Symmetry in reciprocal space

Shmueli, U.; Hall, S. R.; Grosse-Kunstleve, R. W.

doi:10.1107/97809553602060000552

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 1.4, pp. 107-119 | 1 | 2 |
https://doi.org/10.1107/97809553602060000552

Appendix A1.4.2. Space-group symbols for numeric and symbolic computations

U. Shmueli,^a ^* S. R. Hall^b and R. W. Grosse-Kunstleve^c

^aSchool of Chemistry, Tel Aviv University, 69 978 Tel Aviv, Israel, ^bCrystallography Centre, University of Western Australia, Nedlands 6907, WA, Australia, and ^cLawrence Berkeley National Laboratory, 1 Cyclotron Road, Mailstop 4-230, Berkeley, CA 94720, USA
Correspondence e-mail: ushmueli@post.tau.ac.il

A1.4.2.1. Introduction

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This appendix lists two sets of computer-adapted space-group symbols which are implemented in existing crystallographic software and can be employed in the automated generation of space-group representations. The computer generation of space-group symmetry information is of well known importance in many crystallographic calculations, numeric as well as symbolic. A prerequisite for a computer program that generates this information is a set of computer-adapted space-group symbols which are based on the generating elements of the space group to be derived. The sets of symbols to be presented are:

(i) Explicit symbols . These symbols are based on the classification of crystallographic point groups and space groups by Zachariasen (1945). These symbols are termed explicit because they contain in an explicit manner the rotation and translation parts of the space-group generators of the space group to be derived and used. These computer-adapted explicit symbols were proposed by Shmueli (1984), who also describes in detail their implementation in the program SPGRGEN. This program was used for the automatic preparation of the structure-factor tables for the 17 plane groups and 230 space groups, listed in Appendix 1.4.3, and the 230 space groups in reciprocal space, listed in Appendix 1.4.4. The explicit symbols presented in this appendix are adapted to the 306 representations of the 230 space groups as presented in IT A (1983) with regard to the standard settings and choice of space-group origins.

The symmetry-generating algorithm underlying the explicit symbols, and their definition, are given in Section A1.4.2.2 of this appendix and the explicit symbols are listed in Table A1.4.2.1.

