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

International Tables for Crystallography (2010). Vol. B, ch. 1.4, pp. 122-134   | 1 | 2 |
https://doi.org/10.1107/97809553602060000761

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

U. Shmueli,a* S. R. Hallb and R. W. Grosse-Kunstlevec

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[link]). 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[link]), 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 A1.4.3[link], and the 230 space groups in reciprocal space, listed in Appendix A1.4.4[link]. The explicit symbols presented in this appendix are adapted to the 306 representations of the 230 space groups as presented in IT A (1983[link]) 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[link] of this appendix and the explicit symbols are listed in Table A1.4.2.1[link].

    Table A1.4.2.1| top | pdf |
    Explicit symbols

    No.Short Hermann–Mauguin symbolCommentsExplicit symbols
    1 [P1]   PAN$P1A000
    2 [P\overline{1}]   PAC$I1A000
    3 [P2] [P121] PMN$P2B000
    3 [P2] [P112] PMN$P2C000
    4 [P2_1] [P12_11] PMN$P2B060
    4 [P2_1] [P112_1] PMN$P2C006
    5 [C2] [C121] CMN$P2B000
    5 [C2] [A121] AMN$P2B000
    5 [C2] [I121] IMN$P2B000
    5 [C2] [A112] AMN$P2C000
    5 [C2] [B112] BMN$P2C000
    5 [C2] [I112] IMN$P2C000
    6 [Pm] [P1m1] PMN$I2B000
    6 [Pm] [P11m] PMN$I2C000
    7 [Pc] [P1c1] PMN$I2B006
    7 [Pc] [P1n1] PMN$I2B606
    7 [Pc] [P1a1] PMN$I2B600
    7 [Pc] [P11a] PMN$I2C600
    7 [Pc] [P11n] PMN$I2C660
    7 [Pc] [P11b] PMN$I2C060
    8 [Cm] [C1m1] CMN$I2B000
    8 [Cm] [A1m1] AMN$I2B000
    8 [Cm] [I1m1] IMN$I2B000
    8 [Cm] [A11m] AMN$I2C000
    8 [Cm] [B11m] BMN$I2C000
    8 [Cm] [I11m] IMN$I2C000
    9 [Cc] [C1c1] CMN$I2B006
    9 [Cc] [A1n1] AMN$I2B606
    9 [Cc] [I1a1] IMN$I2B600
    9 [Cc] [A11a] AMN$I2C600
    9 [Cc] [B11n] BMN$I2C660
    9 [Cc] [I11b] IMN$I2C060
    10 [P2/m] [P12/m1] PMC$I1A000$P2B000
    10 [P2/m] [P112/m] PMC$I1A000$P2C000
    11 [P2_1/m] [P12_1/m1] PMC$I1A000$P2B060
    11 [P2_1/m] [P112_1/m] PMC$I1A000$P2C006
    12 [C2/m] [C12/m1] CMC$I1A000$P2B000
    12 [C2/m] [A12/m1] AMC$I1A000$P2B000
    12 [C2/m] [I12/m1] IMC$I1A000$P2B000
    12 [C2/m] [A112/m] AMC$I1A000$P2C000
    12 [C2/m] [B112/m] BMC$I1A000$P2C000
    12 [C2/m] [I112/m] IMC$I1A000$P2C000
    13 [P2/c] [P12/c1] PMC$I1A000$P2B006
    13 [P2/c] [P12/n1] PMC$I1A000$P2B606
    13 [P2/c] [P12/a1] PMC$I1A000$P2B600
    13 [P2/c] [P112/a] PMC$I1A000$P2C600
    13 [P2/c] [P112/n] PMC$I1A000$P2C660
    13 [P2/c] [P112/b] PMC$I1A000$P2C060
    14 [P2_1/c] [P12_1/c1] PMC$I1A000$P2B066
    14 [P2_1/c] [P12_1/n1] PMC$I1A000$P2B666
    14 [P2_1/c] [P12_1/a1] PMC$I1A000$P2B660
    14 [P2_1/c] [P112_1/a] PMC$I1A000$P2C606
    14 [P2_1/c] [P112_1/n] PMC$I1A000$P2C666
    14 [P2_1/c] [P112_1/b] PMC$I1A000$P2C066
    15 [C2/c] [C12/c1] CMC$I1A000$P2B006
    15 [C2/c] [A12/n1] AMC$I1A000$P2B606
