# Overview of symmetry information (CCP4: General)

## General references

Definitive guides are the International Tables, particularly volumes A and F.

## Other CCP4 documentation

• alternate_origins - lists of possible alternative origins for spacegroups.
• cheshirecell - a description of the use of Cheshire cells in Molecular Replacement, and a table of Cheshire cells.
• twinning - a description of twinning together with look-up tables for twinning operators
• CSYM library - documentation for the C API to the symmetry library
• symlib library - documentation for the Fortran API to the symmetry library

## syminfo.lib

From CCP4 5.0, information for all spacegroups in a variety of settings is tabulated in syminfo.lib (which is distributed in the directory \$CCP4/lib/data and replaces the previous symop.lib). A description of the contents of this file is given in the documentation of the Fortran API.

## hexagonal vs. rhombohedral settings e.g. H3 vs. R3

From CCP4 4.2, trigonal spacegroups in a centered hexagonal setting are denoted by "H". The symbol "R" is reserved for these spacegroups in a rhombohedral setting.

Explanation: All trigonal spacegroups (groups 143 to 167) can be described by hexagonal axes with a = b, alpha = beta = 90, gamma = 120 (they have hexagonal metric symmetry but are not in the hexagonal crystal system because they lack a 6-fold symmetry operator). A subset of trigonal groups possess two centering operators: x+2/3,y+1/3,z+1/3 and x+1/3,y+2/3,z+2/3 These groups are denoted by "H" e.g. "H 3", as opposed to the primitive spacegroups such as "P 3". In the extended Hermann Mauguin symbol, it is denoted by ":H" e.g. "R 3 :H".

These trigonal spacegroups which are centered in the hexagonal representation can also be represented in terms of rhombohedral axes. The rhombohedral unit cell has a = b = c, alpha = beta = gamma. The spacegroup lacks the centering operators, and the unit cell is a third of the volume of that in the hexagonal representation. This representation of the spacegroup is denoted by "R" e.g. "R 3". In the extended Hermann Mauguin symbol, it is denoted by ":R" e.g. "R 3 :R".

## Symmetry of reflection data

The quoted spacegroup describes the symmetry of the protein crystal in real space, i.e. the spatial distribution of molecules in the crystal. The diffraction pattern produced by the crystal has, in general, point group symmetry about (0,0,0). The point group is derived from the spacegroup by removing the centering letter (P, A, H, etc.), replacing screw axes by rotation axes, and replacing glide planes by mirror planes. For example:
"P 21" -> "2"
"P 21 21 21" -> "222"
"P 43 21 2" -> "422"
"P 3 1 2" -> "312"
There are 32 point groups in total.

If Friedel's Law is obeyed (i.e. if anomalous scattering is neglected) then the diffraction pattern has inversion symmetry about (0,0,0) since I(h, k, l) = I(-h, -k, -l). Adding the inversion operator gives the Laue class:

"P 21" -> "2/m"
"P 21 21 21" -> "2/m 2/m 2/m" or "mmm"
"P 43 21 2" -> "4/mmm"
"P 3 1 2" -> "-31m"
There are 11 possible Laue classes, one each for triclinic ("-1"), monoclinic ("2/m") and orthorhombic ("mmm"), and two each for tetragonal ("4/m" and "4/mmm"), trigonal ("-3" and "-3m"), hexagonal ("6/m" and "6/mmm") and cubic ("m-3" and "m-3m").

However, to take account of different spacegroup settings (e.g. choice of unique axis), further Laue classes may be defined. CCP4 uses the 11 standard classes but divides "-3m" into "-31m" and "-3m1"

The standard MTZ file holds a list of all reflections in the asymmetric unit of the Laue class (see e.g. the program unique).

In the general case, different point groups can be distinguished when anomalous data is available, since there is no longer necessarily inversion symmetry. In practice, for protein crystals, there is only one possible point group for each Laue class, so the choice is automatic.

## Real space asymmetric units

See the gallery of direct-space asymmetric units from Ralf Grosse-Kunstleve et al.

Note that CCP4 uses simple parallelepipeds which may contain more than one true asymmetric unit.