Table A1.4.2.1| top | pdf |
Explicit symbols

No.	Short Hermann–Mauguin symbol	Comments	Explicit symbols
1			`PAN$P1A000`
2	$[P\overline{1}]$		`PAC$I1A000`
3			`PMN$P2B000`
3			`PMN$P2C000`
4			`PMN$P2B060`
4			`PMN$P2C006`
5			`CMN$P2B000`
5			`AMN$P2B000`
5			`IMN$P2B000`
5			`AMN$P2C000`
5			`BMN$P2C000`
5			`IMN$P2C000`
6			`PMN$I2B000`
6			`PMN$I2C000`
7			`PMN$I2B006`
7			`PMN$I2B606`
7			`PMN$I2B600`
7			`PMN$I2C600`
7			`PMN$I2C660`
7			`PMN$I2C060`
8			`CMN$I2B000`
8			`AMN$I2B000`
8			`IMN$I2B000`
8			`AMN$I2C000`
8			`BMN$I2C000`
8			`IMN$I2C000`
9			`CMN$I2B006`
9			`AMN$I2B606`
9			`IMN$I2B600`
9			`AMN$I2C600`
9			`BMN$I2C660`
9			`IMN$I2C060`
10			`PMC$I1A000$P2B000`
10			`PMC$I1A000$P2C000`
11			`PMC$I1A000$P2B060`
11			`PMC$I1A000$P2C006`
12			`CMC$I1A000$P2B000`
12			`AMC$I1A000$P2B000`
12			`IMC$I1A000$P2B000`
12			`AMC$I1A000$P2C000`
12			`BMC$I1A000$P2C000`
12			`IMC$I1A000$P2C000`
13			`PMC$I1A000$P2B006`
13			`PMC$I1A000$P2B606`
13			`PMC$I1A000$P2B600`
13			`PMC$I1A000$P2C600`
13			`PMC$I1A000$P2C660`
13			`PMC$I1A000$P2C060`
14			`PMC$I1A000$P2B066`
14			`PMC$I1A000$P2B666`
14			`PMC$I1A000$P2B660`
14			`PMC$I1A000$P2C606`
14			`PMC$I1A000$P2C666`
14			`PMC$I1A000$P2C066`
15			`CMC$I1A000$P2B006`
15			`AMC$I1A000$P2B606`
15			`IMC$I1A000$P2B600`
15			`AMC$I1A000$P2C600`
15			`BMC$I1A000$P2C660`
15			`IMC$I1A000$P2C060`
16			`PON$P2C000$P2A000`
17			`PON$P2C006$P2A000`
18			`PON$P2C000$P2A660`
19			`PON$P2C606$P2A660`
20			`CON$P2C006$P2A000`
21			`CON$P2C000$P2A000`
22			`FON$P2C000$P2A000`
23			`ION$P2C000$P2A000`
24			`ION$P2C606$P2A660`
25			`PON$P2C000$I2A000`
26			`PON$P2C006$I2A000`
27			`PON$P2C000$I2A006`
28			`PON$P2C000$I2A600`
29			`PON$P2C006$I2A606`
30			`PON$P2C000$I2A066`
31			`PON$P2C606$I2A000`
32			`PON$P2C000$I2A660`
33			`PON$P2C006$I2A666`
34			`PON$P2C000$I2A666`
35			`CON$P2C000$I2A000`
36			`CON$P2C006$I2A000`
37			`CON$P2C000$I2A006`
38			`AON$P2C000$I2A000`
39			`AON$P2C000$I2A060`
40			`AON$P2C000$I2A600`
41			`AON$P2C000$I2A660`
42			`FON$P2C000$I2A000`
43			`FON$P2C000$I2A333`
44			`ION$P2C000$I2A000`
45			`ION$P2C000$I2A660`
46			`ION$P2C000$I2A600`
47			`POC$I1A000$P2C000$P2A000`
48		Origin 1	`POC$I1A666$P2C000$P2A000`
48		Origin 2	`POC$I1A000$P2C660$P2A066`
49			`POC$I1A000$P2C000$P2A006`
50		Origin 1	`POC$I1A660$P2C000$P2A000`
50		Origin 2	`POC$I1A000$P2C660$P2A060`
51			`POC$I1A000$P2C600$P2A600`
52			`POC$I1A000$P2C600$P2A066`
53			`POC$I1A000$P2C606$P2A000`
54			`POC$I1A000$P2C600$P2A606`
55			`POC$I1A000$P2C000$P2A660`
56			`POC$I1A000$P2C660$P2A606`
57			`POC$I1A000$P2C006$P2A060`
58			`POC$I1A000$P2C000$P2A666`
59		Origin 1	`POC$I1A660$P2C000$P2A660`
59		Origin 2	`POC$I1A000$P2C660$P2A600`
60			`POC$I1A000$P2C666$P2A660`
61			`POC$I1A000$P2C606$P2A660`
62			`POC$I1A000$P2C606$P2A666`
63			`COC$I1A000$P2C006$P2A000`
64			`COC$I1A000$P2C066$P2A000`
65			`COC$I1A000$P2C000$P2A000`
66			`COC$I1A000$P2C000$P2A006`
67			`COC$I1A000$P2C060$P2A000`
68		Origin 1	`COC$I1A066$P2C660$P2A660`
68		Origin 2	`COC$I1A000$P2C600$P2A606`
69			`FOC$I1A000$P2C000$P2A000`
70		Origin 1	`FOC$I1A333$P2C000$P2A000`
70		Origin 2	`FOC$I1A000$P2C990$P2A099`
71			`IOC$I1A000$P2C000$P2A000`
72			`IOC$I1A000$P2C000$P2A660`
73			`IOC$I1A000$P2C606$P2A660`
74			`IOC$I1A000$P2C060$P2A000`
75			`PTN$P4C000`
76			`PTN$P4C003`
77			`PTN$P4C006`
78			`PTN$P4C009`
79			`ITN$P4C000`
80			`ITN$P4C063`
81	$[P\overline{4}]$		`PTN$I4C000`
82	$[I\overline{4}]$		`ITN$I4C000`
83			`PTC$I1A000$P4C000`
84			`PTC$I1A000$P4C006`
85		Origin 1	`PTC$I1A660$P4C660`
85		Origin 2	`PTC$I1A000$P4C600`
86		Origin 1	`PTC$I1A666$P4C666`
86		Origin 2	`PTC$I1A000$P4C066`
87			`ITC$I1A000$P4C000`
88		Origin 1	`ITC$I1A063$P4C063`
88		Origin 2	`ITC$I1A000$P4C933`
89			`PTN$P4C000$P2A000`
90			`PTN$P4C660$P2A660`
91			`PTN$P4C003$P2A006`
92			`PTN$P4C663$P2A669`
93			`PTN$P4C006$P2A000`
94			`PTN$P4C666$P2A666`
95			`PTN$P4C009$P2A006`
96			`PTN$P4C669$P2A663`
97			`ITN$P4C000$P2A000`
98			`ITN$P4C063$P2A063`
99			`PTN$P4C000$I2A000`
100			`PTN$P4C000$I2A660`
101			`PTN$P4C006$I2A006`
102			`PTN$P4C666$I2A666`
103			`PTN$P4C000$I2A006`
104			`PTN$P4C000$I2A666`
105			`PTN$P4C006$I2A000`
106			`PTN$P4C006$I2A660`
107			`ITN$P4C000$I2A000`
108			`ITN$P4C000$I2A006`
109			`ITN$P4C063$I2A666`
110			`ITN$P4C063$I2A660`
111	$[P\overline{4}2m]$		`PTN$I4C000$P2A000`
112	$[P\overline{4}2c]$		`PTN$I4C000$P2A006`
113	$[P\overline{4}2_1m]$		`PTN$I4C000$P2A660`
114	$[P\overline{4}2_1c]$		`PTN$I4C000$P2A666`
115	$[P\overline{4}m2]$		`PTN$I4C000$P2D000`
116	$[P\overline{4}c2]$		`PTN$I4C000$P2D006`
117	$[P\overline{4}b2]$		`PTN$I4C000$P2D660`
118	$[P\overline{4}n2]$		`PTN$I4C000$P2D666`
119	$[I\overline{4}m2]$		`ITN$I4C000$P2D000`
120	$[I\overline{4}c2]$		`ITN$I4C000$P2D006`
121	$[I\overline{4}2m]$		`ITN$I4C000$P2A000`
122	$[I\overline{4}2d]$		`ITN$I4C000$P2A609`
123			`PTC$I1A000$P4C000$P2A000`
124			`PTC$I1A000$P4C000$P2A006`
125		Origin 1	`PTC$I1A660$P4C000$P2A000`
125		Origin 2	`PTC$I1A000$P4C600$P2A060`
