Space group - Knowledge (XXG)

1447:

crystal family of a space group is determined by either its lattice system or its point group. In 3 dimensions the only two lattice families that get merged in this way are the hexagonal and rhombohedral lattice systems, which are combined into the hexagonal crystal family. The 6 crystal families in 3 dimensions are called triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and cubic. Crystal families are commonly used in popular books on crystals, where they are sometimes called crystal systems.

3618: 3606: 3569: 3712: 3700: 3783: 3524: 3512: 3429: 3417: 3746: 3663: 3475: 3383: 3371: 3323: 3286: 4086: 4081: 4075: 4251: 3963: 4862: 4657: 5145: 5139: 4652: 3443: 3339: 31: 4070: 4245: 3958: 563:, to describe the degree of rotation, where the number is how many operations must be applied to complete a full rotation (e.g., 3 would mean a rotation one third of the way around the axis each time). The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. So, 2 3890: 3631: 3537: 3254: 5133: 956:

Space group notation with an explicit origin. Rotation, translation and axis-direction symbols are clearly separated and inversion centers are explicitly defined. The construction and format of the notation make it particularly suited to computer generation of symmetry information. For example, group

844:

12. In this space group the twofold axes are not along the a and b-axes but in a direction rotated by 30°. The international symbols and international short symbols for some of the space groups were changed slightly between 1935 and 2002, so several space groups have 4 different international symbols

740:

Only certain combinations of symmetry elements are possible in a space group. Translations are always present, and the space group P1 has only translations and the identity element. The presence of mirrors implies glide planes as well, and the presence of rotation axes implies screw axes as well, but

1169:

There are (at least) 10 different ways to classify space groups into classes. The relations between some of these are described in the following table. Each classification system is a refinement of the ones below it. To understand an explanation given here it may be necessary to understand the next

1557:

It is essential in Bieberbach's theorems to assume that the group acts as isometries; the theorems do not generalize to discrete cocompact groups of affine transformations of Euclidean space. A counter-example is given by the 3-dimensional Heisenberg group of the integers acting by translations on

1446:

The point group of a space group does not quite determine its lattice system, because occasionally two space groups with the same point group may be in different lattice systems. Crystal families are formed from lattice systems by merging the two lattice systems whenever this happens, so that the

1239:

Two space groups, considered as subgroups of the group of affine transformations of space, have the same affine space group type if they are the same up to an affine transformation, even if that inverts orientation. The affine space group type is determined by the underlying abstract group of the

5442:

of the space group is determined by the lattice system together with the initial letter of its name, which for the non-rhombohedral groups is P, I, F, A or C, standing for the principal, body centered, face centered, A-face centered or C-face centered lattices. There are seven rhombohedral space

1413:

Crystal systems are an ad hoc modification of the lattice systems to make them compatible with the classification according to point groups. They differ from crystal families in that the hexagonal crystal family is split into two subsets, called the trigonal and hexagonal crystal systems. The

724:

crystals, not identical to their mirror image; whereas space groups that do include at least one of those give achiral crystals. Achiral molecules sometimes form chiral crystals, but chiral molecules always form chiral crystals, in one of the space groups that permit this.

1250:

Sometimes called Z-classes. These are determined by the point group together with the action of the point group on the subgroup of translations. In other words, the arithmetic crystal classes correspond to conjugacy classes of finite subgroup of the general linear group

1363:

Sometimes called Q-classes. The crystal class of a space group is determined by its point group: the quotient by the subgroup of translations, acting on the lattice. Two space groups are in the same crystal class if and only if their point groups, which are subgroups of

1274:

Arithmetic crystal classes may be interpreted as different orientations of the point groups in the lattice, with the group elements' matrix components being constrained to have integer coefficients in lattice space. This is rather easy to picture in the two-dimensional,

839:

in this example) denotes the symmetry along the major axis (c-axis in trigonal cases), the second (2 in this case) along axes of secondary importance (a and b) and the third symbol the symmetry in another direction. In the trigonal case there also exists a space group

965:

The space groups with given point group are numbered by 1, 2, 3, ... (in the same order as their international number) and this number is added as a superscript to the Schönflies symbol for the point group. For example, groups numbers 3 to 5 whose point group is

252:. What this means is that the action of any element of a given space group can be expressed as the action of an element of the appropriate point group followed optionally by a translation. A space group is thus some combination of the translational symmetry of a 1201:). In three dimensions, for 11 of the affine space groups, there is no chirality-preserving (i.e. orientation-preserving) map from the group to its mirror image, so if one distinguishes groups from their mirror images these each split into two cases (such as P4 657:

that is a basis of the space group; the lattice must be symmetric under that point group, but the crystal structure itself may not be symmetric under that point group as applied to any particular point (that is, without a translation). For example, the

5426:

The lattice system can be found as follows. If the crystal system is not trigonal then the lattice system is of the same type. If the crystal system is trigonal, then the lattice system is hexagonal unless the space group is one of the seven in the

1525:

there are only a finite number of possibilities for the isomorphism class of the underlying group of a space group, and moreover the action of the group on Euclidean space is unique up to conjugation by affine transformations. This answers part of

762:

The International Union of Crystallography publishes tables of all space group types, and assigns each a unique number from 1 to 230. The numbering is arbitrary, except that groups with the same crystal system or point group are given consecutive

1271:

of its point group with its translation subgroup. There are 73 symmorphic space groups, with exactly one in each arithmetic crystal class. There are also 157 nonsymmorphic space group types with varying numbers in the arithmetic crystal classes.

5455:

Convert all symmetry elements with translational components into their respective symmetry elements without translation symmetry (Glide planes are converted into simple mirror planes; Screw axes are converted into simple axes of

2073:)). Frieze groups are magnetic 1D line groups and layer groups are magnetic wallpaper groups, and the axial 3D point groups are magnetic 2D point groups. Number of original and magnetic groups by (overall, lattice) dimension:( 2057:. The time reversal element flips a magnetic spin while leaving all other structure the same and it can be combined with a number of other symmetry elements. Including time reversal there are 1651 magnetic space groups in 3D ( 1433:). In three dimensions the lattice point group can have one of the 7 different orders 2, 4, 8, 12, 16, 24, or 48. The hexagonal crystal family is split into two subsets, called the rhombohedral and hexagonal lattice systems. 1414:

trigonal crystal system is larger than the rhombohedral lattice system, the hexagonal crystal system is smaller than the hexagonal lattice system, and the remaining crystal systems and lattice systems are the same.

6022:Записки Императорского С.-Петербургского Минералогического Общества (Zapiski Imperatorskova Sankt Petersburgskova Mineralogicheskova Obshchestva, Proceedings of the Imperial St. Petersburg Mineralogical Society) 1108:. However, crystallographers would not use Strukturbericht notation to describe the space group, rather it would be used to describe a specific crystal structure (e.g. space group + atomic arrangement (motif)). 6055:Записки Императорского С.-Петербургского Минералогического Общества (Zapiski Imperatorskogo Sant-Petersburgskogo Mineralogicheskogo Obshchestva, Proceedings of the Imperial St. Petersburg Mineralogical Society) 5414:

plane is a double glide plane, one having glides in two different directions. They are found in seven orthorhombic, five tetragonal and five cubic space groups, all with centered lattice. The use of the symbol

2036:

groups). These symmetries contain an element known as time reversal. They treat time as an additional dimension, and the group elements can include time reversal as reflection in it. They are of importance in

164:

which have been known for several centuries, though the proof that the list was complete was only given in 1891, after the much more difficult classification of space groups had largely been completed.