    15 [C2/c] [I12/a1] IMC$I1A000$P2B600
    15 [C2/c] [A112/a] AMC$I1A000$P2C600
    15 [C2/c] [B112/n] BMC$I1A000$P2C660
    15 [C2/c] [I112/b] IMC$I1A000$P2C060
    16 [P222]   PON$P2C000$P2A000
    17 [P222_1]   PON$P2C006$P2A000
    18 [P2_12_12]   PON$P2C000$P2A660
    19 [P2_12_12_1]   PON$P2C606$P2A660
    20 [C222_1]   CON$P2C006$P2A000
    21 [C222]   CON$P2C000$P2A000
    22 [F222]   FON$P2C000$P2A000
    23 [I222]   ION$P2C000$P2A000
    24 [I2_12_12_1]   ION$P2C606$P2A660
    25 [Pmm2]   PON$P2C000$I2A000
    26 [Pmc2_1]   PON$P2C006$I2A000
    27 [Pcc2]   PON$P2C000$I2A006
    28 [Pma2]   PON$P2C000$I2A600
    29 [Pca2_1]   PON$P2C006$I2A606
    30 [Pnc2]   PON$P2C000$I2A066
    31 [Pmn2_1]   PON$P2C606$I2A000
    32 [Pba2]   PON$P2C000$I2A660
    33 [Pna2_1]   PON$P2C006$I2A666
    34 [Pnn2]   PON$P2C000$I2A666
    35 [Cmm2]   CON$P2C000$I2A000
    36 [Cmc2_1]   CON$P2C006$I2A000
    37 [Ccc2]   CON$P2C000$I2A006
    38 [Amm2]   AON$P2C000$I2A000
    39 [Abm2]   AON$P2C000$I2A060
    40 [Ama2]   AON$P2C000$I2A600
    41 [Aba2]   AON$P2C000$I2A660
    42 [Fmm2]   FON$P2C000$I2A000
    43 [Fdd2]   FON$P2C000$I2A333
    44 [Imm2]   ION$P2C000$I2A000
    45 [Iba2]   ION$P2C000$I2A660
    46 [Ima2]   ION$P2C000$I2A600
    47 [Pmmm]   POC$I1A000$P2C000$P2A000
    48 [Pnnn] Origin 1 POC$I1A666$P2C000$P2A000
    48 [Pnnn] Origin 2 POC$I1A000$P2C660$P2A066
    49 [Pccm]   POC$I1A000$P2C000$P2A006
    50 [Pban] Origin 1 POC$I1A660$P2C000$P2A000
    50 [Pban] Origin 2 POC$I1A000$P2C660$P2A060
    51 [Pmma]   POC$I1A000$P2C600$P2A600
    52 [Pnna]   POC$I1A000$P2C600$P2A066
    53 [Pmna]   POC$I1A000$P2C606$P2A000
    54 [Pcca]   POC$I1A000$P2C600$P2A606
    55 [Pbam]   POC$I1A000$P2C000$P2A660
    56 [Pccn]   POC$I1A000$P2C660$P2A606
    57 [Pbcm]   POC$I1A000$P2C006$P2A060
    58 [Pnnm]   POC$I1A000$P2C000$P2A666
    59 [Pmmn] Origin 1 POC$I1A660$P2C000$P2A660
    59 [Pmmn] Origin 2 POC$I1A000$P2C660$P2A600
    60 [Pbcn]   POC$I1A000$P2C666$P2A660
    61 [Pbca]   POC$I1A000$P2C606$P2A660
    62 [Pnma]   POC$I1A000$P2C606$P2A666
    63 [Cmcm]   COC$I1A000$P2C006$P2A000
    64 [Cmca]   COC$I1A000$P2C066$P2A000
    65 [Cmmm]   COC$I1A000$P2C000$P2A000
    66 [Cccm]   COC$I1A000$P2C000$P2A006
    67 [Cmma]   COC$I1A000$P2C060$P2A000
    68 [Ccca] Origin 1 COC$I1A066$P2C660$P2A660
    68 [Ccca] Origin 2 COC$I1A000$P2C600$P2A606
    69 [Fmmm]   FOC$I1A000$P2C000$P2A000
    70 [Fddd] Origin 1 FOC$I1A333$P2C000$P2A000
    70 [Fddd] Origin 2 FOC$I1A000$P2C990$P2A099
    71 [Immm]   IOC$I1A000$P2C000$P2A000
    72 [Ibam]   IOC$I1A000$P2C000$P2A660
    73 [Ibca]   IOC$I1A000$P2C606$P2A660
    74 [Imma]   IOC$I1A000$P2C060$P2A000
    75 [P4]   PTN$P4C000
    76 [P4_1]   PTN$P4C003
    77 [P4_2]   PTN$P4C006
    78 [P4_3]   PTN$P4C009
    79 [I4]   ITN$P4C000
    80 [I4_1]   ITN$P4C063
    81 [P\overline{4}]   PTN$I4C000
    82 [I\overline{4}]   ITN$I4C000
    83 [P4/m]   PTC$I1A000$P4C000