## Summary listing of spacegroups, and associated Laue and Point groups

Below is a full list of the 230 spacegroups. This information is a summary of some of the information in syminfo.lib.

```
SG_number  Laue_symbol         PointGroup_symbol   Old_SpgrpSymbol           xHM_SpgrpSymbol
1       '-P 1' '-1'         ' P 1' '1'          'P 1'                         'P 1'
2       '-P 1' '-1'         '-P 1' '-1'         'P -1'                        'P -1'
3       '-P 2y' '2/m'       ' P 2y' '2'         'P 1 2 1'                     'P 1 2 1'
3       '-P 2y' '2/m'       ' P 2y' '2'         'P 1 1 2'                     'P 1 1 2'
4       '-P 2y' '2/m'       ' P 2y' '2'         'P 1 21 1'                    'P 1 21 1'
4       '-P 2y' '2/m'       ' P 2y' '2'         'P 1 1 21'                    'P 1 1 21'
5       '-P 2y' '2/m'       ' P 2y' '2'         'C 1 2 1'                     'C 1 2 1'
5       '-P 2y' '2/m'       ' P 2y' '2'         'I 1 2 1' 'I 2'               'I 1 2 1'
5       '-P 2y' '2/m'       ' P 2y' '2'         'B 1 1 2' 'B 2'               'B 1 1 2'
6       '-P 2y' '2/m'       ' P -2y' 'm'        'P 1 m 1'                     'P 1 m 1'
6       '-P 2y' '2/m'       ' P -2y' 'm'        'P 1 1 m'                     'P 1 1 m'
7       '-P 2y' '2/m'       ' P -2y' 'm'        'P 1 c 1'                     'P 1 c 1'
7       '-P 2y' '2/m'       ' P -2y' 'm'        'P 1 1 b'                     'P 1 1 b'
8       '-P 2y' '2/m'       ' P -2y' 'm'        'C 1 m 1'                     'C 1 m 1'
9       '-P 2y' '2/m'       ' P -2y' 'm'        'C 1 c 1'                     'C 1 c 1'
9       '-P 2y' '2/m'       ' P -2y' 'm'        'B 1 1 b'                     'B 1 1 b'
10       '-P 2y' '2/m'       '-P 2y' '2/m'       'P 1 2/m 1'                   'P 1 2/m 1'
10       '-P 2y' '2/m'       '-P 2y' '2/m'       'P 1 1 2/m'                   'P 1 1 2/m'
11       '-P 2y' '2/m'       '-P 2y' '2/m'       'P 1 21/m 1'                  'P 1 21/m 1'
11       '-P 2y' '2/m'       '-P 2y' '2/m'       'P 1 1 21/m'                  'P 1 1 21/m'
12       '-P 2y' '2/m'       '-P 2y' '2/m'       'C 1 2/m 1'                   'C 1 2/m 1'
12       '-P 2y' '2/m'       '-P 2y' '2/m'       'B 1 1 2/m'                   'B 1 1 2/m'
13       '-P 2y' '2/m'       '-P 2y' '2/m'       'P 1 2/c 1'                   'P 1 2/c 1'
13       '-P 2y' '2/m'       '-P 2y' '2/m'       'P 1 1 2/b'                   'P 1 1 2/b'
14       '-P 2y' '2/m'       '-P 2y' '2/m'       'P 1 21/c 1'                  'P 1 21/c 1'
14       '-P 2y' '2/m'       '-P 2y' '2/m'       'P 1 1 21/b'                  'P 1 1 21/b'
15       '-P 2y' '2/m'       '-P 2y' '2/m'       'C 1 2/c 1'                   'C 1 2/c 1'
15       '-P 2y' '2/m'       '-P 2y' '2/m'       'B 1 1 2/b'                   'B 1 1 2/b'
16       '-P 2 2' 'mmm'      ' P 2 2' '222'      'P 2 2 2'                     'P 2 2 2'
17       '-P 2 2' 'mmm'      ' P 2 2' '222'      'P 2 2 21'                    'P 2 2 21'
17       '-P 2 2' 'mmm'      ' P 2 2' '222'      'P 21 2 2'                    'P 21 2 2'
17       '-P 2 2' 'mmm'      ' P 2 2' '222'      'P 2 21 2'                    'P 2 21 2'
18       '-P 2 2' 'mmm'      ' P 2 2' '222'      'P 21 21 2'                   'P 21 21 2'
18       '-P 2 2' 'mmm'      ' P 2 2' '222'      'P 2 21 21'                   'P 2 21 21'
18       '-P 2 2' 'mmm'      ' P 2 2' '222'      'P 21 2 21'                   'P 21 2 21'
19       '-P 2 2' 'mmm'      ' P 2 2' '222'      'P 21 21 21'                  'P 21 21 21'
20       '-P 2 2' 'mmm'      ' P 2 2' '222'      'C 2 2 21'                    'C 2 2 21'