5431:

consisting of the 7 trigonal space groups in the table above whose name begins with R. (The term rhombohedral system is also sometimes used as an alternative name for the whole trigonal system.) The

1240:

space group. In three dimensions, Fifty-four of the affine space group types preserve chirality and give chiral crystals. The two enantiomorphs of a chiral crystal have the same affine space group.

755:

There are at least ten methods of naming space groups. Some of these methods can assign several different names to the same space group, so altogether there are many thousands of different names.

5435:

is larger than the hexagonal crystal system, and consists of the hexagonal crystal system together with the 18 groups of the trigonal crystal system other than the seven whose names begin with R.

1914:

In 3D, there are 230 crystallographic space group types, which reduces to 219 affine space group types because of some types being different from their mirror image; these are said to differ by

1127:

As the name suggests, the orbifold notation describes the orbifold, given by the quotient of Euclidean space by the space group, rather than generators of the space group. It was introduced by

823:

21, showing that it exhibits primitive centering of the motif (i.e., once per unit cell), with a threefold screw axis and a twofold rotation axis. Note that it does not explicitly contain the

1570:

This table gives the number of space group types in small dimensions, including the numbers of various classes of space group. The numbers of enantiomorphic pairs are given in parentheses.

1197:. Thus e.g. a change of angle between translation vectors does not affect the space group type if it does not add or remove any symmetry. A more formal definition involves conjugacy (see 1558:

the Heisenberg group of the reals, identified with 3-dimensional Euclidean space. This is a discrete cocompact group of affine transformations of space, but does not contain a subgroup

811:). The next three describe the most prominent symmetry operation visible when projected along one of the high symmetry directions of the crystal. These symbols are the same as used in 5915:], Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften (Textbooks and Monographs from the Fields of the Exact Sciences), vol. 13, Verlag Birkhäuser, Basel, 287:

in a unit cell is thus the number of lattice points in the cell times the order of the point group. This ranges from 1 in the case of space group P1 to 192 for a space group like Fm

6129:; Birman, J.L.; Dénoyer, F.; Koptsik, V.A.; Verger-Gaugry, J.L.; Weigel, D.; Yamamoto, A.; Abrahams, S.C.; Kopsky, V. (2002), "Report of a Subcommittee on the Nomenclature of 5477: 2971: 2816: 2739: 2662: 2505: 2428: 2271: 3172: 2893: 2582: 2348: 2191: 1267:) if there is a point such that all symmetries are the product of a symmetry fixing this point and a translation. Equivalently, a space group is symmorphic if it is a 6038:. American Crystallographic Association Monograph No. 7. Translated by David and Katherine Harker. Buffalo, NY: American Crystallographic Association. pp. 50–131. 1135:, and is not used much outside mathematics. Some of the space groups have several different fibrifolds associated to them, so have several different fibrifold symbols. 3093: 3051: 3009: 2935: 2780: 2703: 2626: 2469: 2392: 2235: 3136: 2857: 2546: 2312: 2155: 280:

symmetry operations. The combination of all these symmetry operations results in a total of 230 different space groups describing all possible crystal symmetries.

445: 425: 405: 385: 365: 447:

glide, which is a fourth of the way along either a face or space diagonal of the unit cell. The latter is called the diamond glide plane as it features in the

228:

c) even though he already had the correct list of 230 groups from Fedorov and Schönflies; the common claim that Barlow was unaware of their work is incorrect.

1209:). So instead of the 54 affine space groups that preserve chirality there are 54 + 11 = 65 space group types that preserve chirality (the 184:

noticed that two of them were really the same. The space groups in three dimensions were first enumerated in 1891 by Fedorov (whose list had two omissions (I

1966:. There are 44 enantiomorphic point groups in 4-dimensional space. If we consider enantiomorphic groups as different, the total number of point groups is 5927:

Burckhardt, Johann Jakob (1967), "Zur Geschichte der Entdeckung der 230 Raumgruppen" [On the history of the discovery of the 230 space groups],

2032:

In addition to crystallographic space groups there are also magnetic space groups (also called two-color (black and white) crystallographic groups or

188:

3d and Fdd2) and one duplication (Fmm2)), and shortly afterwards in 1891 were independently enumerated by Schönflies (whose list had four omissions (I

1393:), where the lattice point group is the group of symmetries of the underlying lattice that fix a point of the lattice, and contains the point group. 1550:

dimensions up to conjugation by affine transformations is essentially the same as classifying isomorphism classes for groups that are extensions of

1472:. They divided the 219 affine space groups into reducible and irreducible groups. The reducible groups fall into 17 classes corresponding to the 17 669:

The lattice dimension can be less than the overall dimension, resulting in a "subperiodic" space group. For (overall dimension, lattice dimension):

6288:

Neubüser, J.; Souvignier, B.; Wondratschek, H. (2002), "Corrections to Crystallographic Groups of Four-Dimensional Space by Brown et al. (1978) ",

783:, which is the one most commonly used in crystallography, and usually consists of a set of four symbols. The first describes the centering of the 1279:

case. Some of the point groups have reflections, and the reflection lines can be along the lattice directions, halfway in between them, or both.

6871: 6727: 6184: 6089: 5929: 5886: 779:

The Hermann–Mauguin (or international) notation describes the lattice and some generators for the group. It has a shortened form called the

5482: 451:

structure. In 17 space groups, due to the centering of the cell, the glides occur in two perpendicular directions simultaneously,

98:

of the pattern that leave it unchanged. In three dimensions, space groups are classified into 219 distinct types, or 230 types if

5507: 2597: 1954:

The 4895 4-dimensional groups were enumerated by Harold Brown, Rolf Bülow, and Joachim Neubüser et al. (

1879: 2062: 6673: 1998:

enumerated the ones of dimension 6, later the corrected figures were found. Initially published number of 826 Lattice types in

1527: 479:. For example, group Abm2 could be also called Acm2, group Ccca could be called Cccb. In 1992, it was suggested to use symbol 6645: 2363: 2283: 2206: 6634: 6561: 5829:"Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich" 6660: 3852: 3221: 773: 3842: 2033: 1022: 241: 6684: 1521:

subgroup of finite index, and is also the unique maximal normal abelian subgroup. He also showed that in any dimension

720:

The 65 "Sohncke" space groups, not containing any mirrors, inversion points, improper rotations or glide planes, yield

7233: 6556: 5428: 1030: 559:

is a rotation about an axis, followed by a translation along the direction of the axis. These are noted by a number,

6456: 6246: 1417:

The lattice system of a space group is determined by the conjugacy class of the lattice point group (a subgroup of GL

6720: 6168: 201: 200:

m)). The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies.