    84 [P4_2/m]   PTC$I1A000$P4C006
    85 [P4/n] Origin 1 PTC$I1A660$P4C660
    85 [P4/n] Origin 2 PTC$I1A000$P4C600
    86 [P4_2/n] Origin 1 PTC$I1A666$P4C666
    86 [P4_2/n] Origin 2 PTC$I1A000$P4C066
    87 [I4/m]   ITC$I1A000$P4C000
    88 [I4_1/a] Origin 1 ITC$I1A063$P4C063
    88 [I4_1/a] Origin 2 ITC$I1A000$P4C933
    89 [P422]   PTN$P4C000$P2A000
    90 [P42_12]   PTN$P4C660$P2A660
    91 [P4_122]   PTN$P4C003$P2A006
    92 [P4_12_12]   PTN$P4C663$P2A669
    93 [P4_222]   PTN$P4C006$P2A000
    94 [P4_22_12]   PTN$P4C666$P2A666
    95 [P4_322]   PTN$P4C009$P2A006
    96 [P4_32_12]   PTN$P4C669$P2A663
    97 [I422]   ITN$P4C000$P2A000
    98 [I4_122]   ITN$P4C063$P2A063
    99 [P4mm]   PTN$P4C000$I2A000
    100 [P4bm]   PTN$P4C000$I2A660
    101 [P4_2cm]   PTN$P4C006$I2A006
    102 [P4_2nm]   PTN$P4C666$I2A666
    103 [P4cc]   PTN$P4C000$I2A006
    104 [P4nc]   PTN$P4C000$I2A666
    105 [P4_2mc]   PTN$P4C006$I2A000
    106 [P4_2bc]   PTN$P4C006$I2A660
    107 [I4mm]   ITN$P4C000$I2A000
    108 [I4cm]   ITN$P4C000$I2A006
    109 [I4_1md]   ITN$P4C063$I2A666
    110 [I4_1cd]   ITN$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$P2A609
    123 [P4/mmm]   PTC$I1A000$P4C000$P2A000
    124 [P4/mcc]   PTC$I1A000$P4C000$P2A006
    125 [P4/nbm] Origin 1 PTC$I1A660$P4C000$P2A000
    125 [P4/nbm] Origin 2 PTC$I1A000$P4C600$P2A060
    126 [P4/nnc] Origin 1 PTC$I1A666$P4C000$P2A000
    126 [P4/nnc] Origin 2 PTC$I1A000$P4C600$P2A066
    127 [P4/mbm]   PTC$I1A000$P4C000$P2A660
    128 [P4/mnc]   PTC$I1A000$P4C000$P2A666
    129 [P4/nmm] Origin 1 PTC$I1A660$P4C660$P2A660
    129 [P4/nmm] Origin 2 PTC$I1A000$P4C600$P2A600
    130 [P4/ncc] Origin 1 PTC$I1A660$P4C660$P2A666
    130 [P4/ncc] Origin 2 PTC$I1A000$P4C600$P2A606
    131 [P4_2/mmc]   PTC$I1A000$P4C006$P2A000
    132 [P4_2/mcm]   PTC$I1A000$P4C006$P2A006
    133 [P4_2/nbc] Origin 1 PTC$I1A666$P4C666$P2A006
    133 [P4_2/nbc] Origin 2 PTC$I1A000$P4C606$P2A060
    134 [P4_2/nnm] Origin 1 PTC$I1A666$P4C666$P2A000
    134 [P4_2/nnm] Origin 2 PTC$I1A000$P4C606$P2A066
    135 [P4_2/mbc]   PTC$I1A000$P4C006$P2A660
    136 [P4_2/mnm]   PTC$I1A000$P4C666$P2A666
    137 [P4_2/nmc] Origin 1 PTC$I1A666$P4C666$P2A666
    137 [P4_2/nmc] Origin 2 PTC$I1A000$P4C606$P2A600
    138 [P4_2/ncm] Origin 1 PTC$I1A666$P4C666$P2A660
    138 [P4_2/ncm] Origin 2 PTC$I1A000$P4C606$P2A606
    139 [I4/mmm]   ITC$I1A000$P4C000$P2A000
    140 [I4/mcm]   ITC$I1A000$P4C000$P2A006
    141 [I4_1/amd] Origin 1 ITC$I1A063$P4C063$P2A063
    141 [I4_1/amd] Origin 2 ITC$I1A000$P4C393$P2A000
    142 [I4_1/acd] Origin 1 ITC$I1A063$P4C063$P2A069
    142 [I4_1/acd] Origin 2 ITC$I1A000$P4C393$P2A006
    143 [P3]   PRN$P3C000
    144 [P3_1]   PRN$P3C004
    145 [P3_2]   PRN$P3C008
    146 [R3] Hexagonal axes RRN$P3C000
    146 [R3] Rhombohedral axes PRN$P3Q000
    147 [P\overline{3}]   PRC$I3C000
    148 [R\overline{3}] Hexagonal axes RRC$I3C000
    148 [R\overline{3}] Rhombohedral axes PRC$I3Q000