126		Origin 1	`PTC$I1A666$P4C000$P2A000`
126		Origin 2	`PTC$I1A000$P4C600$P2A066`
127			`PTC$I1A000$P4C000$P2A660`
128			`PTC$I1A000$P4C000$P2A666`
129		Origin 1	`PTC$I1A660$P4C660$P2A660`
129		Origin 2	`PTC$I1A000$P4C600$P2A600`
130		Origin 1	`PTC$I1A660$P4C660$P2A666`
130		Origin 2	`PTC$I1A000$P4C600$P2A606`
131			`PTC$I1A000$P4C006$P2A000`
132			`PTC$I1A000$P4C006$P2A006`
133		Origin 1	`PTC$I1A666$P4C666$P2A006`
133		Origin 2	`PTC$I1A000$P4C606$P2A060`
134		Origin 1	`PTC$I1A666$P4C666$P2A000`
134		Origin 2	`PTC$I1A000$P4C606$P2A066`
135			`PTC$I1A000$P4C006$P2A660`
136			`PTC$I1A000$P4C666$P2A666`
137		Origin 1	`PTC$I1A666$P4C666$P2A666`
137		Origin 2	`PTC$I1A000$P4C606$P2A600`
138		Origin 1	`PTC$I1A666$P4C666$P2A660`
138		Origin 2	`PTC$I1A000$P4C606$P2A606`
139			`ITC$I1A000$P4C000$P2A000`
140			`ITC$I1A000$P4C000$P2A006`
141		Origin 1	`ITC$I1A063$P4C063$P2A063`
141		Origin 2	`ITC$I1A000$P4C393$P2A000`
142		Origin 1	`ITC$I1A063$P4C063$P2A069`
142		Origin 2	`ITC$I1A000$P4C393$P2A006`
143			`PRN$P3C000`
144			`PRN$P3C004`
145			`PRN$P3C008`
146		Hexagonal axes	`RRN$P3C000`
146		Rhombohedral axes	`PRN$P3Q000`
147	$[P\overline{3}]$		`PRC$I3C000`
148	$[R\overline{3}]$	Hexagonal axes	`RRC$I3C000`
148	$[R\overline{3}]$	Rhombohedral axes	`PRC$I3Q000`
149			`PRN$P3C000$P2G000`
150			`PRN$P3C000$P2F000`
151			`PRN$P3C004$P2G000`
152			`PRN$P3C004$P2F008`
153			`PRN$P3C008$P2G000`
154			`PRN$P3C008$P2F004`
155		Hexagonal axes	`RRN$P3C000$P2F000`
155		Rhombohedral axes	`PRN$P3Q000$P2E000`
156			`PRN$P3C000$I2F000`
157			`PRN$P3C000$I2G000`
158			`PRN$P3C000$I2F006`
159			`PRN$P3C000$I2G006`
160		Hexagonal axes	`RRN$P3C000$I2F000`
160		Rhombohedral axes	`PRN$P3Q000$I2E000`
161		Hexagonal axes	`RRN$P3C000$I2F006`
161		Rhombohedral axes	`PRN$P3Q000$I2E666`
162	$[P\overline{3}1m]$		`PRC$I3C000$P2G000`
163	$[P\overline{3}1c]$		`PRC$I3C000$P2G006`
164	$[P\overline{3}m1]$		`PRC$I3C000$P2F000`
165	$[P\overline{3}c1]$		`PRC$I3C000$P2F006`
166	$[R\overline{3}m]$	Hexagonal axes	`RRC$I3C000$P2F000`
166	$[R\overline{3}m]$	Rhombohedral axes	`PRC$I3Q000$P2E000`
167	$[R\overline{3}c]$	Hexagonal axes	`RRC$I3C000$P2F006`
167	$[R\overline{3}c]$	Rhombohedral axes	`PRC$I3Q000$P2E666`
168			`PHN$P6C000`
169			`PHN$P6C002`
170			`PHN$P6C005`
171			`PHN$P6C004`
172			`PHN$P6C008`
173			`PHN$P6C006`
174	$[P\overline{6}]$		`PHN$I6C000`
175			`PHC$I1A000$P6C000`
176			`PHC$I1A000$P6C006`
177			`PHN$P6C000$P2F000`
178			`PHN$P6C002$P2F000`
179			`PHN$P6C005$P2F000`
180			`PHN$P6C004$P2F000`
181			`PHN$P6C008$P2F000`
182			`PHN$P6C006$P2F000`
183			`PHN$P6C000$I2F000`
184			`PHN$P6C000$I2F006`
185			`PHN$P6C006$I2F006`
186			`PHN$P6C006$I2F000`
187	$[P\overline{6}m2]$		`PHN$I6C000$P2G000`
188	$[P\overline{6}c2]$		`PHN$I6C006$P2G000`
189	$[P\overline{6}2m]$		`PHN$I6C000$P2F000`
190	$[P\overline{6}2c]$		`PHN$I6C006$P2F000`
191			`PHC$I1A000$P6C000$P2F000`
192			`PHC$I1A000$P6C000$P2F006`
193			`PHC$I1A000$P6C006$P2F006`
194			`PHC$I1A000$P6C006$P2F000`
195			`PCN$P3Q000$P2C000$P2A000`
196			`FCN$P3Q000$P2C000$P2A000`
197			`ICN$P3Q000$P2C000$P2A000`
198			`PCN$P3Q000$P2C606$P2A660`
199			`ICN$P3Q000$P2C606$P2A660`
200	$[Pm\overline{3}]$		`PCC$I3Q000$P2C000$P2A000`
201	$[Pn\overline{3}]$	Origin 1	`PCC$I3Q666$P2C000$P2A000`
201	$[Pn\overline{3}]$	Origin 2	`PCC$I3Q000$P2C660$P2A066`
202	$[Fm\overline{3}]$		`FCC$I3Q000$P2C000$P2A000`
203	$[Fd\overline{3}]$	Origin 1	`FCC$I3Q333$P2C000$P2A000`
203	$[Fd\overline{3}]$	Origin 2	`FCC$I3Q000$P2C330$P2A033`
204	$[Im\overline{3}]$		`ICC$I3Q000$P2C000$P2A000`
205	$[Pa\overline{3}]$		`PCC$I3Q000$P2C606$P2A660`
206	$[Ia\overline{3}]$		`ICC$I3Q000$P2C606$P2A660`
207			`PCN$P3Q000$P4C000$P2D000`
208			`PCN$P3Q000$P4C666$P2D666`
209			`FCN$P3Q000$P4C000$P2D000`
210			`FCN$P3Q000$P4C993$P2D939`
211			`ICN$P3Q000$P4C000$P2D000`
212			`PCN$P3Q000$P4C939$P2D399`
213			`PCN$P3Q000$P4C393$P2D933`
214			`ICN$P3Q000$P4C393$P2D933`
215	$[P\overline{4}3m]$		`PCN$P3Q000$I4C000$I2D000`
216	$[F\overline{4}3m]$		`FCN$P3Q000$I4C000$I2D000`
217	$[I\overline{4}3m]$		`ICN$P3Q000$I4C000$I2D000`
218	$[P\overline{4}3n]$		`PCN$P3Q000$I4C666$I2D666`
219	$[F\overline{4}3c]$		`FCN$P3Q000$I4C666$I2D666`
220	$[I\overline{4}3d]$		`ICN$P3Q000$I4C939$I2D399`
221	$[Pm\overline{3}m]$		`PCC$I3Q000$P4C000$P2D000`
222	$[Pn\overline{3}n]$	Origin 1	`PCC$I3Q666$P4C000$P2D000`
222	$[Pn\overline{3}n]$	Origin 2	`PCC$I3Q000$P4C600$P2D006`
223	$[Pm\overline{3}n]$		`PCC$I3Q000$P4C666$P2D666`
224	$[Pn\overline{3}m]$	Origin 1	`PCC$I3Q666$P4C666$P2D666`
224	$[Pn\overline{3}m]$	Origin 2	`PCC$I3Q000$P4C066$P2D660`
225	$[Fm\overline{3}m]$		`FCC$I3Q000$P4C000$P2D000`
226	$[Fm\overline{3}c]$		`FCC$I3Q000$P4C666$P2D666`
227	$[Fd\overline{3}m]$	Origin 1	`FCC$I3Q333$P4C993$P2D939`
227	$[Fd\overline{3}m]$	Origin 2	`FCC$I3Q000$P4C693$P2D936`
228	$[Fd\overline{3}c]$	Origin 1	`FCC$I3Q999$P4C993$P2D939`
228	$[Fd\overline{3}c]$	Origin 2	`FCC$I3Q000$P4C093$P2D930`
229	$[Im\overline{3}m]$		`ICC$I3Q000$P4C000$P2D000`
230	$[Ia\overline{3}d]$		`ICC$I3Q000$P4C393$P2D933`