5695: 1194: 181: 7228: 7218: 6782: 5432: 4854: 261: 6046: 5597: 6133:-Dimensional Crystallography. II. Symbols for arithmetic crystal classes, Bravais classes and space groups", 6013: 2061:, p.428). It has also been possible to construct magnetic versions for other overall and lattice dimensions ( 303:

The elements of the space group fixing a point of space are the identity element, reflections, rotations and

7182: 6551: 6350: 5551: 1916: 1875: 815:, with the addition of glide planes and screw axis, described above. By way of example, the space group of 328: 99: 7223: 7067: 6476: 3857: 3226: 960: 103: 5541: 7053: 6713: 6468: 5837: 5791: 1190: 1182: 1151: 7005: 3617: 3605: 5696:"On the minimum number of beams needed to distinguish enantiomorphs in X-ray and electron diffraction" 3568: 7029: 7023: 6787: 6513: 6359: 6327: 6261: 6218: 6107: 5910: 5832: 5786: 5707: 5668: 5125: 3817: 2828: 2027: 1477: 750: 95: 6469:"Enantiomorphism of crystallographic groups in higher dimensions with results in dimensions up to 6" 5785:

Bieberbach, Ludwig (1911), "Über die Bewegungsgruppen der Euklidischen Räume" [On the groups of

1088:

Silicates. Some structure designation share the same space groups. For example, space group 225 is A

6762: 6736: 6700: 3437: 3333: 2054: 663: 106: 6695: 6348:

Palistrant, A. F. (2012), "Complete Scheme of Four-Dimensional Crystallographic Symmetry Groups",

6962: 6778: 6772: 6608: 6572: 6539: 6445: 6375: 6198: 5973: 5962: 5862: 5816: 5774: 5499: 2050: 2038: 1461: 1453: 1268: 1128: 1120: 347:

is a reflection in a plane, followed by a translation parallel with that plane. This is noted by

91: 2944: 2789: 2712: 2635: 2478: 2401: 2244: 176:. More accurately, he listed 66 groups, but both the Russian mathematician and crystallographer 3711: 3699: 3145: 2866: 2555: 2321: 2164: 7213: 7124: 6885: 6592: 6493: 6429: 6307: 6277: 6234: 6180: 6152: 6085: 5993: 5946: 5882: 5854: 5808: 4223:

Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma

3862: 3782: 3625: 3523: 3511: 3428: 3416: 3231: 1539: 1498: 1157: 1112: 741:

the converses are not true. An inversion and a mirror implies two-fold screw axes, and so on.

304: 269: 6701:

The Geometry Center: 10.1 Formulas for Symmetries in Cartesian Coordinates (three dimensions)

5757:[On the geometric properties of rigid structures and their application to crystals], 2014:

counted the enantiomorphs, but that paper relied on old erroneous CARAT data for dimension 6.

7192: 7129: 7104: 7096: 7088: 7080: 7072: 7059: 7041: 7035: 6584: 6529: 6521: 6485: 6421: 6395: 6367: 6335: 6297: 6269: 6226: 6172: 6142: 6115: 6077: 5977: 5938: 5846: 5800: 5766: 5715: 5676: 5598:"Zusammenstellung der kirstallographischen Resultate des Herrn Schoenflies und der meinigen" 5491: 4085: 4080: 4074: 3867: 3745: 3662: 3474: 3382: 3370: 3322: 3285: 3236: 3071: 3029: 2987: 2913: 2758: 2681: 2604: 2447: 2370: 2213: 1962:

corrected the number of enantiomorphic groups from 112 to 111, so total number of groups is

1438: 1139: 1132: 331:

of the space group by the Bravais lattice is a finite group which is one of the 32 possible

169: 87: 17: 6604: 6441: 6209:

Litvin, D.B. (May 2008), "Tables of crystallographic properties of magnetic space groups",

6194: 6005: 5958: 5920: 5896: 3114: 2835: 2524: 2290: 2133: 7145: 6979: 6848: 6757: 6696:

The Geometry Center: 2.1 Formulas for Symmetries in Cartesian Coordinates (two dimensions)

6677: 6638: 6600: 6568: 6437: 6190: 6073: 6001: 5954: 5916: 5892: 5874: 5755:"Über die geometrischen Eigenschaften starrer Strukturen und ihre Anwendung auf Kristalle" 5439: 4250: 3962: 3248: 3188: 2517: 1897: 1473: 1399: 1352: 1276: 784: 690: 622: 324: 320: 308: 284: 257: 245: 161: 127: 114: 1144:

Spatial and point symmetry groups, represented as modifications of the pure reflectional

6517: 6363: 6331: 6265: 6222: 6111: 5711: 5672: 7150: 6992: 6945: 6938: 6931: 6924: 6896: 6863: 6833: 6828: 6818: 6767: 4861: 4656: 3832: 3531: 3197: 2046: 2042: 1405: 1198: 1014: 824: 729: 430: 410: 390: 370: 350: 292: 249: 208:) later enumerated the groups with a different method, but omitted four groups (Fdd2, I 177: 136: 83: 6400: 6318:

Opgenorth, J; Plesken, W; Schulz, T (1998), "Crystallographic Algorithms and Tables",

5981: 5653: 5600:[Compilation of the crystallographic results of Mr. Schoenflies and of mine]. 7207: 7160: 7119: 7047: 6968: 6953: 6901: 6853: 6808: 6612: 6202: 5966: 5866: 5820: 5778: 1518: 1210: 1145: 659: 448: 6670: 6543: 6449: 6379: 7187: 7114: 6914: 6876: 6823: 6813: 6803: 6744: 6425: 5144: 5138: 4062: 2440: 1543: 1214: 684: 117:

in any number of dimensions. In dimensions other than 3, they are sometimes called

6504:

Souvignier, Bernd (2006), "The four-dimensional magnetic point and space groups",

5835:

in Euclidean spaces (Second essay.) Groups with a finite fundamental domain],

4651: 30: 6655: 6081: 1505:-dimensional Euclidean space with a compact fundamental domain. Bieberbach ( 6843: 6631: 6409: 6126: 3872: 3442: 3338: 3241: 3206: 2751: 812: 706: 654: 344: 332: 277: 145:

of the crystal. A definitive source regarding 3-dimensional space groups is the

67: 6525: 957:

number 3 has three Hall symbols: P 2y (P 1 2 1), P 2 (P 1 1 2), P 2x (P 2 1 1).

172:

listed the 65 space groups (called Sohncke groups) whose elements preserve the

7155: 6489: 6459:(1923), "Theorie der Kristallstruktur" [Theory of Crystal Structure], 6386:

Plesken, Wilhelm; Hanrath, W (1984), "The lattices of six-dimensional space",

6371: 6340: 6302: 6273: 6230: 6147: 6119: 5719: 5620: 4237: 3950: 696: 680: 674: 567:

is a twofold rotation followed by a translation of 1/2 of the lattice vector.

556: 273: 62:

indicates the glide planes (b) and (c). The black boxes outline the unit cell.

6665: 6596: 6433: 6176: 5997: 5950: 5858: 5812: 5770: 7015: 3882: 2674: 1465: 700: 253: 173: 75: 6497: 6311: 6281: 6238: 6156: 4069: 835:

21, it is trigonal). In the international short symbol the first symbol (3

240:

The space groups in three dimensions are made from combinations of the 32

6752: 5733: 4644: 1469: 265: 142: 110: 6069:

International Tables for Crystallography, Volume A: Space Group Symmetry

5459:

Axes of rotation, rotoinversion axes and mirror planes remain unchanged.

427:

glide, which is a glide along the half of a diagonal of a face, and the

6588: 5942: 5850: 5804: 5503: 4244: 3957: 483:

for such planes. The symbols for five space groups have been modified:

71: 54:

indicates the mirror plane perpendicular to the c-axis (a), the second

6534: 5680: 3889: 1546:

group. Combining these results shows that classifying space groups in

1513:) proved that the subgroup of translations of any such group contains 847:

The viewing directions of the 7 crystal systems are shown as follows.

232:

describes the history of the discovery of the space groups in detail.

