I m 1 1        I -2x
     9:b1     C 1 c 1        C -2yc
     9:b2     A 1 n 1        A -2yac
     9:b3     I 1 a 1        I -2ya
     9:-b1    A 1 a 1        A -2ya
     9:-b2    C 1 n 1        C -2ybc
     9:-b3    I 1 c 1        I -2yc
     9:c1     A 1 1 a        A -2a
     9:c2     B 1 1 n        B -2bc
     9:c3     I 1 1 b        I -2b
     9:-c1    B 1 1 b        B -2b
     9:-c2    A 1 1 n        A -2ac
     9:-c3    I 1 1 a        I -2a
     9:a1     B b 1 1        B -2xb
     9:a2     C n 1 1        C -2xbc
     9:a3     I c 1 1        I -2xc
     9:-a1    C c 1 1        C -2xc
     9:-a2    B n 1 1        B -2xbc
     9:-a3    I b 1 1        I -2xb
    10:b      P 1 2/m 1     -P 2y
    10:c      P 1 1 2/m     -P 2
    10:a      P 2/m 1 1     -P 2x
    11:b      P 1 21/m 1    -P 2yb
    11:c      P 1 1 21/m    -P 2c
    11:a      P 21/m 1 1    -P 2xa
    12:b1     C 1 2/m 1     -C 2y
    12:b2     A 1 2/m 1     -A 2y
    12:b3     I 1 2/m 1     -I 2y
    12:c1     A 1 1 2/m     -A 2
    12:c2     B 1 1 2/m     -B 2
    12:c3     I 1 1 2/m     -I 2
    12:a1     B 2/m 1 1     -B 2x
    12:a2     C 2/m 1 1     -C 2x
    12:a3     I 2/m 1 1     -I 2x
    13:b1     P 1 2/c 1     -P 2yc
    13:b2     P 1 2/n 1     -P 2yac
    13:b3     P 1 2/a 1     -P 2ya
    13:c1     P 1 1 2/a     -P 2a
    13:c2     P 1 1 2/n     -P 2ab
    13:c3     P 1 1 2/b     -P 2b
    13:a1     P 2/b 1 1     -P 2xb
    13:a2     P 2/n 1 1     -P 2xbc
    13:a3     P 2/c 1 1     -P 2xc
    14:b1     P 1 21/c 1    -P 2ybc
    14:b2     P 1 21/n 1    -P 2yn
    14:b3     P 1 21/a 1    -P 2yab
    14:c1     P 1 1 21/a    -P 2ac
    14:c2     P 1 1 21/n    -P 2n
    14:c3     P 1 1 21/b    -P 2bc
    14:a1     P 21/b 1 1    -P 2xab
    14:a2     P 21/n 1 1    -P 2xn
    14:a3     P 21/c 1 1    -P 2xac
    15:b1     C 1 2/c 1     -C 2yc
    15:b2     A 1 2/n 1     -A 2yac
    15:b3     I 1 2/a 1     -I 2ya
    15:-b1    A 1 2/a 1     -A 2ya
    15:-b2    C 1 2/n 1     -C 2ybc
    15:-b3    I 1 2/c 1     -I 2yc
    15:c1     A 1 1 2/a     -A 2a
    15:c2     B 1 1 2/n     -B 2bc
    15:c3     I 1 1 2/b     -I 2b
    15:-c1    B 1 1 2/b     -B 2b
    15:-c2    A 1 1 2/n     -A 2ac
    15:-c3    I 1 1 2/a     -I 2a
    15:a1     B 2/b 1 1     -B 2xb
    15:a2     C 2/n 1 1     -C 2xbc
    15:a3     I 2/c 1 1     -I 2xc
    15:-a1    C 2/c 1 1     -C 2xc
    15:-a2    B 2/n 1 1     -B 2xbc
    15:-a3    I 2/b 1 1     -I 2xb
    16        P 2 2 2        P 2 2
    17        P 2 2 21       P 2c 2
    17:cab    P 21 2 2       P 2a 2a
    17:bca    P 2 21 2       P 2 2b
    18        P 21 21 2      P 2 2ab
    18:cab    P 2 21 21      P 2bc 2
    18:bca    P 21 2 21      P 2ac 2ac
    19        P 21 21 21     P 2ac 2ab
    20        C 2 2 21       C 2c 2
    20:cab    A 21 2 2       A 2a 2a
    20:bca    B 2 21 2       B 2 2b
    21        C 2 2 2        C 2 2
    21:cab    A 2 2 2        A 2 2
    21:bca    B 2 2 2        B 2 2
    22        F 2 2 2        F 2 2
    23        I 2 2 2        I 2 2
    24        I 21 21 21     I 2b 2c
    25        P m m 2        P 2 -2