(ii) Hall symbols . These symbols are based on the implied-origin notation of Hall (1981a,b), who also describes in detail the algorithm implemented in the program SGNAME (Hall, 1981a). In the first edition of IT B (1993), the term `concise space-group symbols' was used for this notation. In recent years, however, the term `Hall symbols' has come into use in symmetry papers (Altermatt & Brown, 1987; Grosse-Kunstleve, 1999), software applications (Hovmöller, 1992; Grosse-Kunstleve, 1995; Larine et al., 1995; Dowty, 1997) and data-handling approaches (Bourne et al., 1998). This term has therefore been adopted for the second edition.

The main difference in the definition of the Hall symbols between this edition and the first edition of IT B is the generalization of the origin-shift vector to a full change-of-basis matrix. The examples have been expanded to show how this matrix is applied. The notation has also been made more consistent, and a typographical error in a default axis direction has been corrected.¹ The lattice centring symbol `H' has been added to Table A1.4.2.2. In addition, Hall symbols are now provided for 530 settings to include all settings from Table 4.3.1 of IT A (1983). Namely, all non-standard symbols for the monoclinic and orthorhombic space groups are included.

Table A1.4.2.2| top | pdf |
Lattice symbol L

The lattice symbol L implies Seitz matrices for the lattice translations. For noncentrosymmetric lattices the rotation parts of the Seitz matrices are for 1 (see Table A1.4.2.4). For centrosymmetric lattices the rotation parts are 1 and −1. The translation parts in the fourth columns of the Seitz matrices are listed in the last column of the table. The total number of matrices implied by each symbol is given by nS.

Noncentrosymmetric		Centrosymmetric		Implied lattice translation(s)
Symbol	nS	Symbol	nS	Implied lattice translation(s)
P	1	−P	2
A	2	−A	4	$[0,{1 \over 2},{1 \over 2}]$
B	2	−B	4	$[{1 \over 2},0,{1 \over 2}]$
C	2	−C	4	$[{1 \over 2},{1 \over 2},0]$
I	2	−I	4	$[{1 \over 2},{1 \over 2},{1\over 2}]$
R	3	−R	6	$[{2 \over 3},{1 \over 3},{1 \over 3}]$ $[{1 \over 3},{2 \over 3},{2 \over 3}]$
H	3	−H	6	$[{2 \over 3},{1 \over 3},0]$ $[{1 \over 3},{2 \over 3},0]$
F	4	−F	8	$[0,{1 \over 2},{1 \over 2}]$ $[{1 \over 2},0,{1 \over 2}]$ $[{1 \over 2},{1 \over 2},0]$

Some of the space-group symbols listed in Table A1.4.2.7 differ from those listed in Table B.6 (p. 119) of the first edition of IT B. This is because the symmetry of many space groups can be represented by more than one subset of `generator' elements and these lead to different Hall symbols. The symbols listed in this edition have been selected after first sorting the symmetry elements into a strictly prescribed order based on the shape of their Seitz matrices, whereas those in Table B.6 were selected from symmetry elements in the order of IT I (1965). Software for selecting the Hall symbols listed in Table A1.4.2.7 is freely available (Hall, 1997). These symbols and their equivalents in the first edition of IT B will generate identical symmetry elements, but the former may be used as a reference table in a strict mapping procedure between different symmetry representations (Hall et al., 2000).

The Hall symbols are defined in Section A1.4.2.3 of this appendix and are listed in Table A1.4.2.7.