6671:

Interactive 3D visualization of all 230 crystallographic space groups

1615:

Geometric crystal classes, Q-classes, crystallographic point groups,

1457: 1226: 816: 721: 575:

The general formula for the action of an element of a space group is

35: 6165:

Group theoretical methods and applications to molecules and crystals

5828: 5754: 5495: 1036:

A related notation for crystal structures given a letter and index:

6098:

Hall, S.R. (1981), "Space-Group Notation with an Explicit Origin",

5873:

Brown, Harold; Bülow, Rolf; Neubüser, Joachim; Wondratschek, Hans;

1385:

These correspond to conjugacy classes of lattice point groups in GL

4594:

P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P4

1186: 6575:[On an algorithm for the determination of space groups], 5634: 1643: 1634: 1625: 1616: 1607: 1598: 1589: 1580: 407:, depending on which axis the glide is along. There is also the 5132: 3807:

Both: reflection lines both along and between lattice directions

3630: 3536: 3253: 1222: 58:

indicates the mirror planes parallel to the c-axis (b), and the

6709: 3804:

Between: reflection lines halfway in between lattice directions

1176:(Crystallographic) space group types (230 in three dimensions) 323:

3, called the Bravais lattice (so named after French physicist

3794:

For each geometric class, the possible arithmetic classes are

1959: 1955: 1476:, and the remaining 35 irreducible groups are the same as the 1185:

of space, have the same space group type if they are the same

827:, although this is unique to each space group (in the case of 6686:

The Fibrifold Notation and Classification for 3D Space Groups

6650: 1456:, Delgado Friedrichs, and Huson et al. ( 1383:

These are determined by the underlying Bravais lattice type.

3191:

using the classification of the 2-dimensional space groups:

1460:) gave another classification of the space groups, called a 6643: 2007: 1646: 1637: 1628: 1619: 1610: 1601: 1592: 1583: 1217:

belong to the same crystallographic space group, such as P2

1534:

showed that conversely any group that is the extension of

1181:

Two space groups, considered as subgroups of the group of

102:

copies are considered distinct. Space groups are discrete

6705: 6067: 5654:"The Crystallographic Space Groups in Geometric Algebra" 3183:

Table of space groups in 2 dimensions (wallpaper groups)

6573:"Über einen Algorithmus zur Bestimmung der Raumgruppen" 327:). There are 14 possible types of Bravais lattice. The 6626: 2003: 605:

is its vector, and where the element transforms point

6247:"Tables of properties of magnetic subperiodic groups" 3148: 3117: 3074: 3032: 2990: 2947: 2916: 2869: 2838: 2792: 2761: 2715: 2684: 2638: 2607: 2558: 2527: 2481: 2450: 2404: 2373: 2324: 2293: 2247: 2216: 2167: 2136: 433: 413: 393: 373: 353: 6410:"Counting crystallographic groups in low dimensions" 5447:

Derivation of the crystal class from the space group

1933:

refers to 3D. They were enumerated independently by

1245:

Arithmetic crystal classes (73 in three dimensions)