21       '-P 2 2' 'mmm'      ' P 2 2' '222'      'C 2 2 2'                     'C 2 2 2'
22       '-P 2 2' 'mmm'      ' P 2 2' '222'      'F 2 2 2'                     'F 2 2 2'
23       '-P 2 2' 'mmm'      ' P 2 2' '222'      'I 2 2 2'                     'I 2 2 2'
24       '-P 2 2' 'mmm'      ' P 2 2' '222'      'I 21 21 21'                  'I 21 21 21'
25       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'P m m 2'                     'P m m 2'
26       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'P m c 21'                    'P m c 21'
27       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'P c c 2'                     'P c c 2'
28       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'P m a 2'                     'P m a 2'
29       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'P c a 21'                    'P c a 21'
30       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'P n c 2'                     'P n c 2'
31       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'P m n 21'                    'P m n 21'
32       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'P b a 2'                     'P b a 2'
33       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'P n a 21'                    'P n a 21'
34       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'P n n 2'                     'P n n 2'
35       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'C m m 2'                     'C m m 2'
36       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'C m c 21'                    'C m c 21'
37       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'C c c 2'                     'C c c 2'
38       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'A m m 2'                     'A m m 2'
39       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'A b m 2'                     'A b m 2'
40       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'A m a 2'                     'A m a 2'
41       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'A b a 2'                     'A b a 2'
42       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'F m m 2'                     'F m m 2'
43       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'F d d 2'                     'F d d 2'
44       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'I m m 2'                     'I m m 2'
45       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'I b a 2'                     'I b a 2'
46       '-P 2 2' 'mmm'      ' P 2 -2' 'mm2'     'I m a 2'                     'I m a 2'
47       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'P 2/m 2/m 2/m' 'P m m m'     'P m m m'
48       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'P 2/n 2/n 2/n'  'P n n n'    'P n n n   1'
49       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'P 2/c 2/c 2/m' 'P c c m'     'P c c m'
50       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'P 2/b 2/a 2/n' 'P b a n'     'P b a n   1'
51       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'P 21/m 2/m 2/a'  'P m m a'   'P m m a'
52       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'P 2/n 21/n 2/a'  'P n n a'   'P n n a'
53       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'P 2/m 2/n 21/a' 'P m n a'    'P m n a'
54       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'P 21/c 2/c 2/a' 'P c c a'    'P c c a'
55       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'P 21/b 21/a 2/m' 'P b a m'   'P b a m'