A1.4.2.2. Explicit symbols

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As shown elsewhere (Shmueli, 1984), the set of representative operators of a crystallographic space group [i.e. the set that is listed for each space group in the symmetry tables of IT A (1983) and automatically regenerated for the purpose of compiling the symmetry tables in the present chapter] may have one of the following forms: $[\eqalignno{& \{({\bf Q},{\bf u})\},&\cr&\{({\bf Q},{\bf u})\} \times \{({\bf R},{\bf v})\},\quad\hbox{\rm or }&({\rm A}1.4.2.1)\cr&\{({\bf P},{\bf t})\} \times [\{({\bf Q},{\bf u})\} \times \{({\bf R},{\bf v})\}],&\cr}]$ where P, Q and R are point-group operators, and t, u and v are zero vectors or translations not belonging to the lattice-translations subgroup. Each of the forms in (A1.4.2.1), enclosed in braces, is evaluated as, e.g., $[\{({\bf P},{\bf t})\}=\{({\bf I},{\bf 0}),({\bf P},{\bf t}),({\bf P},{\bf t})^{2},\ldots,({\bf P},{\bf t})^{g-1}\}, \eqno({\rm A}1.4.2.2)]$ where I is a unit operator and g is the order of the rotation operator P (i.e. P ^g = I). The representative operations of the space group are evaluated by expanding the generators into cyclic groups, as in (A1.4.2.2), and forming, as needed, ordered products of the expanded groups as indicated in (A1.4.2.1) and explained in detail in the original article (Shmueli, 1984). The rotation and translation parts of the generators (P, t), (Q, u) and (R, v) presented here were adapted to the settings and choices of origin used in the main symmetry tables of IT A (1983).

The general structure of a three-generator symbol, corresponding to the last line of (A1.4.2.1), as represented in Table A1.4.2.1, is $[{\rm LSC}\${\rm r}_1{\rm Pt}_1{\rm t}_2{\rm t}_3\${\rm r}_2{\rm Qu}_1{\rm u}_2{\rm u}_3\${\rm r}_3{\rm Rv}_1{\rm v}_2{\rm v}_3, \eqno({\rm A}1.4.2.3)]$ where

L – lattice type; can be P, A, B, C, I, F, or R. The symbol R is used only for the seven rhombohedral space groups in their representations in rhombohedral and hexagonal axes [obverse setting (IT I, 1952)].
S – crystal system; can be A (triclinic), M (monoclinic), O (orthorhombic), T (tetragonal), R (trigonal), H (hexagonal) or C (cubic).
C – status of centrosymmetry ; can be C or N according as the space group is centrosymmetric or noncentrosymmetric, respectively.
$ – this character is followed by six characters that define a generator of the space group.
r_i – indicator of the type of rotation that follows: r_i is P or I according as the rotation part of the ith generator is proper or improper , respectively.
P, Q, R – two-character symbols of matrix representations of the point-group rotation operators P, Q and R, respectively (see below).
t₁t₂t₃, u₁u₂u₃, v₁v₂v₃ – components of the translation parts of the generators, given in units of $[{{1}\over{12}}]$ ; e.g. the translation part (0 $[{{1}\over{2}}]$ $[{{3}\over{4}}]$ ) is given in Table A1.4.2.1 as 069. An exception: (0 0 $[{{5}\over{6}}]$ ) is denoted by 005 and not by 0010.

The two-character symbols for the matrices of rotation, which appear in the explicit space-group symbols in Table A1.4.2.1, are defined as follows: $[\displaylines{{\tt 1A}=\pmatrix{1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1\cr}\quad{\tt 2A}=\pmatrix{1 & 0 & 0 \cr 0 & \overline{1} & 0 \cr 0 & 0 & \overline{1}\cr}\quad{\tt 2B}=\pmatrix{\overline{1} & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & \overline{1}}\cr{\tt 2C}=\pmatrix{\overline{1} & 0 & 0 \cr 0 & \overline{1} & 0 \cr 0 & 0 & 1}\quad{\tt 2D}=\pmatrix{0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & \overline{1}}\quad{\tt 2E}=\pmatrix{0 & \overline{1} & 0 \cr \overline{1} & 0 & 0 \cr 0 & 0 & \overline{1}}\cr{ \tt 2F}=\pmatrix{1 & \overline{1} & 0 \cr 0 & \overline{1} & 0 \cr 0 & 0 & \overline{1}}\quad{\tt 2G}=\pmatrix{1 & 0 & 0 \cr 1 & \overline{1} & 0 \cr 0 & 0 & \overline{1}}\quad{\tt 3Q}=\pmatrix{0 & 0 & 1 \cr 1 & 0 & 0 \cr 0 & 1 & 0}\cr{\tt 3C}=\pmatrix{0 & \overline{1} & 0 \cr 1 & \overline{1} & 0 \cr 0 & 0 & 1}\quad{\tt 4C}=\pmatrix{0 & \overline{1} & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1}\quad{\tt 6C}=\pmatrix{1 & \overline{1} & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1},}]$ where only matrices of proper rotation are given (and required), since the corresponding matrices of improper rotation are created by the program for appropriate value of the r_i indicator. The first character of a symbol is the order of the axis of rotation and the second character specifies its orientation: in terms of direct-space lattice vectors, we have $[\displaylines{{\tt A} = [100], {\tt B} = [010], {\tt C} = [001], {\tt D} = [110],\cr{\tt E} = [1\overline{1}0], {\tt F} = [100], {\tt G} = [210]\hbox{ and }{\tt Q} = [111]}]$ for the standard orientations of the axes of rotation. Note that the axes 2F, 2G, 3C and 6C appear in trigonal and hexagonal space groups.

In the above scheme a space group is determined by one, two or at most three generators [see (A1.4.2.1)]. It should be pointed out that a convenient way of achieving a representation of the space group in any setting and relative to any origin is to start from the standard generators in Table A1.4.2.1 and let the computer program perform the appropriate transformation of the generators only, as in equations (1.4.4.4) and (1.4.4.5). The subsequent expansion of the transformed generators and the formation of the required products [see (A1.4.2.1) and (A1.4.2.2)] leads to the new representation of the space group.