    25:cab    P 2 m m        P -2 2
    25:bca    P m 2 m        P -2 -2
    26        P m c 21       P 2c -2
    26:ba-c   P c m 21       P 2c -2c
    26:cab    P 21 m a       P -2a 2a
    26:-cba   P 21 a m       P -2 2a
    26:bca    P b 21 m       P -2 -2b
    26:a-cb   P m 21 b       P -2b -2
    27        P c c 2        P 2 -2c
    27:cab    P 2 a a        P -2a 2
    27:bca    P b 2 b        P -2b -2b
    28        P m a 2        P 2 -2a
    28:ba-c   P b m 2        P 2 -2b
    28:cab    P 2 m b        P -2b 2
    28:-cba   P 2 c m        P -2c 2
    28:bca    P c 2 m        P -2c -2c
    28:a-cb   P m 2 a        P -2a -2a
    29        P c a 21       P 2c -2ac
    29:ba-c   P b c 21       P 2c -2b
    29:cab    P 21 a b       P -2b 2a
    29:-cba   P 21 c a       P -2ac 2a
    29:bca    P c 21 b       P -2bc -2c
    29:a-cb   P b 21 a       P -2a -2ab
    30        P n c 2        P 2 -2bc
    30:ba-c   P c n 2        P 2 -2ac
    30:cab    P 2 n a        P -2ac 2
    30:-cba   P 2 a n        P -2ab 2
    30:bca    P b 2 n        P -2ab -2ab
    30:a-cb   P n 2 b        P -2bc -2bc
    31        P m n 21       P 2ac -2
    31:ba-c   P n m 21       P 2bc -2bc
    31:cab    P 21 m n       P -2ab 2ab
    31:-cba   P 21 n m       P -2 2ac
    31:bca    P n 21 m       P -2 -2bc
    31:a-cb   P m 21 n       P -2ab -2
    32        P b a 2        P 2 -2ab
    32:cab    P 2 c b        P -2bc 2
    32:bca    P c 2 a        P -2ac -2ac
    33        P n a 21       P 2c -2n
    33:ba-c   P b n 21       P 2c -2ab
    33:cab    P 21 n b       P -2bc 2a
    33:-cba   P 21 c n       P -2n 2a
    33:bca    P c 21 n       P -2n -2ac
    33:a-cb   P n 21 a       P -2ac -2n
    34        P n n 2        P 2 -2n
    34:cab    P 2 n n        P -2n 2
    34:bca    P n 2 n        P -2n -2n
    35        C m m 2        C 2 -2
    35:cab    A 2 m m        A -2 2
    35:bca    B m 2 m        B -2 -2
    36        C m c 21       C 2c -2
    36:ba-c   C c m 21       C 2c -2c
    36:cab    A 21 m a       A -2a 2a
    36:-cba   A 21 a m       A -2 2a
    36:bca    B b 21 m       B -2 -2b
    36:a-cb   B m 21 b       B -2b -2
    37        C c c 2        C 2 -2c
    37:cab    A 2 a a        A -2a 2
    37:bca    B b 2 b        B -2b -2b
    38        A m m 2        A 2 -2
    38:ba-c   B m m 2        B 2 -2
    38:cab    B 2 m m        B -2 2
    38:-cba   C 2 m m        C -2 2
    38:bca    C m 2 m        C -2 -2
    38:a-cb   A m 2 m        A -2 -2
    39        A b m 2        A 2 -2c
    39:ba-c   B m a 2        B 2 -2c
    39:cab    B 2 c m        B -2c 2
    39:-cba   C 2 m b        C -2b 2
    39:bca    C m 2 a        C -2b -2b
    39:a-cb   A c 2 m        A -2c -2c
    40        A m a 2        A 2 -2a
    40:ba-c   B b m 2        B 2 -2b
    40:cab    B 2 m b        B -2b 2
    40:-cba   C 2 c m        C -2c 2
    40:bca    C c 2 m        C -2c -2c
    40:a-cb   A m 2 a        A -2a -2a
    41        A b a 2        A 2 -2ac
    41:ba-c   B b a 2        B 2 -2bc
    41:cab    B 2 c b        B -2bc 2
    41:-cba   C 2 c b        C -2bc 2
    41:bca    C c 2 a        C -2bc -2bc