7138: 7014: 6862: 6796: 6743: 6020:, The symmetry of regular systems of figures], 1234:Affine space group types (219 in three dimensions) 728:Among the 65 Sohncke groups are 22 that come in 11 319:The translations form a normal abelian subgroup of 5585:] (in German). Leipzig, Germany: B.G. Teubner. 5543:Die Entwicklung einer Theorie der Krystallstruktur 3166: 3130: 3087: 3045: 3003: 2965: 2929: 2887: 2851: 2810: 2774: 2733: 2697: 2656: 2620: 2576: 2540: 2499: 2463: 2422: 2386: 2342: 2306: 2265: 2229: 2185: 2149: 1874:One is the group of integers and the other is the 1229:, they belong to two enantiomorphic space groups. 439: 419: 399: 379: 359: 5879:Crystallographic groups of four-dimensional space 1517:linearly independent translations, and is a free 711:(3,3): The space groups discussed in this article 699:; with the 3D crystallographic point groups, the 6661:Crystal Lattice Structures: Index by Space Group 6072:, vol. A (5th ed.), Berlin, New York: 5602:Zeitschrift für Krystallographie und Mineralogie 5548:The Development of a Theory of Crystal Structure 3801:Along: reflection lines along lattice directions 1501:group, is a discrete subgroup of isometries of 6666:Full list of 230 crystallographic space groups 5976:; Delgado Friedrichs, Olaf; Huson, Daniel H.; 1999: 1960:Neubüser, Souvignier & Wondratschek (2002) 1259:) over the integers. A space group is called 6721: 1995: 1979: 8: 5652:David Hestenes; Jeremy Holt (January 2007). 662:structure does not have any point where the 86:of a repeating pattern in space, usually in 248:, each of the latter belonging to one of 7 6728: 6714: 6706: 5619:Sydney R. Hall; Ralf W. Grosse-Kunstleve. 5478:"Crystallography and cohomology of groups" 3847:Space groups (international short symbol) 3826: 3109:Four-dimensional discrete symmetry groups 2078: 2074: 2041:that contain ordered unpaired spins, i.e. 2011: 1983: 1942: 1531: 1510: 1506: 260:), the point group symmetry operations of 229: 192:3d, Pc, Cc, ?) and one duplication (P 27:Symmetry group of a configuration in space 6533: 6408:Plesken, Wilhelm; Schulz, Tilman (2000), 6399: 6339: 6301: 6146: 5905:Die Bewegungsgruppen der Kristallographie 3158: 3153: 3147: 3122: 3116: 3079: 3073: 3037: 3031: 2995: 2989: 2957: 2952: 2946: 2921: 2915: 2879: 2874: 2868: 2843: 2837: 2802: 2797: 2791: 2766: 2760: 2725: 2720: 2714: 2689: 2683: 2648: 2643: 2637: 2612: 2606: 2568: 2563: 2557: 2532: 2526: 2491: 2486: 2480: 2455: 2449: 2414: 2409: 2403: 2378: 2372: 2334: 2329: 2323: 2298: 2292: 2257: 2252: 2246: 2221: 2215: 2177: 2172: 2166: 2141: 2135: 432: 412: 392: 372: 352: 5564: 5527: 3821: 3193: 2284:One-dimensional discrete symmetry groups 2083: 1938: 1606:Abstract crystallographic point groups, 1572: 1358:Bravais flocks (14 in three dimensions) 1172: 849: 485: 160:Space groups in 2 dimensions are the 17 147:International Tables for Crystallography 119: 29: 5468: 1858: 1624:Arithmetic crystal classes, Z-classes, 1372:), are conjugate in the larger group GL 6872:Classification of finite simple groups 6627:International Union of Crystallography 6014:"Симметрія правильныхъ системъ фигуръ" 2070: 2066: 2004:Opgenorth, Plesken & Schulz (1998) 1934: 1497:dimensions, an affine space group, or 205: 5930:Archive for History of Exact Sciences 5635:"Strukturbericht - Wikimedia Commons" 5583:Crystal Systems and Crystal Structure 5550:] (in German). Leipzig, Germany: 3841: 3831: 3812:Table of space groups in 3 dimensions 3215: 2099: 2096: 1554:by a finite group acting faithfully. 272:(also called rotoinversion), and the 141:, and represent a description of the 90:. The elements of a space group (its 7: 6066:Hahn, Th. (2002), Hahn, Theo (ed.), 5579:Krystallsysteme und Krystallstruktur 5420: 3824: 1982:enumerated the ones of dimension 5. 1642:Crystallographic space group types, 1213:).For most chiral crystals, the two 149: 6018:Simmetriya pravil'nykh sistem figur 5982:"On three-dimensional space groups" 5694:J.C.H. Spence and J.M. Zuo (1994). 2058: 455:the same glide plane can be called 130:, space groups are also called the 5986:Beiträge zur Algebra und Geometrie 4225:Cmcm, Cmce, Cmmm, Cccm, Cmme, Ccce 1566:Classification in small dimensions 25: 6632:Point Groups and Bravais Lattices 6577:Commentarii Mathematici Helvetici 6401:10.1090/s0025-5718-1984-0758205-5 5903:Burckhardt, Johann Jakob (1947), 5881:, New York: Wiley-Interscience , 5483:The American Mathematical Monthly 2022:Magnetic groups and time reversal 649:being the identity. The matrices 168:In 1879 the German mathematician 6506:Zeitschrift für Kristallographie 5759:Zeitschrift für Kristallographie 5700:Acta Crystallographica Section A 5143: 5137: 5131: 4860: 4655: 4650: 4249: 4243: 4084: 4079: 4073: 4068: 3961: 3956: 3888: 3781: 3744: 3710: 3698: 3661: 3629: 3616: 3604: 3567: 3535: 3522: 3510: 3473: 3441: 3427: 3415: 3381: 3369: 3337: 3321: 3284: 3252: 3216:Wallpaper groups (cell diagram) 2128:Zero-dimensional symmetry group 1880:symmetry groups in one dimension 1468:structures on the corresponding 283:The number of replicates of the 5661:Journal of Mathematical Physics 5443:groups, with initial letter R. 4795: 4639: 1480:and are classified separately. 6646:Bilbao Crystallographic Server 6426:10.1080/10586458.2000.10504417 6053:, Symmetry in the plane]. 5577:Schönflies, Arthur M. (1891). 4194:, Ccc2, Amm2, Aem2, Ama2, Aea2 2908:Four-dimensional point groups 2829:Three-dimensional space groups 2598:Three-dimensional point groups 1: 5621:"Concise Space-Group Symbols" 3825: 768:International symbol notation 242:crystallographic point groups 180:and the German mathematician 6082:10.1107/97809553602060000100 3754: 3717: 3671: 3623: 3577: 3529: 3483: 3434: 3388: 3331: 3294: 3246: 3219: 2364:Two-dimensional point groups 2207:One-dimensional point groups 2000:Plesken & Hanrath (1984) 1528:Hilbert's eighteenth problem 18:Crystallographic space group 7139:Infinite dimensional groups 6557:Encyclopedia of Mathematics 6461:Gebrüder Bornträger, Berlin 6057:. 2nd series (in Russian). 