    149 [P312]   PRN$P3C000$P2G000
    150 [P321]   PRN$P3C000$P2F000
    151 [P3_112]   PRN$P3C004$P2G000
    152 [P3_121]   PRN$P3C004$P2F008
    153 [P3_212]   PRN$P3C008$P2G000
    154 [P3_221]   PRN$P3C008$P2F004
    155 [R32] Hexagonal axes RRN$P3C000$P2F000
    155 [R32] Rhombohedral axes PRN$P3Q000$P2E000
    156 [P3m1]   PRN$P3C000$I2F000
    157 [P31m]   PRN$P3C000$I2G000
    158 [P3c1]   PRN$P3C000$I2F006
    159 [P31c]   PRN$P3C000$I2G006
    160 [R3m] Hexagonal axes RRN$P3C000$I2F000
    160 [R3m] Rhombohedral axes PRN$P3Q000$I2E000
    161 [R3c] Hexagonal axes RRN$P3C000$I2F006
    161 [R3c] Rhombohedral axes PRN$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$P2E666
    168 [P6]   PHN$P6C000
    169 [P6_1]   PHN$P6C002
    170 [P6_5]   PHN$P6C005
    171 [P6_2]   PHN$P6C004
    172 [P6_4]   PHN$P6C008
    173 [P6_3]   PHN$P6C006
    174 [P\overline{6}]   PHN$I6C000
    175 [P6/m]   PHC$I1A000$P6C000
    176 [P6_3/m]   PHC$I1A000$P6C006
    177 [P622]   PHN$P6C000$P2F000
    178 [P6_122]   PHN$P6C002$P2F000
    179 [P6_522]   PHN$P6C005$P2F000
    180 [P6_222]   PHN$P6C004$P2F000
    181 [P6_422]   PHN$P6C008$P2F000
    182 [P6_322]   PHN$P6C006$P2F000
    183 [P6mm]   PHN$P6C000$I2F000
    184 [P6cc]   PHN$P6C000$I2F006
    185 [P6_3cm]   PHN$P6C006$I2F006
    186 [P6_3mc]   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 [P6/mmm]   PHC$I1A000$P6C000$P2F000
    192 [P6/mcc]   PHC$I1A000$P6C000$P2F006
    193 [P6_3/mcm]   PHC$I1A000$P6C006$P2F006
    194 [P6_3/mmc]   PHC$I1A000$P6C006$P2F000
    195 [P23]   PCN$P3Q000$P2C000$P2A000
    196 [F23]   FCN$P3Q000$P2C000$P2A000
    197 [I23]   ICN$P3Q000$P2C000$P2A000
    198 [P2_13]   PCN$P3Q000$P2C606$P2A660
    199 [I2_13]   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 [P432]   PCN$P3Q000$P4C000$P2D000
    208 [P4_232]   PCN$P3Q000$P4C666$P2D666
    209 [F432]   FCN$P3Q000$P4C000$P2D000
    210 [F4_132]   FCN$P3Q000$P4C993$P2D939
    211 [I432]   ICN$P3Q000$P4C000$P2D000
    212 [P4_332]   PCN$P3Q000$P4C939$P2D399
    213 [P4_132]   PCN$P3Q000$P4C393$P2D933
    214 [I4_132]   ICN$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[link],b[link]), who also describes in detail the algorithm implemented in the program SGNAME (Hall, 1981a[link]). In the first edition of IT B (1993[link]), 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[link]; Grosse-Kunstleve, 1999[link]), software applications (Hovmöller, 1992[link]; Grosse-Kunstleve, 1995[link]; Larine et al., 1995[link]; Dowty, 1997[link]) and data-handling approaches (Bourne et al., 1998[link]). 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[link]. In addition, Hall symbols are now provided for 530 settings to include all settings from Table 4.3.1 of IT A (1983[link]). Namely, all nonstandard 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[link]). 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.