In order to illustrate an explicit space-group symbol consider, for example, the symbol for the space group $[Ia\overline{3}d]$ , as given in Table A1.4.2.1: $[{\tt ICC\$I3Q000\$P4C393\$P2D933}.]$ The first three characters tell us that the Bravais lattice of this space group is of type I, that the space group is centrosymmetric and that it belongs to the cubic system. We then see that the generators are (i) an improper threefold axis along [111] (I3Q) with a zero translation part, (ii) a proper fourfold axis along [001] (P4C) with translation part (1/4, 3/4, 1/4) and (iii) a proper twofold axis along [110] (P2D) with translation part (3/4, 1/4, 1/4).

If we make use of the above-outlined interpretation of the explicit symbol (A1.4.2.3), the space-group symmetry transformations in direct space, corresponding to these three generators of the space group $[Ia\overline{3}d]$ , become $[\displaylines{\left[\pmatrix{0 & 0 & \overline{1} \cr \overline{1} & 0 & 0 \cr 0 & \overline{1} & 0}\pmatrix{x \cr y \cr z}+\pmatrix{0 \cr 0 \cr 0}\right]=\pmatrix{\overline{z} \cr \overline{x} \cr \overline{y}},\cr\left[\pmatrix{0 & \overline{1} & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1}\pmatrix{x \cr y \cr z}+\pmatrix{{{1}\over{4}} \cr {{3}\over{4}} \cr {{1}\over{4}}}\right]=\pmatrix{{{1}\over{4}}-y \cr {{3}\over{4}}+x \cr {{1}\over{4}}+z}, \cr \left[\pmatrix{0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & \overline{1}}\pmatrix{x \cr y \cr z}+\pmatrix{{{3}\over{4}} \cr {{1}\over{4}} \cr {{1}\over{4}}}\right]=\pmatrix{{{3}\over{4}}+y \cr{{1}\over{4}}+x \cr {{1}\over{4}}-z}.}]$

The corresponding symmetry transformations in reciprocal space, in the notation of Section 1.4.4, are $[\left[(hkl)\pmatrix{0 & 0 & \overline{1} \cr \overline{1} & 0 & 0 \cr 0 & \overline{1} & 0}:-(hkl)\pmatrix{0 \cr 0 \cr 0}\right]=[\overline{k} \overline{l} \overline{h}: 0]\hbox{;}]$ similarly, $[[k \overline{h} l: -131/4]]$ and $[[k h \overline{l}: -311/4]]$ are obtained from the second and third generator of $[Ia\overline{3}d]$ , respectively.

The first column of Table A1.4.2.1 lists the conventional space-group number. The second column shows the conventional short Hermann–Mauguin or international space-group symbol, and the third column, Comments, shows the full international space-group symbol only for the different settings of the monoclinic space groups that are given in the main space-group tables of IT A (1983). Other comments pertain to the choice of the space-group origin – where there are alternatives – and to axial systems. The fourth column shows the explicit space-group symbols described above for each of the settings considered in IT A (1983).

A1.4.2.3. Hall symbols

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The explicit-origin space-group notation proposed by Hall (1981a) is based on a subset of the symmetry operations, in the form of Seitz matrices, sufficient 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.

Table A1.4.2.7 lists space-group notation in several formats. The first column of Table A1.4.2.7 lists the space-group numbers with axis codes appended to identify the non-standard settings. The second column lists the Hermann–Mauguin symbols in computer-entry format with appended codes to identify the origin and cell choice when there are alternatives. The general forms of the Hall notation are listed in the fourth column and the computer-entry representations of these symbols are listed in the third column. The computer-entry format is the general notation expressed as case-insensitive ASCII characters with the overline (bar) symbol replaced by a minus sign.

The Hall notation has the general form: $[{\bf L}[{\bf N}_{\bf T}^{\bf A}]_{\bf 1}\ldots{}[{\bf N}_{\bf T}^{\bf A}]_{\bf p}{\bf V}.\eqno({\rm A}1.4.2.4)]$ L is the symbol specifying the lattice translational symmetry (see Table A1.4.2.2). The integral translations are implicitly included in the set of generators. If L has a leading minus sign, it also specifies an inversion centre at the origin. $[[{\bf N}_{\bf T}^{\bf A}]_{\bf n}]$ specifies the 4 × 4 Seitz matrix S _n of a symmetry element in the minimum set which defines the space-group symmetry (see Tables A1.4.2.3 to A1.4.2.6 ), and p is the number of elements in the set. V is a change-of-basis operator needed for less common descriptions of the space-group symmetry.

Table A1.4.2.3| top | pdf |
Translation symbol T

The symbol T specifies the translation elements of a Seitz matrix. Alphabetical symbols (given in the first column) specify translations along a fixed direction. Numerical symbols (given in the third column) 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	$[{1 \over 2},0,0]$	1 in 3₁	$[{1 \over 3}]$
b	$[0,{1 \over 2},0]$	2 in 3₂	$[{2 \over 3}]$
c	$[0,0,{1 \over 2}]$	1 in 4₁	$[{1 \over 4}]$
n	$[{1 \over 2}, {1 \over 2}, {1\over 2}]$	3 in 4₃	$[{3 \over 4}]$
u	$[{1 \over 4},0,0]$	1 in 6₁	$[{1 \over 6}]$
v	$[0,{1 \over 4},0]$	2 in 6₂	$[{1 \over 3}]$
w	$[0,0,{1 \over 4}]$	4 in 6₄	$[{2 \over 3}]$
d	$[{1\over 4},{1\over 4},{1\over 4}]$	5 in 6₅	$[{5 \over 6}]$