6024:, 2nd series (in Russian), 5827:Bieberbach, Ludwig (1912), 5429:rhombohedral lattice system 1996:Plesken & Schulz (2000) 1980:Plesken & Schulz (2000) 1031:Strukturbericht designation 7250: 6526:10.1524/zkri.2006.221.1.77 6467:Souvignier, Bernd (2003), 6169:Cambridge University Press 5789:in Euclidean spaces], 5540:Sohncke, Leonhard (1879). 5452:Leave out the Bravais type 3815: 2966:{\displaystyle G_{40}^{1}} 2811:{\displaystyle G_{32}^{1}} 2734:{\displaystyle G_{31}^{1}} 2657:{\displaystyle G_{30}^{1}} 2500:{\displaystyle G_{21}^{1}} 2423:{\displaystyle G_{20}^{1}} 2266:{\displaystyle G_{10}^{1}} 2025: 1986:counted the enantiomorphs. 1633:Affine space group types, 1225:, but for others, such as 781:international short symbol 748: 641:) is a unique function of 7178: 6683:Huson, Daniel H. (1999), 6490:10.1107/S0108767303004161 6457:Schönflies, Arthur Moritz 6372:10.1134/S1063774512040104 6341:10.1107/S010876739701547X 6303:10.1107/S0108767302001368 6274:10.1107/S010876730500406X 6245:Litvin, D.B. (May 2005), 6231:10.1107/S010876730800768X 6148:10.1107/S010876730201379X 6120:10.1107/s0567739481001228 5720:10.1107/S0108767394002850 5124: 4853: 4643: 4236: 4061: 3949: 3881: 3846: 3798:None: no reflection lines 3787: 3750: 3667: 3624: 3573: 3530: 3479: 3435: 3332: 3327: 3290: 3247: 3210: 3204: 3196: 3167:{\displaystyle G_{4}^{1}} 2901: 2888:{\displaystyle G_{3}^{1}} 2590: 2577:{\displaystyle G_{2}^{1}} 2356: 2343:{\displaystyle G_{1}^{1}} 2199: 2186:{\displaystyle G_{0}^{1}} 2091: 2086: 2053:structures as studied by 1445: 1437: 1425:)) in the larger group GL 1355:(32 in three dimensions) 1249: 1244: 1238: 1233: 1180: 1175: 1076:More complex compounds), 695:(3,1): Three-dimensional 182:Arthur Moritz Schoenflies 7042:Special orthogonal group 6552:"Crystallographic group" 6477:Acta Crystallographica A 6414:Experimental Mathematics 6320:Acta Crystallographica A 6290:Acta Crystallographica A 6254:Acta Crystallographica A 6211:Acta Crystallographica A 6177:10.1017/CBO9780511534867 6135:Acta Crystallographica A 6100:Acta Crystallographica A 6047:"Симметрія на плоскости" 6045:Fedorov, E. S. (1891b). 6012:Fedorov, E. S. (1891a), 5771:10.1524/zkri.1894.23.1.1 5596:von Fedorow, E. (1892). 5433:hexagonal lattice system 2002:was corrected to 841 in 1488: 1441:(6 in three dimensions) 1408:(7 in three dimensions) 1402:(7 in three dimensions) 1193:of space that preserves 973:have Schönflies symbols 774:Hermann–Mauguin notation 6656:Space Group Info (new) 6651:Space Group Info (old) 6351:Crystallography Reports 6051:Simmetrija na ploskosti 6034:Fedorov, E. S. (1971). 5476:Hiller, Howard (1986). 4229:Immm, Ibam, Ibca, Imma 1876:infinite dihedral group 853:Position in the symbol 679:(2,1): Two-dimensional 673:(1,1): One-dimensional 299:Elements fixing a point 7068:Exceptional Lie groups 6163:Kim, Shoon K. (1999), 5831:[On the groups of 4790:P3m1, P31m, P3c1, P31c 4136:, C222, F222, I222, I2 3168: 3132: 3089: 3088:{\displaystyle G_{43}} 3047: 3046:{\displaystyle G_{42}} 3005: 3004:{\displaystyle G_{41}} 2967: 2931: 2930:{\displaystyle G_{40}} 2889: 2853: 2812: 2776: 2775:{\displaystyle G_{32}} 2735: 2699: 2698:{\displaystyle G_{31}} 2658: 2622: 2621:{\displaystyle G_{30}} 2578: 2542: 2501: 2465: 2464:{\displaystyle G_{21}} 2424: 2388: 2387:{\displaystyle G_{20}} 2344: 2308: 2267: 2231: 2230:{\displaystyle G_{10}} 2187: 2151: 2063:Daniel Litvin's papers 1183:affine transformations 1165:Classification systems 1100:. Space group 221 is A 1040:Elements (monatomic), 441: 421: 401: 381: 361: 63: 7054:Special unitary group 6550:Vinberg, E. (2001) , 6032:English translation: 5911:Rigid Transformations 5838:Mathematische Annalen 5833:rigid transformations 5792:Mathematische Annalen 5787:rigid transformations 5639:commons.wikimedia.org 5419:became official with 3816:Further information: 3169: 3133: 3131:{\displaystyle G_{4}} 3090: 3048: 3006: 2968: 2932: 2890: 2854: 2852:{\displaystyle G_{3}} 2813: 2777: 2736: 2700: 2659: 2623: 2579: 2543: 2541:{\displaystyle G_{2}} 2502: 2466: 2425: 2389: 2345: 2309: 2307:{\displaystyle G_{1}} 2268: 2232: 2188: 2152: 2150:{\displaystyle G_{0}} 2008:Janssen et al. (2002) 1647:sequence A006227 1638:sequence A004029 1629:sequence A004027 1620:sequence A004028 1611:sequence A006226 1602:sequence A256413 1593:sequence A004031 1584:sequence A004032 1489:Bieberbach's theorems 1191:affine transformation 749:Further information: 442: 422: 402: 382: 362: 96:rigid transformations 33: 7151:Diffeomorphism group 7030:Special linear group 7024:General linear group 6036:Symmetry of Crystals 5978:Thurston, William P. 5734:"The CARAT Homepage" 3818:List of space groups 3146: 3115: 3072: 3030: 2988: 2945: 2914: 2867: 2836: 2790: 2759: 2713: 2682: 2636: 2605: 2556: 2525: 2479: 2448: 2402: 2371: 2322: 2291: 2245: 2214: 2165: 2134: 2028:Magnetic space group 751:List of space groups 431: 411: 391: 371: 351: 6976:Other finite groups 6763:Commutator subgroup 6518:2006ZK....221...77S 6364:2012CryRp..57..471P 6332:1998AcCrA..54..517O 6266:2005AcCrA..61..382L 6223:2008AcCrA..64..419L 6112:1981AcCrA..37..517H 5974:Conway, John Horton 5712:1994AcCrA..50..647S 5673:2007JMP....48b3514H 3163: 2962: 2884: 2807: 2730: 2653: 2573: 2496: 2419: 2339: 2262: 2182: 2055:neutron diffraction 2039:magnetic structures 1484:In other dimensions 1464:, according to the 1084:Organic compounds, 961:Schönflies notation 92:symmetry operations 34:The space group of 7234:Molecular geometry 7006:Rubik's Cube group 6963:Baby monster group 6773:Group homomorphism 6676:2021-04-18 at the 6637:2012-07-16 at the 6589:10.1007/BF02568029 5943:10.1007/BF00412962 5913:in Crystallography 5851:10.1007/BF01456724 5805:10.1007/BF01564500 5753:Barlow, W (1894), 5109:P6/mmm, P6/mcc, P6 4628:I4/mmm, I4/mcm, I4 4465:nm, P4cc, P4nc, P4 4176:, Pcc2, Pma2, Pca2 3164: 3149: 3128: 3085: 3043: 3001: 2963: 2948: 2927: 2885: 2870: 2849: 2808: 2793: 2772: 2731: 2716: 2695: 2654: 2639: 2618: 2574: 2559: 2538: 2497: 2482: 2461: 2420: 2405: 2384: 2340: 2325: 2304: 2263: 2248: 2227: 2183: 2168: 2147: 