    41:a-cb   A c 2 a        A -2ac -2ac
    42        F m m 2        F 2 -2
    42:cab    F 2 m m        F -2 2
    42:bca    F m 2 m        F -2 -2
    43        F d d 2        F 2 -2d
    43:cab    F 2 d d        F -2d 2
    43:bca    F d 2 d        F -2d -2d
    44        I m m 2        I 2 -2
    44:cab    I 2 m m        I -2 2
    44:bca    I m 2 m        I -2 -2
    45        I b a 2        I 2 -2c
    45:cab    I 2 c b        I -2a 2
    45:bca    I c 2 a        I -2b -2b
    46        I m a 2        I 2 -2a
    46:ba-c   I b m 2        I 2 -2b
    46:cab    I 2 m b        I -2b 2
    46:-cba   I 2 c m        I -2c 2
    46:bca    I c 2 m        I -2c -2c
    46:a-cb   I m 2 a        I -2a -2a
    47        P m m m       -P 2 2
    48:1      P n n n:1      P 2 2 -1n
    48:2      P n n n:2     -P 2ab 2bc
    49        P c c m       -P 2 2c
    49:cab    P m a a       -P 2a 2
    49:bca    P b m b       -P 2b 2b
    50:1      P b a n:1      P 2 2 -1ab
    50:2      P b a n:2     -P 2ab 2b
    50:1cab   P n c b:1      P 2 2 -1bc
    50:2cab   P n c b:2     -P 2b 2bc
    50:1bca   P c n a:1      P 2 2 -1ac
    50:2bca   P c n a:2     -P 2a 2c
    51        P m m a       -P 2a 2a
    51:ba-c   P m m b       -P 2b 2
    51:cab    P b m m       -P 2 2b
    51:-cba   P c m m       -P 2c 2c
    51:bca    P m c m       -P 2c 2
    51:a-cb   P m a m       -P 2 2a
    52        P n n a       -P 2a 2bc
    52:ba-c   P n n b       -P 2b 2n
    52:cab    P b n n       -P 2n 2b
    52:-cba   P c n n       -P 2ab 2c
    52:bca    P n c n       -P 2ab 2n
    52:a-cb   P n a n       -P 2n 2bc
    53        P m n a       -P 2ac 2
    53:ba-c   P n m b       -P 2bc 2bc
    53:cab    P b m n       -P 2ab 2ab
    53:-cba   P c n m       -P 2 2ac
    53:bca    P n c m       -P 2 2bc
    53:a-cb   P m a n       -P 2ab 2
    54        P c c a       -P 2a 2ac
    54:ba-c   P c c b       -P 2b 2c
    54:cab    P b a a       -P 2a 2b
    54:-cba   P c a a       -P 2ac 2c
    54:bca    P b c b       -P 2bc 2b
    54:a-cb   P b a b       -P 2b 2ab
    55        P b a m       -P 2 2ab
    55:cab    P m c b       -P 2bc 2
    55:bca    P c m a       -P 2ac 2ac
    56        P c c n       -P 2ab 2ac
    56:cab    P n a a       -P 2ac 2bc
    56:bca    P b n b       -P 2bc 2ab
    57        P b c m       -P 2c 2b
    57:ba-c   P c a m       -P 2c 2ac
    57:cab    P m c a       -P 2ac 2a
    57:-cba   P m a b       -P 2b 2a
    57:bca    P b m a       -P 2a 2ab
    57:a-cb   P c m b       -P 2bc 2c
    58        P n n m       -P 2 2n
    58:cab    P m n n       -P 2n 2
    58:bca    P n m n       -P 2n 2n
    59:1      P m m n:1      P 2 2ab -1ab
    59:2      P m m n:2     -P 2ab 2a
    59:1cab   P n m m:1      P 2bc 2 -1bc
    59:2cab   P n m m:2     -P 2c 2bc
    59:1bca   P m n m:1      P 2ac 2ac -1ac
    59:2bca   P m n m:2     -P 2c 2a
    60        P b c n       -P 2n 2ab
    60:ba-c   P c a n       -P 2n 2c
    60:cab    P n c a       -P 2a 2n
    60:-cba   P n a b       -P 2bc 2n
    60:bca    P b n a       -P 2ac 2b
    60:a-cb   P c n b       -P 2b 2ac
    61        P b c a       -P 2ac 2ab