56       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'P 21/c 21/c 2/n' 'P c c n'   'P c c n'
57       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'P 2/b 21/c 21/m' 'P b c m'   'P b c m'
58       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'P 21/n 21/n 2/m' 'P n n m'   'P n n m'
59       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'P 21/m 21/m 2/n' 'P m m n'   'P m m n   1'
59       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'P 21/m 21/m 2/n a'           'P m m n   2'
60       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'P 21/b 2/c 21/n' 'P b c n'   'P b c n'
61       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'P 21/b 21/c 21/a' 'P b c a'  'P b c a'
62       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'P 21/n 21/m 21/a' 'P n m a'  'P n m a'
63       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'C 2/m 2/c 21/m' 'C m c m'    'C m c m'
64       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'C 2/m 2/c 21/a' 'C m c a'    'C m c a'
65       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'C 2/m 2/m 2/m'  'C m m m'    'C m m m'
66       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'C 2/c 2/c 2/m'  'C c c m'    'C c c m'
67       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'C 2/m 2/m 2/a' 'C m m a'     'C m m a'
68       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'C c c a' 'C 2/c 2/c 2/a'     'C c c a   1'
69       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'F 2/m 2/m 2/m' 'F m m m'     'F m m m'
70       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'F 2/d 2/d 2/d' 'F d d d'     'F d d d   1'
71       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'I 2/m 2/m 2/m' 'I m m m'     'I m m m'
72       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'I 2/b 2/a 2/m' 'I b a m'     'I b a m'
73       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'I 21/b 21/c 21/a' 'I b c a'  'I b c a'
74       '-P 2 2' 'mmm'      '-P 2 2' 'mmm'      'I 21/m 21/m 21/a' 'I m m a'  'I m m a'
75       '-P 4' '4/m'        ' P 4' '4'          'P 4'                         'P 4'
76       '-P 4' '4/m'        ' P 4' '4'          'P 41'                        'P 41'
77       '-P 4' '4/m'        ' P 4' '4'          'P 42'                        'P 42'
78       '-P 4' '4/m'        ' P 4' '4'          'P 43'                        'P 43'
79       '-P 4' '4/m'        ' P 4' '4'          'I 4'                         'I 4'
80       '-P 4' '4/m'        ' P 4' '4'          'I 41'                        'I 41'
81       '-P 4' '4/m'        ' P -4' '-4'        'P -4'                        'P -4'
82       '-P 4' '4/m'        ' P -4' '-4'        'I -4'                        'I -4'
83       '-P 4' '4/m'        '-P 4' '4/m'        'P 4/m'                       'P 4/m'
84       '-P 4' '4/m'        '-P 4' '4/m'        'P 42/m'                      'P 42/m'
85       '-P 4' '4/m'        '-P 4' '4/m'        'P 4/n'                       'P 4/n   1'
85       '-P 4' '4/m'        '-P 4' '4/m'        'P 4/n'                       'P 4/n   2'
86       '-P 4' '4/m'        '-P 4' '4/m'        'P 42/n'                      'P 42/n   1'
86       '-P 4' '4/m'        '-P 4' '4/m'        'P 42/n'                      'P 42/n   2'
87       '-P 4' '4/m'        '-P 4' '4/m'        'I 4/m'                       'I 4/m'
88       '-P 4' '4/m'        '-P 4' '4/m'        'I 41/a'                      'I 41/a   1'
88       '-P 4' '4/m'        '-P 4' '4/m'        'I 41/a'                      'I 41/a   2'
89       '-P 4 2' '4/mmm'    ' P 4 2' '422'      'P 4 2 2'                     'P 4 2 2'