    NoncentrosymmetricCentrosymmetricImplied lattice translation(s)
    SymbolnSSymbolnS
    P 1 −P 2 [0,0,0]
    A 2 −A 4 [0,0,0] [0,{1 \over 2},{1 \over 2}]
    B 2 −B 4 [0,0,0] [{1 \over 2},0,{1 \over 2}]
    C 2 −C 4 [0,0,0] [{1 \over 2},{1 \over 2},0]
    I 2 −I 4 [0,0,0] [{1 \over 2},{1 \over 2},{1\over 2}]
    R 3 −R 6 [0,0,0] [{2 \over 3},{1 \over 3},{1 \over 3}] [{1 \over 3},{2 \over 3},{2 \over 3}]
    H 3 −H 6 [0,0,0] [{2 \over 3},{1 \over 3},0] [{1 \over 3},{2 \over 3},0]
    F 4 −F 8 [0,0,0] [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[link] 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[link]). Software for selecting the Hall symbols listed in Table A1.4.2.7[link] is freely available (Hall, 1997[link]). 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[link]).

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

A1.4.2.2. Explicit symbols

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As shown elsewhere (Shmueli, 1984[link]), 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[link]) 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[link]), 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. Pg = I). The representative operations of the space group are evaluated by expanding the generators into cyclic groups, as in (A1.4.2.2[link]), and forming, as needed, ordered products of the expanded groups as indicated in (A1.4.2.1[link]) and explained in detail in the original article (Shmueli, 1984[link]). 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[link]).

The general structure of a three-generator symbol, corresponding to the last line of (A1.4.2.1[link]), as represented in Table A1.4.2.1[link], 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[link])].

  • 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.

  • ri – indicator of the type of rotation that follows: ri 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[link] 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[link], 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 ri 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[link])]. 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[link] and let the computer program perform the appropriate transformation of the generators only, as in equations (1.4.4.4[link]) and (1.4.4.5[link]). The subsequent expansion of the transformed generators and the formation of the required products [see (A1.4.2.1[link]) and (A1.4.2.2[link])] 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[link]:[{\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[link]), 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[link], 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]\semi]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[link] 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[link]). 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[link]).

A1.4.2.3. Hall symbols

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The explicit-origin space-group notation proposed by Hall (1981a[link]) 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[link] lists space-group notation in several formats. The first column of Table A1.4.2.7[link] lists the space-group numbers with axis codes appended to identify the nonstandard 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[link]). 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 Sn of a symmetry element in the minimum set which defines the space-group symmetry (see Tables A1.4.2.3[link] to A1.4.2.6[link][link][link]), 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 symbolTranslation vectorSubscript symbolFractional 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.

AxisSymbol ARotation order
1234 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 Nx, Ny or Nz. Note that the single prime ′ is the default and may be omitted.

Preceding rotationRotationAxisMatrix
Nx 2 bc [\pmatrix{\bar{1}&0&0\cr0&0&\bar{1}\cr0&\bar{1}&0\cr}]
2′′ b + c [\pmatrix{\bar{1}&0&0\cr0&0&1\cr0&1&0\cr}]
Ny 2 ac [\pmatrix{0&0&\bar{1}\cr0&\bar{1}&0\cr\bar{1}&0&0\cr}]
2′′ a + c [\pmatrix{0&0&1\cr0&\bar{1}&0\cr1&0&0\cr}]
Nz 2 ab [\pmatrix{0&\bar{1}&0\cr\bar{1}&0&0\cr0&0&\bar{1}\cr}]
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.