    61:ba-c   P c a b       -P 2bc 2ac
    62        P n m a       -P 2ac 2n
    62:ba-c   P m n b       -P 2bc 2a
    62:cab    P b n m       -P 2c 2ab
    62:-cba   P c m n       -P 2n 2ac
    62:bca    P m c n       -P 2n 2a
    62:a-cb   P n a m       -P 2c 2n
    63        C m c m       -C 2c 2
    63:ba-c   C c m m       -C 2c 2c
    63:cab    A m m a       -A 2a 2a
    63:-cba   A m a m       -A 2 2a
    63:bca    B b m m       -B 2 2b
    63:a-cb   B m m b       -B 2b 2
    64        C m c a       -C 2bc 2
    64:ba-c   C c m b       -C 2bc 2bc
    64:cab    A b m a       -A 2ac 2ac
    64:-cba   A c a m       -A 2 2ac
    64:bca    B b c m       -B 2 2bc
    64:a-cb   B m a b       -B 2bc 2
    65        C m m m       -C 2 2
    65:cab    A m m m       -A 2 2
    65:bca    B m m m       -B 2 2
    66        C c c m       -C 2 2c
    66:cab    A m a a       -A 2a 2
    66:bca    B b m b       -B 2b 2b
    67        C m m a       -C 2b 2
    67:ba-c   C m m b       -C 2b 2b
    67:cab    A b m m       -A 2c 2c
    67:-cba   A c m m       -A 2 2c
    67:bca    B m c m       -B 2 2c
    67:a-cb   B m a m       -B 2c 2
    68:1      C c c a:1      C 2 2 -1bc
    68:2      C c c a:2     -C 2b 2bc
    68:1ba-c  C c c b:1      C 2 2 -1bc
    68:2ba-c  C c c b:2     -C 2b 2c
    68:1cab   A b a a:1      A 2 2 -1ac
    68:2cab   A b a a:2     -A 2a 2c
    68:1-cba  A c a a:1      A 2 2 -1ac
    68:2-cba  A c a a:2     -A 2ac 2c
    68:1bca   B b c b:1      B 2 2 -1bc
    68:2bca   B b c b:2     -B 2bc 2b
    68:1a-cb  B b a b:1      B 2 2 -1bc
    68:2a-cb  B b a b:2     -B 2b 2bc
    69        F m m m       -F 2 2
    70:1      F d d d:1      F 2 2 -1d
    70:2      F d d d:2     -F 2uv 2vw
    71        I m m m       -I 2 2
    72        I b a m       -I 2 2c
    72:cab    I m c b       -I 2a 2
    72:bca    I c m a       -I 2b 2b
    73        I b c a       -I 2b 2c
    73:ba-c   I c a b       -I 2a 2b
    74        I m m a       -I 2b 2
    74:ba-c   I m m b       -I 2a 2a
    74:cab    I b m m       -I 2c 2c
    74:-cba   I c m m       -I 2 2b
    74:bca    I m c m       -I 2 2a
    74:a-cb   I m a m       -I 2c 2
    75        P 4            P 4
    76        P 41           P 4w
    77        P 42           P 4c
    78        P 43           P 4cw
    79        I 4            I 4
    80        I 41           I 4bw
    81        P -4           P -4
    82        I -4           I -4
    83        P 4/m         -P 4
    84        P 42/m        -P 4c
    85:1      P 4/n:1        P 4ab -1ab
    85:2      P 4/n:2       -P 4a
    86:1      P 42/n:1       P 4n -1n
    86:2      P 42/n:2      -P 4bc
    87        I 4/m         -I 4
    88:1      I 41/a:1       I 4bw -1bw
    88:2      I 41/a:2      -I 4ad
    89        P 4 2 2        P 4 2
    90        P 42 1 2       P 4ab 2ab
    91        P 41 2 2       P 4w 2c
    92        P 41 21 2      P 4abw 2nw
    93        P 42 2 2       P 4c 2
    94        P 42 21 2      P 4n 2n
    95        P 43 2 2       P 4cw 2c
    96        P 43 21 2      P 4nw 2abw
    97        I 4 2 2        I 4 2