90       '-P 4 2' '4/mmm'    ' P 4 2' '422'      'P 4 21 2'                    'P 4 21 2'
91       '-P 4 2' '4/mmm'    ' P 4 2' '422'      'P 41 2 2'                    'P 41 2 2'
92       '-P 4 2' '4/mmm'    ' P 4 2' '422'      'P 41 21 2'                   'P 41 21 2'
93       '-P 4 2' '4/mmm'    ' P 4 2' '422'      'P 42 2 2'                    'P 42 2 2'
94       '-P 4 2' '4/mmm'    ' P 4 2' '422'      'P 42 21 2'                   'P 42 21 2'
95       '-P 4 2' '4/mmm'    ' P 4 2' '422'      'P 43 2 2'                    'P 43 2 2'
96       '-P 4 2' '4/mmm'    ' P 4 2' '422'      'P 43 21 2'                   'P 43 21 2'
97       '-P 4 2' '4/mmm'    ' P 4 2' '422'      'I 4 2 2'                     'I 4 2 2'
98       '-P 4 2' '4/mmm'    ' P 4 2' '422'      'I 41 2 2'                    'I 41 2 2'
99       '-P 4 2' '4/mmm'    ' P 4 -2' '4mm'     'P 4 m m'                     'P 4 m m'
100       '-P 4 2' '4/mmm'    ' P 4 -2' '4mm'     'P 4 b m'                     'P 4 b m'
101       '-P 4 2' '4/mmm'    ' P 4 -2' '4mm'     'P 42 c m'                    'P 42 c m'
102       '-P 4 2' '4/mmm'    ' P 4 -2' '4mm'     'P 42 n m'                    'P 42 n m'
103       '-P 4 2' '4/mmm'    ' P 4 -2' '4mm'     'P 4 c c'                     'P 4 c c'
104       '-P 4 2' '4/mmm'    ' P 4 -2' '4mm'     'P 4 n c'                     'P 4 n c'
105       '-P 4 2' '4/mmm'    ' P 4 -2' '4mm'     'P 42 m c'                    'P 42 m c'
106       '-P 4 2' '4/mmm'    ' P 4 -2' '4mm'     'P 42 b c'                    'P 42 b c'
107       '-P 4 2' '4/mmm'    ' P 4 -2' '4mm'     'I 4 m m'                     'I 4 m m'
108       '-P 4 2' '4/mmm'    ' P 4 -2' '4mm'     'I 4 c m'                     'I 4 c m'
109       '-P 4 2' '4/mmm'    ' P 4 -2' '4mm'     'I 41 m d'                    'I 41 m d'
110       '-P 4 2' '4/mmm'    ' P 4 -2' '4mm'     'I 41 c d'                    'I 41 c d'
111       '-P 4 2' '4/mmm'    ' P 4 -2' '4mm'     'P -4 2 m'                    'P -4 2 m'
112       '-P 4 2' '4/mmm'    ' P 4 -2' '4mm'     'P -4 2 c'                    'P -4 2 c'
113       '-P 4 2' '4/mmm'    ' P 4 -2' '4mm'     'P -4 21 m'                   'P -4 21 m'
114       '-P 4 2' '4/mmm'    ' P 4 -2' '4mm'     'P -4 21 c'                   'P -4 21 c'
115       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P -4 m 2'                    'P -4 m 2'
116       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P -4 c 2'                    'P -4 c 2'
117       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P -4 b 2'                    'P -4 b 2'
118       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P -4 n 2'                    'P -4 n 2'
119       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'I -4 m 2'                    'I -4 m 2'
120       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'I -4 c 2'                    'I -4 c 2'
121       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'I -4 2 m'                    'I -4 2 m'
122       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'I -4 2 d'                    'I -4 2 d'
123       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P 4/m 2/m 2/m' 'P4/m m m'    'P 4/m m m'
124       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P 4/m 2/c 2/c' 'P4/m c c'    'P 4/m c c'
125       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P 4/n 2/b 2/m' 'P4/n b m'    'P 4/n b m   1'
125       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P 4/n 2/b 2/m' 'P4/n b m'    'P 4/n b m   1'