1597:Bravais lattices, 1579:Crystal families, 1538:by a finite group 1462:fibrifold notation 1314:: pmm, pmg, pgg; D 1269:semidirect product 1152:Geometric notation 1121:Fibrifold notation 1044:for AB compounds, 437: 417: 397: 377: 357: 305:improper rotations 202:William Barlow 64: 7201: 7200: 6886:Alternating group 6326:(Pt 5): 517–531, 6186:978-0-521-64062-6 6141:(Pt 6): 605–621, 6091:978-0-7923-6590-7 5888:978-0-471-03095-9 5681:10.1063/1.2426416 5408: 5407: 4198:Imm2, Iba2, Ima2 3792: 3791: 3205:Geometric class, 3180: 3179: 2051:antiferromagnetic 2012:Souvignier (2003) 1984:Souvignier (2003) 1964:4783 + 111 = 4894 1943:Schönflies (1891) 1855: 1854: 1588:Crystal systems, 1540:acting faithfully 1532:Zassenhaus (1948) 1451: 1450: 1158:geometric algebra 1113:Orbifold notation 949: 948: 664:cubic point group 645:that is zero for 548: 547: 440:{\displaystyle d} 420:{\displaystyle n} 400:{\displaystyle c} 380:{\displaystyle b} 360:{\displaystyle a} 270:improper rotation 258:lattice centering 230:Burckhardt (1967) 120:Bieberbach groups 16:(Redirected from 7241: 7193:Abstract algebra 7130:Quaternion group 7060:Symplectic group 7036:Orthogonal group 6730: 6723: 6716: 6707: 6692: 6691: 6615: 6569:Zassenhaus, Hans 6564: 6546: 6537: 6500: 6473: 6463: 6452: 6404: 6403: 6394:(168): 573–587, 6382: 6344: 6343: 6314: 6305: 6284: 6251: 6241: 6217:(Pt 3): 419–24, 6205: 6159: 6150: 6122: 6094: 6062: 6039: 6029: 6008: 5969: 5923: 5899: 5875:Zassenhaus, Hans 5869: 5823: 5781: 5745: 5744: 5742: 5740: 5730: 5724: 5723: 5691: 5685: 5684: 5658: 5649: 5643: 5642: 5631: 5625: 5624: 5616: 5610: 5609: 5593: 5587: 5586: 5574: 5568: 5562: 5556: 5555: 5537: 5531: 5525: 5519: 5518: 5516: 5515: 5506:. Archived from 5473: 5403: 5399: 5393: 5389: 5385: 5381: 5375: 5371: 5367: 5363: 5342: 5330: 5326: 5322: 5316: 5312: 5308: 5287: 5229: 5225: 5221: 5217: 5213: 5209: 5205: 5185: 5147: 5141: 5135: 5083: 5079: 5075: 5071: 5050: 4933: 4913: 4864: 4845: 4841: 4835: 4831: 4827: 4823: 4802: 4722: 4718: 4698: 4659: 4654: 4568: 4564: 4560: 4556: 4550: 4546: 4542: 4538: 4530: 4522: 4518: 4514: 4493: 4322: 4318: 4298: 4253: 4247: 4088: 4083: 4077: 4072: 3965: 3960: 3942: 3922: 3892: 3839:Bravais lattice 3828: 3822: 3785: 3748: 3714: 3702: 3665: 3633: 3620: 3608: 3571: 3539: 3526: 3514: 3477: 3445: 3431: 3419: 3385: 3373: 3341: 3325: 3288: 3256: 3202:Bravais lattice 3194: 3189:wallpaper groups 3173: 3171: 3170: 3165: 3162: 3157: 3137: 3135: 3134: 3129: 3127: 3126: 3094: 3092: 3091: 3086: 3084: 3083: 3052: 3050: 3049: 3044: 3042: 3041: 3010: 3008: 3007: 3002: 3000: 2999: 2972: 2970: 2969: 2964: 2961: 2956: 2936: 2934: 2933: 2928: 2926: 2925: 2894: 2892: 2891: 2886: 2883: 2878: 2858: 2856: 2855: 2850: 2848: 2847: 2817: 2815: 2814: 2809: 2806: 2801: 2781: 2779: 2778: 2773: 2771: 2770: 2740: 2738: 2737: 2732: 2729: 2724: 2704: 2702: 2701: 2696: 2694: 2693: 2663: 2661: 2660: 2655: 2652: 2647: 2627: 2625: 2624: 2619: 2617: 2616: 2583: 2581: 2580: 2575: 2572: 2567: 2547: 2545: 2544: 2539: 2537: 2536: 2518:Wallpaper groups 2506: 2504: 2503: 2498: 2495: 2490: 2470: 2468: 2467: 2462: 2460: 2459: 2429: 2427: 2426: 2421: 2418: 2413: 2393: 2391: 2390: 2385: 2383: 2382: 2349: 2347: 2346: 2341: 2338: 2333: 2313: 2311: 2310: 2305: 2303: 2302: 2272: 2270: 2269: 2264: 2261: 2256: 2236: 2234: 2233: 2228: 2226: 2225: 2192: 2190: 2189: 2184: 2181: 2176: 2156: 2154: 2153: 2148: 2146: 2145: 2100:Magnetic groups 2097:Ordinary groups 2084: 2015: 1993: 1987: 1977: 1971: 1969: 1965: 1952: 1946: 1912: 1906: 1898:wallpaper groups 1895:are also called 1889: 1883: 1872: 1866: 1863: 1645: 1636: 1627: 1618: 1609: 1600: 1591: 1582: 1573: 1474:wallpaper groups 1439:Crystal families 1173: 1140:Coxeter notation 1008: 1007: 996: 995: 984: 983: 850: 691:Wallpaper groups 486: 446: 444: 443: 438: 426: 424: 423: 418: 406: 404: 403: 398: 386: 384: 383: 378: 366: 364: 363: 358: 309:inversion points 290: 246:Bravais lattices 223: 215: 211: 195: 191: 187: 170:Leonhard Sohncke 162:wallpaper groups 132:crystallographic 88:three dimensions 21: 7249: 7248: 7244: 7243: 7242: 7240: 7239: 7238: 7229:Discrete groups 7219:Crystallography 7204: 7203: 7202: 7197: 7174: 7146:Conformal group 7134: 7108: 7100: 7092: 7084: 7076: 7010: 7002: 6989: 6980:Symmetric group 6959: 6949: 6942: 6935: 6928: 6920: 6911: 6907: 6897:Sporadic groups 6891: 6882: 6864:Discrete groups 6858: 6849:Wallpaper group 6829:Solvable groups 6797:Types of groups 6792: 6758:Normal subgroup 6739: 6734: 6689: 6682: 6678:Wayback Machine 6639:Wayback Machine 6623: 6618: 6567: 6549: 6503: 6471: 6466: 6455: 6407: 6385: 6347: 6317: 6287: 6260:(Pt 3): 382–5, 6249: 6244: 6208: 6187: 6162: 6125: 6097: 6092: 6074:Springer-Verlag 6065: 6044: 6033: 6011: 5972: 5926: 5902: 5889: 5872: 5826: 5784: 5752: 5748: 5738: 5736: 5732: 5731: 5727: 5693: 5692: 5688: 5656: 5651: 5650: 5646: 5633: 5632: 5628: 5618: 5617: 5613: 5595: 5594: 5590: 5576: 5575: 5571: 5565:Fedorov (1891a) 5563: 5559: 5539: 5538: 5534: 5528:Fedorov (1891b) 5526: 5522: 5513: 5511: 5496:10.2307/2322930 5490:(10): 765–779. 5475: 5474: 5470: 5466: 5449: 5440:Bravais lattice 5401: 5397: 5395: 5391: 5387: 5383: 5379: 5377: 5373: 5369: 5365: 5361: 5349: 5340: 5328: 5324: 5320: 5318: 5314: 5310: 5306: 5294: 5285: 5276: 5272: 5268: 5264: 5262: 5260: 5256: 5254: 5227: 5223: 5219: 5215: 5211: 5207: 5203: 5191: 5183: 5173: 5169: 5165: 5142: 5136: 5130: 5128: 5116: 5112: 5098: 5081: 5077: 5073: 5069: 5057: 5048: 5039: 5035: 5021: 5006: 5002: 4998: 4994: 4990: 4976: 4961: 4947: 4931: 4919: 4911: 4903: 4899: 4895: 4891: 4887: 4873: 4859: 4857: 4843: 4839: 4837: 4833: 4829: 4825: 4821: 4809: 4800: 4791: 4779: 4764: 4762: 4758: 4754: 4750: 4736: 4720: 4716: 4704: 4696: 4687: 4686: 4682: 4668: 4649: 4647: 4635: 4631: 4627: 4625: 4621: 4617: 4613: 4609: 4605: 4601: 4597: 4583: 4566: 4562: 4558: 4554: 4552: 4548: 4544: 4540: 4536: 4534: 4528: 4526: 4520: 4516: 4512: 4500: 4491: 4482: 4478: 4474: 4472: 4468: 4464: 4460: 4446: 4431: 4427: 4425: 4421: 4417: 4413: 4409: 4405: 4401: 4397: 4393: 4389: 4375: 4360: 