    98        I 41 2 2       I 4bw 2bw
    99        P 4 m m        P 4 -2
   100        P 4 b m        P 4 -2ab
   101        P 42 c m       P 4c -2c
   102        P 42 n m       P 4n -2n
   103        P 4 c c        P 4 -2c
   104        P 4 n c        P 4 -2n
   105        P 42 m c       P 4c -2
   106        P 42 b c       P 4c -2ab
   107        I 4 m m        I 4 -2
   108        I 4 c m        I 4 -2c
   109        I 41 m d       I 4bw -2
   110        I 41 c d       I 4bw -2c
   111        P -4 2 m       P -4 2
   112        P -4 2 c       P -4 2c
   113        P -4 21 m      P -4 2ab
   114        P -4 21 c      P -4 2n
   115        P -4 m 2       P -4 -2
   116        P -4 c 2       P -4 -2c
   117        P -4 b 2       P -4 -2ab
   118        P -4 n 2       P -4 -2n
   119        I -4 m 2       I -4 -2
   120        I -4 c 2       I -4 -2c
   121        I -4 2 m       I -4 2
   122        I -4 2 d       I -4 2bw
   123        P 4/m m m     -P 4 2
   124        P 4/m c c     -P 4 2c
   125:1      P 4/n b m:1    P 4 2 -1ab
   125:2      P 4/n b m:2   -P 4a 2b
   126:1      P 4/n n c:1    P 4 2 -1n
   126:2      P 4/n n c:2   -P 4a 2bc
   127        P 4/m b m     -P 4 2ab
   128        P 4/m n c     -P 4 2n
   129:1      P 4/n m m:1    P 4ab 2ab -1ab
   129:2      P 4/n m m:2   -P 4a 2a
   130:1      P 4/n c c:1    P 4ab 2n -1ab
   130:2      P 4/n c c:2   -P 4a 2ac
   131        P 42/m m c    -P 4c 2
   132        P 42/m c m    -P 4c 2c
   133:1      P 42/n b c:1   P 4n 2c -1n
   133:2      P 42/n b c:2  -P 4ac 2b
   134:1      P 42/n n m:1   P 4n 2 -1n
   134:2      P 42/n n m:2  -P 4ac 2bc
   135        P 42/m b c    -P 4c 2ab
   136        P 42/m n m    -P 4n 2n
   137:1      P 42/n m c:1   P 4n 2n -1n
   137:2      P 42/n m c:2  -P 4ac 2a
   138:1      P 42/n c m:1   P 4n 2ab -1n
   138:2      P 42/n c m:2  -P 4ac 2ac
   139        I 4/m m m     -I 4 2
   140        I 4/m c m     -I 4 2c
   141:1      I 41/a m d:1   I 4bw 2bw -1bw
   141:2      I 41/a m d:2  -I 4bd 2
   142:1      I 41/a c d:1   I 4bw 2aw -1bw
   142:2      I 41/a c d:2  -I 4bd 2c
   143        P 3            P 3
   144        P 31           P 31
   145        P 32           P 32
   146:H      R 3:H          R 3
   146:R      R 3:R          P 3*
   147        P -3          -P 3
   148:H      R -3:H        -R 3
   148:R      R -3:R        -P 3*
   149        P 3 1 2        P 3 2
   150        P 3 2 1        P 3 2"
   151        P 31 1 2       P 31 2c (0 0 1)
   152        P 31 2 1       P 31 2"
   153        P 32 1 2       P 32 2c (0 0 -1)
   154        P 32 2 1       P 32 2"
   155:H      R 32:H         R 3 2"
   155:R      R 32:R         P 3* 2
   156        P 3 m 1        P 3 -2"
   157        P 3 1 m        P 3 -2
   158        P 3 c 1        P 3 -2"c
   159        P 3 1 c        P 3 -2c
   160:H      R 3 m:H        R 3 -2"
   160:R      R 3 m:R        P 3* -2
   161:H      R 3 c:H        R 3 -2"c
   161:R      R 3 c:R        P 3* -2n
   162        P -3 1 m      -P 3 2
   163        P -3 1 c      -P 3 2c
   164        P -3 m 1      -P 3 2"
   165        P -3 c 1      -P 3 2"c