126       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P 4/n 2/n 2/c' 'P4/n n c'    'P 4/n n c   1'
126       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P 4/n 2/n 2/c' 'P4/n n c'    'P 4/n n c   1'
127       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P 4/m 21/b 2/m' 'P4/m b m'   'P 4/m b m'
128       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P 4/m 21/n 2/c' 'P4/m n c'   'P 4/m n c'
129       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P 4/n 21/m 2/m' 'P4/n m m'   'P 4/n m m   1'
130       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P 4/n 2/c 2/c' 'P4/n c c'    'P 4/n c c   1'
131       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P 42/m 2/m 2/c' 'P42/m m c'  'P 42/m m c'
132       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P 42/m 2/c 2/m' 'P42/m c m'  'P 42/m c m'
133       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P 42/n 2/b 2/c' 'P42/n b c'  'P 42/n b c   1'
134       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P 42/n 2/n 2/m' 'P42/n n m'  'P 42/n n m   1'
135       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P 42/m 21/b 2/c' 'P42/m b c' 'P 42/m b c'
136       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P 42/m 21/n 2/m' 'P42/m n m' 'P 42/m n m'
137       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P 42/n 21/m 2/c' 'P42/n m c' 'P 42/n m c   1'
138       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'P 42/n 21/c 2/m' 'P42/n c m' 'P 42/n c m   1'
139       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'I 4/m 2/m 2/m' 'I4/m m m'    'I 4/m m m'
140       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'I 4/m 2/c 2/m' 'I4/m c m'    'I 4/m c m'
141       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'I 41/a 2/m 2/d' 'I41/a m d'  'I 41/a m d   1'
142       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'I41/a c d' 'I 41/a 2/c 2/d'  'I 41/a c d   1'
142       '-P 4 2' '4/mmm'    ' P -4 -2' '-42m'   'I41/a c d' 'I 41/a 2/c 2/d'  'I 41/a c d   2'
143       '-P 3' '-3'         ' P 3' '3'          'P 3'                         'P 3'
144       '-P 3' '-3'         ' P 3' '3'          'P 31'                        'P 31'
145       '-P 3' '-3'         ' P 3' '3'          'P 32'                        'P 32'
146       '-P 3' '-3'         ' P 3' '3'          'H 3'                         'R 3   H'
146       '-P 3*' '-3'        ' P 3*' '3'         'R 3'                         'R 3   R'
147       '-P 3' '-3'         '-P 3' '-3'         'P -3'                        'P -3'
148       '-P 3' '-3'         '-P 3' '-3'         'H -3'                        'R -3   H'
148       '-P 3*' '-3'        '-P 3*' '-3'        'R -3'                        'R -3   R'
149       '-P 3 2' '-31m'     ' P 3 2' '312'      'P 3 1 2'                     'P 3 1 2'
150       '-P 3 2"' '-3m1'    ' P 3 2"' '321'     'P 3 2 1'                     'P 3 2 1'
151       '-P 3 2' '-31m'     ' P 3 2' '312'      'P 31 1 2'                    'P 31 1 2'
152       '-P 3 2"' '-3m1'    ' P 3 2"' '321'     'P 31 2 1'                    'P 31 2 1'
153       '-P 3 2' '-31m'     ' P 3 2' '312'      'P 32 1 2'                    'P 32 1 2'
154       '-P 3 2"' '-3m1'    ' P 3 2"' '321'     'P 32 2 1'                    'P 32 2 1'
155       '-P 3 2"' '-3m1'    ' P 3 2"' '321'     'H 3 2'                       'R 3 2   H'
155       '-P 3* 2' '-3m'     ' P 3* 2' '32'      'R 3 2'                       'R 3 2   R'
156       '-P 3 2"' '-3m1'    ' P 3 -2"' '3m1'    'P 3 m 1'                     'P 3 m 1'
157       '-P 3 2' '-31m'     ' P 3 -2' '31m'     'P 3 1 m'                     'P 3 1 m'