Table A1.4.2.4| top | pdf |
Rotation matrices for principal axes

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

Axis	Symbol A	Rotation order
Axis	Symbol A	1	2	3	4	6
a	x	$[\pmatrix{1&0&0\cr0&1&0\cr0&0&1\cr}]$	$[\pmatrix{1&0&0\cr0&\bar{1}&0\cr0&0&\bar{1}\cr}]$	$[\pmatrix{1&0&0\cr0&0&\bar{1}\cr0&1&\bar{1}\cr}]$	$[\pmatrix{1&0&0\cr0&0&\bar{1}\cr0&1&0\cr}]$	$[\pmatrix{1&0&0\cr0&1&\bar{1}\cr0&1&0\cr}]$
b	y	$[\pmatrix{1&0&0\cr0&1&0\cr0&0&1\cr}]$	$[\pmatrix{\bar{1}&0&0\cr0&1&0\cr0&0&\bar{1}\cr}]$	$[\pmatrix{\bar{1}&0&1\cr0&1&0\cr\bar{1}&0&0\cr}]$	$[\pmatrix{0&0&1\cr0&1&0\cr\bar{1}&0&0\cr}]$	$[\pmatrix{0&0&1\cr0&1&0\cr\bar{1}&0&1\cr}]$
c	z	$[\pmatrix{1&0&0\cr0&1&0\cr0&0&1\cr}]$	$[\pmatrix{\bar{1}&0&0\cr0&\bar{1}&0\cr0&0&1\cr}]$	$[\pmatrix{0&\bar{1}&0\cr1&\bar{1}&0\cr0&0&1\cr}]$	$[\pmatrix{0&\bar{1}&0\cr1&0&0\cr0&0&1\cr}]$	$[\pmatrix{1&\bar{1}&0\cr1&0&0\cr0&0&1\cr}]$

Table A1.4.2.5| top | pdf |
Rotation matrices for face-diagonal axes

The symbols for face-diagonal twofold rotations are 2′ and 2′′. The face-diagonal axis direction is determined by the axis of the preceding rotation N^x, N^y or N^z. Note that the single prime ′ is the default and may be omitted.

Preceding rotation	Rotation	Axis	Matrix
N^x	2′	b − c	$[\pmatrix{\bar{1}&0&0\cr0&0&\bar{1}\cr0&\bar{1}&0\cr}]$
N^x	2′′	b + c	$[\pmatrix{\bar{1}&0&0\cr0&0&1\cr0&1&0\cr}]$
N^y	2′	a − c	$[\pmatrix{0&0&\bar{1}\cr0&\bar{1}&0\cr\bar{1}&0&0\cr}]$
N^y	2′′	a + c	$[\pmatrix{0&0&1\cr0&\bar{1}&0\cr1&0&0\cr}]$
N^z	2′	a − b	$[\pmatrix{0&\bar{1}&0\cr\bar{1}&0&0\cr0&0&\bar{1}\cr}]$
N^z	2′′	a + b	$[\pmatrix{0&1&0\cr1&0&0\cr0&0&\bar{1}\cr}]$

Table A1.4.2.6| top | pdf |
Rotation matrix for the body-diagonal axis

The symbol for the threefold 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	Rotation	Matrix
a + b + c	3*	$[\pmatrix{0&0&1\cr1&0&0\cr0&1&0\cr}]$

The matrix symbol $[{\bf N}_{\bf T}^{\bf A}]$ is composed of three parts: N is the symbol denoting the |N|-fold order of the rotation matrix (see Tables A1.4.2.4, A1.4.2.5 and A1.4.2.6), T is a subscript symbol denoting the translation vector (see Table A1.4.2.3) and A is a superscript symbol denoting the axis of rotation.

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 as negative integers −1, −2, −3, −4 or −6 for improper rotations . The T translation symbols 1, 2, 3, 4, 5, 6, a, b, c, n, u, v, w, d are described in Table A1.4.2.3. These translations apply additively [e.g. ad signifies a ( $[{3 \over 4}, {1\over 4}, {1 \over 4}]$ ) translation]. The A axis symbols x, y, z denote rotations about the axes a, b and c, respectively (see Table A1.4.2.4). 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 A1.4.2.5). The axis symbol * always refers to a threefold rotation along a + b + c (see Table A1.4.2.6).

The change-of-basis operator V has the general form (v _x, v _y, v _z). The vectors v _x, v _y and v _z are specified by $[\eqalign{{v}_x &={r}_{1, \, 1}X+{r}_{1, \, 2}Y+{r}_{1, \, 3}Z+{\bf t}_1\cr {v}_y &={r}_{2, \, 1}X+{r}_{2, \, 2}Y+{r}_{2, \, 3}Z+{\bf t}_2\cr{v}_z &={r}_{3, \, 1}X+{r}_{3, \, 2}Y+{r}_{3, \, 3}Z+{\bf t}_3\cr},]$ where $[{r}_{i, \, j}]$ and $[{\bf t}_i]$ are fractions or real numbers. Terms in which $[{r}_{i, \, j}]$ or $[{\bf t}_i]$ are zero need not be specified. The 4 × 4 change-of-basis matrix operator V is defined as $[{\bf V} = \pmatrix{{r}_{1, \, 1}&{r}_{1, \, 2}&{r}_{1, \, 3}&{\bf t}_1\cr {r}_{2, \, 1}&{r}_{2, \, 2}&{r}_{2, \, 3}&{\bf t}_2\cr {r}_{3, \, 1}&{r}_{3, \, 2}& {r}_{3, \, 3}& {\bf t}_3\cr0&0&0&1\cr}.]$ The transformed symmetry operations are derived from the specified Seitz matrices S _n as $[{\bf S}_{\bf n}^{\prime}={\bf V}\cdot{\bf S}_{\bf n}\cdot{\bf V}^{-1}]$ and from the integral translations t(1, 0, 0), t(0, 1, 0) and t(0, 0, 1) as $[({\bf t}_{\bf n}^{\prime}, {\bf 1})^{T}={\bf V}\cdot({\bf t}_{\bf n}, {\bf 1})^{T}.]$

A shorthand form of V may be used when the change-of-basis operator only translates the origin of the basis system. In this form v _x, v _y and v _z are specified simply as shifts in twelfths, implying the matrix operator $[{\bf V}=\pmatrix{1&0&0&{{{v}_x}/12}\cr 0&1&0&{{{v}_y}/12}\cr 0 & 0 & 1 & {{{v}_z}/12}\cr 0 & 0 & 0 & 1\cr}.]$ In the shorthand form of V, the commas separating the vectors may be omitted.