AxisRotationMatrix
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[link], A1.4.2.5[link] and A1.4.2.6[link]), T is a subscript symbol denoting the translation vector (see Table A1.4.2.3[link]) 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[link]. 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[link]). The axis symbols ′′ and ′ signal rotations about the body-diagonal vectors a + b (or alternatively b + c or c + a) and ab (or alternatively bc or ca) (see Table A1.4.2.5[link]). The axis symbol * always refers to a threefold rotation along a + b + c (see Table A1.4.2.6[link]).

The change-of-basis operator V has the general form (vx, vy, vz). The vectors vx, vy and vz 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 Sn 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 vx, vy and vz 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, ab 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 ab 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&{5 \over 6}\cr0&0&0&1\cr}.\cr}]The change-of-basis vector (0 0 −1) could also be entered as (xyz − 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[link]). 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[link]–A1.4.2.6[link][link][link][link][link].

n:cH–M entryHall entryHall 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[link]) Table 4.3.2.1[link] ] b, -b, c, -c, a, -a; cell choices [cf. IT A (2005[link]) Table 4.3.2.1[link] ] 1, 2, 3.

  • Orthorhombic. Code = <origin choice><setting>: origin choices 1, 2; setting choices [cf. IT A (2005[link]) Table 4.3.2.1[link] ] 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.

References

International Tables for Crystallography (1983). Vol. A, Space-Group Symmetry, edited by Th. Hahn. Dordrecht: Reidel.
International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, edited by Th. Hahn, 5th ed., corrected reprint. Heidelberg: Springer.
International Tables for Crystallography (1993). Vol. B, Reciprocal Space, edited by U. Shmueli, 1st ed. Dordrecht: Kluwer Academic Publishers.
International Tables for X-ray Crystallography (1952). Vol. I, Symmetry Groups, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press.
International Tables for X-ray Crystallography (1965). Vol. I, Symmetry Groups, edited by N. F. M. Henry & K. Lonsdale, 2nd ed. Birmingham: Kynoch Press.
Altermatt, U. D. & Brown, I. D. (1987). A real-space computer-based symmetry algebra. Acta Cryst. A43, 125–130.
Bourne, P. E., Berman, H. M., McMahon, B., Watenpaugh, K. D., Westbrook, J. D. & Fitzgerald, P. M. D. (1997). mmCIF: macromolecular crystallographic information file. Methods Enzymol. 277, 571–590.
Dowty, E. (2000). ATOMS: a program for the display of atomic structures. Commercial software available from http://www.shapesoftware.com .
Grosse-Kunstleve, R. W. (1995). SGINFO: a comprehensive collection of ANSI C routines for the handling of space-group symmetry. Freely available from http://cci.lbl.gov/sginfo/ .
Grosse-Kunstleve, R. W. (1999). Algorithms for deriving crystallographic space-group information. Acta Cryst. A55, 383–395.
Hall, S. R. (1981a). Space-group notation with an explicit origin. Acta Cryst. A37, 517–525.
Hall, S. R. (1981b). Space-group notation with an explicit origin: erratum. Acta Cryst. A37, 921.
Hall, S. R. (1997). LoopX: a script used to loop Xtal. Copyright University of Western Australia. Freely available from http://www.crystal.uwa.edu.au/~syd/symmetry/ .
Hall, S. R., du Boulay, D. J. & Olthof-Hazekamp, R. (2000). Xtal: a system of crystallographic programs. Copyright University of Western Australia. Freely available from http://xtal.crystal.uwa.edu.au .
Hovmöller, S. (1992). CRISP: crystallographic image processing on a personal computer. Ultramicroscopy, 41, 121–135.
Larine, M., Klimkovich, S., Farrants, G., Hovmöller, S. & Xiaodong, Z. (1995). Space Group Explorer: a computer program for obtaining extensive information about any of the 230 space groups. Freely available from http://www.calidris-em.com/ .
Shmueli, U. (1984). Space-group algorithms. I. The space group and its symmetry elements. Acta Cryst. A40, 559–567.
Zachariasen, W. H. (1945). Theory of X-ray Diffraction in Crystals. New York: John Wiley.








































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