158       '-P 3 2"' '-3m1'    ' P 3 -2"' '3m1'    'P 3 c 1'                     'P 3 c 1'
159       '-P 3 2' '-31m'     ' P 3 -2' '31m'     'P 3 1 c'                     'P 3 1 c'
160       '-P 3 2"' '-3m1'    ' P 3 -2"' '3m1'    'H 3 m'                       'R 3 m   H'
160       '-P 3* 2' '-3m'     ' P 3* -2' '3m'     'R 3 m'                       'R 3 m   R'
161       '-P 3 2"' '-3m1'    ' P 3 -2"' '3m1'    'H 3 c'                       'R 3 c   H'
161       '-P 3* 2' '-3m'     ' P 3* -2' '3m'     'R 3 c'                       'R 3 c   R'
162       '-P 3 2' '-31m'     '-P 3 2' '-31m'     'P -3 1 2/m' 'P -3 1 m'       'P -3 1 m'
163       '-P 3 2' '-31m'     '-P 3 2' '-31m'     'P -3 1 2/c' 'P -3 1 c'       'P -3 1 c'
164       '-P 3 2"' '-3m1'    '-P 3 2"' '-3m1'    'P -3 2/m 1' 'P -3 m 1'       'P -3 m 1'
165       '-P 3 2"' '-3m1'    '-P 3 2"' '-3m1'    'P -3 2/c 1' 'P -3 c 1'       'P -3 c 1'
166       '-P 3 2"' '-3m1'    '-P 3 2"' '-3m1'    'H -3 2/m' 'H -3 m'           'R -3 m   H'
166       '-P 3* 2' '-3m'     '-P 3* 2' '-3m'     'R -3 2/m' 'R -3 m'           'R -3 m   R'
167       '-P 3 2"' '-3m1'    '-P 3 2"' '-3m1'    'H -3 2/c' 'H -3 c'           'R -3 c   H'
167       '-P 3* 2' '-3m'     '-P 3* 2' '-3m'     'R -3 2/c' 'R -3 c'           'R -3 c   R'
168       '-P 6' '6/m'        ' P 6' '6'          'P 6'                         'P 6'
169       '-P 6' '6/m'        ' P 6' '6'          'P 61'                        'P 61'
170       '-P 6' '6/m'        ' P 6' '6'          'P 65'                        'P 65'
171       '-P 6' '6/m'        ' P 6' '6'          'P 62'                        'P 62'
172       '-P 6' '6/m'        ' P 6' '6'          'P 64'                        'P 64'
173       '-P 6' '6/m'        ' P 6' '6'          'P 63'                        'P 63'
174       '-P 6' '6/m'        ' P -6' '-6'        'P -6'                        'P -6'
175       '-P 6' '6/m'        '-P 6' '6/m'        'P 6/m'                       'P 6/m'
176       '-P 6' '6/m'        '-P 6' '6/m'        'P 63/m'                      'P 63/m'
177       '-P 6 2' '6/mmm'    ' P 6 2' '622'      'P 6 2 2'                     'P 6 2 2'
178       '-P 6 2' '6/mmm'    ' P 6 2' '622'      'P 61 2 2'                    'P 61 2 2'
179       '-P 6 2' '6/mmm'    ' P 6 2' '622'      'P 65 2 2'                    'P 65 2 2'
180       '-P 6 2' '6/mmm'    ' P 6 2' '622'      'P 62 2 2'                    'P 62 2 2'
181       '-P 6 2' '6/mmm'    ' P 6 2' '622'      'P 64 2 2'                    'P 64 2 2'
182       '-P 6 2' '6/mmm'    ' P 6 2' '622'      'P 63 2 2'                    'P 63 2 2'
183       '-P 6 2' '6/mmm'    ' P 6 -2' '6mm'     'P 6 m m'                     'P 6 m m'
184       '-P 6 2' '6/mmm'    ' P 6 -2' '6mm'     'P 6 c c'                     'P 6 c c'
185       '-P 6 2' '6/mmm'    ' P 6 -2' '6mm'     'P 63 c m'                    'P 63 c m'
186       '-P 6 2' '6/mmm'    ' P 6 -2' '6mm'     'P 63 m c'                    'P 63 m c'
187       '-P 6 2' '6/mmm'    ' P -6 2' '-62m'    'P -6 m 2'                    'P -6 m 2'
188       '-P 6 2' '6/mmm'    ' P -6 2' '-62m'    'P -6 c 2'                    'P -6 c 2'
189       '-P 6 2' '6/mmm'    ' P -6 -2' '-6m2'   'P -6 2 m'                    'P -6 2 m'
190       '-P 6 2' '6/mmm'    ' P -6 -2' '-6m2'   'P -6 2 c'                    'P -6 2 c'
191       '-P 6 2' '6/mmm'    '-P 6 2' '6/mmm'    'P 6/m 2/m 2/m' 'P 6/m m m'   'P 6/m m m'
192       '-P 6 2' '6/mmm'    '-P 6 2' '6/mmm'    'P 6/m 2/c 2/c' 'P 6/m c c'   'P 6/m c c'