A1.4.2.3.1. Default axes

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For most 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:

(i) the first rotation or roto-inversion has an axis direction of c;
(ii) the second rotation (if |N| is 2) has an axis direction of a if preceded by an |N| of 2 or 4, a−b if preceded by an |N| of 3 or 6;
(iii) the third rotation (if |N| is 3) has an axis direction of a + b + c.

A1.4.2.3.2. Example matrices

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The following examples show how the notation expands to Seitz matrices.

The notation $[\bar{\it 2}_c^{x}]$ represents an improper twofold rotation along a and a c/2 translation: $[-2{\rm xc}=\pmatrix{-1&0&0&0\cr0&1&0&0\cr0&0&1&{1 \over 2}\cr0&0&0&1}.]$

The notation $[{\it 3}^*]$ represents a threefold rotation along a + b + c: $[3^*=\pmatrix{0&0&1&0\cr1&0&0&0\cr0&1&0&0\cr0&0&0&1}.]$

The notation $[{\it 4}_{vw}]$ represents a fourfold rotation along c (implied) and translation of b/4 and c/4: $[4{\rm vw}=\pmatrix{0&-1&0&0\cr1&0&0&{1 \over 4}\cr0&0&1&{1 \over 4}\cr0&0&0&1}.]$

The notation 6₁ 2 (0 0 −1) represents a 6₁ screw along c, a twofold rotation along a − b and an origin shift of −c/12. Note that the 6₁ matrix is unchanged by the shifted origin whereas the 2 matrix is changed by −c/6. $[\eqalign{&61\;2\;(0\;0\;{-1})\cr&\quad{}=\pmatrix{1&-1&0&0\cr1&0&0&0\cr0&0&1&{1 \over 6}\cr 0&0&0&1\cr},\pmatrix{0&-1&0&0\cr-1&0&0&0\cr 0&0&-1&#38>;{5 \over 6}\cr0&0&0&1\cr}.\cr}]$ The change-of-basis vector (0 0 −1) could also be entered as (x, y, z − 1/12).

The reverse setting of the R-centred lattice (hexagonal axes) is specified using a change-of-basis transformation applied to the standard obverse setting (see Table A1.4.2.2). The obverse Seitz matrices are $[\openup3pt{\rm R}\;3=\pmatrix{1&0&0&{1 \over 3}\cr0&1&0&{2 \over 3}\cr0&0&1&{2 \over 3}\cr0&0&0&1\cr},\pmatrix{1&0&0&{2 \over 3}\cr0&1&0&{1 \over 3}\cr0&0&1&{1 \over 3}\cr0&0&0&1\cr},\pmatrix{0&-1&0&0\cr1&-1&0&0\cr0&0&1&0\cr0&0&0&1}.]$ The reverse-setting Seitz matrices are $[\displaylines{{\rm R}\;3\;({\rm -x,-y,z})\hfill\cr\quad= \pmatrix{1&0&0&{1 \over 3}\cr0&1&0&{2 \over 3}\cr0&0&1&{1 \over 3}\cr0&0&0&1\cr},\pmatrix{1&0&0&{2 \over 3}\cr0&1&0&{1 \over 3}\cr0&0&1&{2 \over 3}\cr0&0&0&1\cr},\pmatrix{0&-1&0&0\cr1&-1&0&0\cr0&0&1&0\cr0&0&0&1}.\cr}]$

Table A1.4.2.7| top | pdf |
Hall symbols

The first column, n:c, lists the space-group numbers and axis codes^† separated by a colon. The second column lists the Hermann–Mauguin symbols in computer-entry format. The third column lists the Hall symbols in computer-entry format and the fourth column lists the Hall symbols as described in Tables A1.4.2.2–A1.4.2.6 .

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

The codes appended to the space-group numbers listed in the first column identify the relationship between the symmetry elements and the crystal cell. Where no code is given the first choice listed below applies.

Monoclinic. Code = <unique axis><cell choice>: unique axis choices [cf. IT A (2005) Table 4.3.2.1 ] b, -b, c, -c, a, -a; cell choices [cf. IT A (2005) Table 4.3.2.1 ] 1, 2, 3.
Orthorhombic. Code = <origin choice><setting>: origin choices 1, 2; setting choices [cf. IT A (2005) Table4.3.2.1 ] abc, ba-c, cab, -cba, bca, a-cb.
Tetragonal, cubic. Code = <origin choice>: origin choices 1, 2.
Trigonal. Code = <cell choice>: cell choices h (hexagonal), r (rhombohedral).

The conventional primitive hexagonal lattice may be transformed to a C-centred orthohexagonal setting using the change-of-basis operator $[\openup3pt{\rm P}\;6\;({\rm x-1/2y, 1/2y, z})=\pmatrix{{1 \over 2}&-{3 \over 2}&0&0\cr{1 \over 2}&{1 \over 2}&0&0\cr0&0&1&0\cr0&0&0&1\cr}.]$ In this case the lattice translation for the C centring is obtained by transforming the integral translation t(0, 1, 0): $[\eqalign{{\bi V}\cdot\pmatrix{0&1&0&1\cr}^T&=\pmatrix{1&-{1\over 2}&0&0\cr0&{1 \over 2}&0&0\cr0&0&1&0\cr0&0&0&1\cr}\pmatrix{0\cr1\cr0\cr1\cr}\cr&=\pmatrix{-{1\over 2}&{1\over 2}&0&1\cr}^T.\cr}]$

The standard setting of an I-centred tetragonal space group may be transformed to a primitive setting using the change-of-basis operator $[{\rm I }\;4\;({\rm y+z,x+z,x+y})=\pmatrix{0&1&0&0\cr0&1&-1&0\cr-1&1&0&0\cr0&0&0&1\cr}.]$ Note that in the primitive setting, the fourfold axis is along a + b.

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