   166:H      R -3 m:H      -R 3 2"
   166:R      R -3 m:R      -P 3* 2
   167:H      R -3 c:H      -R 3 2"c
   167:R      R -3 c:R      -P 3* 2n
   168        P 6            P 6
   169        P 61           P 61
   170        P 65           P 65
   171        P 62           P 62
   172        P 64           P 64
   173        P 63           P 6c
   174        P -6           P -6
   175        P 6/m         -P 6
   176        P 63/m        -P 6c
   177        P 6 2 2        P 6 2
   178        P 61 2 2       P 61 2 (0 0 -1)
   179        P 65 2 2       P 65 2 (0 0 1)
   180        P 62 2 2       P 62 2c (0 0 1)
   181        P 64 2 2       P 64 2c (0 0 -1)
   182        P 63 2 2       P 6c 2c
   183        P 6 m m        P 6 -2
   184        P 6 c c        P 6 -2c
   185        P 63 c m       P 6c -2
   186        P 63 m c       P 6c -2c
   187        P -6 m 2       P -6 2
   188        P -6 c 2       P -6c 2
   189        P -6 2 m       P -6 -2
   190        P -6 2 c       P -6c -2c
   191        P 6/m m m     -P 6 2
   192        P 6/m c c     -P 6 2c
   193        P 63/m c m    -P 6c 2
   194        P 63/m m c    -P 6c 2c
   195        P 2 3          P 2 2 3
   196        F 2 3          F 2 2 3
   197        I 2 3          I 2 2 3
   198        P 21 3         P 2ac 2ab 3
   199        I 21 3         I 2b 2c 3
   200        P m -3        -P 2 2 3
   201:1      P n -3:1       P 2 2 3 -1n
   201:2      P n -3:2      -P 2ab 2bc 3
   202        F m -3        -F 2 2 3
   203:1      F d -3:1       F 2 2 3 -1d
   203:2      F d -3:2      -F 2uv 2vw 3
   204        I m -3        -I 2 2 3
   205        P a -3        -P 2ac 2ab 3
   206        I a -3        -I 2b 2c 3
   207        P 4 3 2        P 4 2 3
   208        P 42 3 2       P 4n 2 3
   209        F 4 3 2        F 4 2 3
   210        F 41 3 2       F 4d 2 3
   211        I 4 3 2        I 4 2 3
   212        P 43 3 2       P 4acd 2ab 3
   213        P 41 3 2       P 4bd 2ab 3
   214        I 41 3 2       I 4bd 2c 3
   215        P -4 3 m       P -4 2 3
   216        F -4 3 m       F -4 2 3
   217        I -4 3 m       I -4 2 3
   218        P -4 3 n       P -4n 2 3
   219        F -4 3 c       F -4c 2 3
   220        I -4 3 d       I -4bd 2c 3
   221        P m -3 m      -P 4 2 3
   222:1      P n -3 n:1     P 4 2 3 -1n
   222:2      P n -3 n:2    -P 4a 2bc 3
   223        P m -3 n      -P 4n 2 3
   224:1      P n -3 m:1     P 4n 2 3 -1n
   224:2      P n -3 m:2    -P 4bc 2bc 3
   225        F m -3 m      -F 4 2 3
   226        F m -3 c      -F 4c 2 3
   227:1      F d -3 m:1     F 4d 2 3 -1d
   227:2      F d -3 m:2    -F 4vw 2vw 3
   228:1      F d -3 c:1     F 4d 2 3 -1cd
   228:2      F d -3 c:2    -F 4cvw 2vw 3
   229        I m -3 m      -I 4 2 3
   230        I a -3 d      -I 4bd 2c 3

References

[1] S.R. Hall; Space-Group Notation with an Explicit Origin ; Acta Cryst. (1981). A37, 517-525
[2] International Tables Volume B 1994, Section 1.4. Symmetry in reciprocal space

Sydney R. Hall <syd@crystal.uwa.edu.au>
Crystallography Centre, University of Western Australia

Ralf W. Grosse-Kunstleve <rwgk@cci.lbl.gov>