193       '-P 6 2' '6/mmm'    '-P 6 2' '6/mmm'    'P 63/m 2/c 2/m' 'P 63/m c m' 'P 63/m c m'
194       '-P 6 2' '6/mmm'    '-P 6 2' '6/mmm'    'P 63/m 2/m 2/c' 'P 63/m m c' 'P 63/m m c'
195       '-P 2 2 3' 'm-3'    ' P 2 2 3' '23'     'P 2 3'                       'P 2 3'
196       '-P 2 2 3' 'm-3'    ' P 2 2 3' '23'     'F 2 3'                       'F 2 3'
197       '-P 2 2 3' 'm-3'    ' P 2 2 3' '23'     'I 2 3'                       'I 2 3'
198       '-P 2 2 3' 'm-3'    ' P 2 2 3' '23'     'P 21 3'                      'P 21 3'
199       '-P 2 2 3' 'm-3'    ' P 2 2 3' '23'     'I 21 3'                      'I 21 3'
200       '-P 2 2 3' 'm-3'    '-P 2 2 3' 'm-3'    'P 2/m -3' 'P m -3'           'P m -3'
201       '-P 2 2 3' 'm-3'    '-P 2 2 3' 'm-3'    'P 2/n -3' 'P n -3'           'P n -3   1'
202       '-P 2 2 3' 'm-3'    '-P 2 2 3' 'm-3'    'F 2/m -3' 'F m -3'           'F m -3'
203       '-P 2 2 3' 'm-3'    '-P 2 2 3' 'm-3'    'F 2/d -3' 'F d -3'           'F d -3   1'
203       '-P 2 2 3' 'm-3'    '-P 2 2 3' 'm-3'    'F 2/d -3' 'F d -3'           'F d -3   1'
204       '-P 2 2 3' 'm-3'    '-P 2 2 3' 'm-3'    'I 2/m -3' 'I m -3'           'I m -3'
205       '-P 2 2 3' 'm-3'    '-P 2 2 3' 'm-3'    'P 21/a -3' 'P a -3'          'P a -3'
206       '-P 2 2 3' 'm-3'    '-P 2 2 3' 'm-3'    'I 21/a -3' 'I a -3'          'I a -3'
207       '-P 4 2 3' 'm-3m'   ' P 4 2 3' '432'    'P 4 3 2'                     'P 4 3 2'
208       '-P 4 2 3' 'm-3m'   ' P 4 2 3' '432'    'P 42 3 2'                    'P 42 3 2'
209       '-P 4 2 3' 'm-3m'   ' P 4 2 3' '432'    'F 4 3 2'                     'F 4 3 2'
210       '-P 4 2 3' 'm-3m'   ' P 4 2 3' '432'    'F 41 3 2'                    'F 41 3 2'
211       '-P 4 2 3' 'm-3m'   ' P 4 2 3' '432'    'I 4 3 2'                     'I 4 3 2'
212       '-P 4 2 3' 'm-3m'   ' P 4 2 3' '432'    'P 43 3 2'                    'P 43 3 2'
213       '-P 4 2 3' 'm-3m'   ' P 4 2 3' '432'    'P 41 3 2'                    'P 41 3 2'
214       '-P 4 2 3' 'm-3m'   ' P 4 2 3' '432'    'I 41 3 2'                    'I 41 3 2'
215       '-P 4 2 3' 'm-3m'   ' P -4 2 3' '-43m'  'P -4 3 m'                    'P -4 3 m'
216       '-P 4 2 3' 'm-3m'   ' P -4 2 3' '-43m'  'F -4 3 m'                    'F -4 3 m'
217       '-P 4 2 3' 'm-3m'   ' P -4 2 3' '-43m'  'I -4 3 m'                    'I -4 3 m'
218       '-P 4 2 3' 'm-3m'   ' P -4 2 3' '-43m'  'P -4 3 n'                    'P -4 3 n'
219       '-P 4 2 3' 'm-3m'   ' P -4 2 3' '-43m'  'F -4 3 c'                    'F -4 3 c'
220       '-P 4 2 3' 'm-3m'   ' P -4 2 3' '-43m'  'I -4 3 d'                    'I -4 3 d'
221       '-P 4 2 3' 'm-3m'   '-P 4 2 3' 'm-3m'   'P 4/m -3 2/m' 'P m -3 m'     'P m -3 m'
222       '-P 4 2 3' 'm-3m'   '-P 4 2 3' 'm-3m'   'P 4/n -3 2/n' P n -3 n'      'P n -3 n   1'
223       '-P 4 2 3' 'm-3m'   '-P 4 2 3' 'm-3m'   'P 42/m -3 2/n' 'P m -3 n'    'P m -3 n'
224       '-P 4 2 3' 'm-3m'   '-P 4 2 3' 'm-3m'   'P 42/n -3 2/m' 'P n -3 m'    'P n -3 m   1'
225       '-P 4 2 3' 'm-3m'   '-P 4 2 3' 'm-3m'   'F 4/m -3 2/m' 'F m -3 m'     'F m -3 m'
226       '-P 4 2 3' 'm-3m'   '-P 4 2 3' 'm-3m'   'F 4/m -3 2/c' 'F m -3 c'     'F m -3 c'
227       '-P 4 2 3' 'm-3m'   '-P 4 2 3' 'm-3m'   'F 41/d -3 2/m' 'F d -3 m'    'F d -3 m   1'
228       '-P 4 2 3' 'm-3m'   '-P 4 2 3' 'm-3m'   'F d -3 c' 'F 41/d -3 2/c'    'F d -3 c   1'
229       '-P 4 2 3' 'm-3m'   '-P 4 2 3' 'm-3m'   'I 4/m -3 2/m' 'I m -3 m'     'I m -3 m'
230       '-P 4 2 3' 'm-3m'   '-P 4 2 3' 'm-3m'   'I 41/a -3 2/d' 'I a -3 d'    'I a -3 d'

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Martyn Winn, Daresbury