4356: 4354: 4350: 4336: 4320: 4316: 4304: 4296: 4288: 4284: 4280: 4276: 4262: 4248: 4242: 4240: 4228: 4226: 4224: 4212: 4197: 4195: 4193: 4189: 4187: 4183: 4179: 4175: 4161: 4147: 4143: 4139: 4135: 4131: 4127: 4123: 4119: 4115: 4111: 4097: 4078: 4067: 4065: 4053: 4051: 4048:C2/m, P2/c, P2 4047: 4045: 4031: 4016: 4004: 3989: 3988: 3974: 3955: 3953: 3940: 3928: 3920: 3901: 3887: 3885: 3838: 3836: 3820: 3814: 3777: 3762: 3740: 3725: 3706: 3694: 3679: 3657: 3642: 3628: 3612: 3600: 3585: 3563: 3548: 3534: 3518: 3506: 3491: 3469: 3454: 3440: 3423: 3411: 3396: 3377: 3365: 3350: 3336: 3317: 3302: 3280: 3265: 3251: 3212: 3201: 3185: 3144: 3143: 3118: 3113: 3112: 3075: 3070: 3069: 3033: 3028: 3027: 2991: 2986: 2985: 2943: 2942: 2917: 2912: 2911: 2865: 2864: 2839: 2834: 2833: 2788: 2787: 2762: 2757: 2756: 2711: 2710: 2685: 2680: 2679: 2634: 2633: 2608: 2603: 2602: 2554: 2553: 2528: 2523: 2522: 2477: 2476: 2451: 2446: 2445: 2400: 2399: 2374: 2369: 2368: 2320: 2319: 2294: 2289: 2288: 2243: 2242: 2217: 2212: 2211: 2163: 2162: 2137: 2132: 2131: 2093: 2088: 2079:Souvignier 2006 2075:Palistrant 2012 2030: 2024: 2019: 2018: 1994: 1990: 1978: 1974: 1967: 1963: 1953: 1949: 1939:Fedorov (1891a) 1928: 1924: 1917:enantiomorphous 1913: 1909: 1893:2D space groups 1890: 1886: 1873: 1869: 1864: 1860: 1568: 1491: 1486: 1428: 1420: 1406:Lattice systems 1400:Crystal systems 1388: 1375: 1367: 1353:Crystal classes 1343: 1339: 1332: 1328: 1324: 1317: 1313: 1309: 1302: 1298: 1294: 1290: 1286: 1277:wallpaper group 1254: 1220: 1208: 1204: 1167: 1124: 1116: 1107: 1103: 1099: 1095: 1091: 1063: 1059: 1051: 1033: 1026: 1018: 1006: 1003: 1002: 1001: 994: 991: 990: 989: 982: 979: 978: 977: 972: 843: 838: 834: 822: 785:Bravais lattice 776: 769: 753: 747: 738: 718: 601:is its matrix, 573: 571:General formula 566: 553: 489:Space group no. 429: 428: 409: 408: 389: 388: 369: 368: 349: 348: 341: 325:Auguste Bravais 317: 301: 288: 285:asymmetric unit 250:lattice systems 238: 227: 221: 219: 213: 209: 199: 193: 189: 185: 158: 128:crystallography 115:Euclidean space 113:of an oriented 45: 39: 28: 23: 22: 15: 12: 11: 5: 7247: 7245: 7237: 7236: 7231: 7226: 7221: 7216: 7206: 7205: 7199: 7198: 7196: 7195: 7190: 7185: 7179: 7176: 7175: 7173: 7172: 7169: 7166: 7163: 7158: 7153: 7148: 7142: 7140: 7136: 7135: 7133: 7132: 7127: 7125:Poincaré group 7122: 7117: 7111: 7110: 7106: 7102: 7098: 7094: 7090: 7086: 7082: 7078: 7074: 7070: 7064: 7063: 7057: 7051: 7045: 7039: 7033: 7027: 7020: 7018: 7012: 7011: 7009: 7008: 7003: 6998: 6993:Dihedral group 6990: 6985: 6977: 6973: 6972: 6966: 6960: 6957: 6951: 6947: 6940: 6933: 6926: 6921: 6918: 6912: 6909: 6905: 6899: 6893: 6892: 6889: 6883: 6880: 6874: 6868: 6866: 6860: 6859: 6857: 6856: 6851: 6846: 6841: 6836: 6834:Symmetry group 6831: 6826: 6821: 6819:Infinite group 6816: 6811: 6809:Abelian groups 6806: 6800: 6798: 6794: 6793: 6791: 6790: 6785: 6783:direct product 6775: 6770: 6768:Quotient group 6765: 6760: 6755: 6749: 6747: 6741: 6740: 6735: 6733: 6732: 6725: 6718: 6710: 6704: 6703: 6698: 6693: 6680: 6668: 6663: 6658: 6653: 6648: 6641: 6629: 6622: 6621:External links 6619: 6617: 6616: 6565: 6547: 6501: 6484:(3): 210–220, 6464: 6453: 6420:(3): 407–411, 6405: 6383: 6358:(4): 471–477, 6345: 6315: 6285: 6242: 6206: 6185: 6160: 6123: 6106:(4): 517–525, 6095: 6090: 6063: 6042: 6041: 6040: 6009: 5992:(2): 475–507, 5970: 5937:(3): 235–246, 5924: 5900: 5887: 5870: 5845:(3): 400–412, 5824: 5799:(3): 297–336, 5782: 5749: 5747: 5746: 5725: 5706:(5): 647–650. 5686: 5644: 5626: 5611: 5588: 5569: 5557: 5532: 5520: 5467: 5465: 5462: 5461: 5460: 5457: 5453: 5448: 5445: 5406: 5405: 5358: 5355: 5353: 5350: 5347: 5344: 5337: 5333: 5332: 5303: 5300: 5298: 5295: 5292: 5289: 5283: 5279: 5278: 5274: 5270: 5266: 5258: 5252: 5249: 5246: 5244: 5241: 5238: 5235: 5231: 5230: 5200: 5197: 5195: 5192: 5189: 5186: 5180: 5176: 5175: 5171: 5167: 5162: 5159: 5157: 5154: 5151: 5148: 5123: 5119: 5118: 5114: 5110: 5107: 5104: 5102: 5099: 5096: 5093: 5090: 5086: 5085: 5066: 5063: 5061: 5058: 5055: 5052: 5046: 5042: 5041: 5037: 5033: 5032:P6mm, P6cc, P6 5030: 5027: 5025: 5022: 5019: 5016: 5013: 5009: 5008: 5004: 5000: 4996: 4992: 4988: 4985: 4982: 4980: 4977: 4974: 4971: 4968: 4964: 4963: 4959: 4956: 4953: 4951: 4948: 4945: 4942: 4939: 4935: 4934: 4928: 4925: 4923: 4920: 4917: 4914: 4909: 4905: 4904: 4901: 4897: 4893: 4889: 4885: 4882: 4879: 4877: 4874: 4871: 4868: 4865: 4852: 4848: 4847: 4818: 4815: 4813: 4810: 4807: 4804: 4798: 4794: 4793: 4788: 4785: 4783: 4780: 4777: 4774: 4771: 4767: 4766: 4760: 4756: 4752: 4748: 4747:P312, P321, P3 4745: 4742: 4740: 4737: 4734: 4731: 4728: 4724: 4723: 4713: 4710: 4708: 4705: 4702: 4699: 4694: 4690: 4689: 4684: 4680: 4677: 4674: 4672: 4669: 4666: 4663: 4660: 4642: 4638: 4637: 4633: 4629: 4623: 4619: 4615: 4611: 4607: 4603: 4599: 4595: 4592: 4589: 4587: 4584: 4581: 4578: 4575: 4571: 4570: 4532: 4524: 4509: 4506: 4504: 4501: 4498: 4495: 4489: 4485: 4484: 4480: 4476: 4475:I4mm, I4cm, I4 4470: 4466: 4462: 4458: 4457:P4mm, P4bm, P4 4455: 4452: 4450: 4447: 4444: 4441: 4438: 4434: 4433: 4429: 4423: 4419: 4415: 4411: 4407: 4403: 4399: 4395: 4391: 4387: 4384: 4381: 4379: 4376: 4373: 4370: 4367: 4363: 4362: 4358: 4352: 4348: 4345: 4342: 4340: 4337: 4334: 4331: 4328: 4324: 4323: 4313: 4310: 4308: 4305: 4302: 4299: 4294: 4290: 4289: 4286: 4282: 4278: 4274: 4271: 4268: 4266: 4263: 4260: 4257: 4254: 4235: 4231: 4230: 4221: 4218: 4216: 4213: 4210: 4207: 4204: 4200: 4199: 4191: 4185: 4181: 4177: 4173: 4170: 4167: 4165: 4162: 4159: 4156: 4153: 4149: 4148: 4145: 4141: 4137: 4133: 4129: 4125: 4121: 4117: 4113: 4109: 4106: 4103: 4101: 4098: 4095: 4092: 4089: 4060: 4056: 4055: 4049: 4043: 4040: 4037: 4035: 4032: 4029: 4026: 4023: 4019: 4018: 4013: 4010: 4008: 4005: 4002: 3999: 3996: 3992: 3991: 3986: 3983: 3980: 3978: 3975: 3972: 3969: 3966: 3948: 3944: 3943: 3937: 3934: 3932: 3929: 3926: 3923: 3918: 3914: 3913: 3910: 3907: 3905: 3902: 3899: 3896: 3893: 3880: 3876: 3875: 3870: 3865: 3860: 3855: 3849: 3848: 3845: 3840: 3833:Crystal system 3830: 3813: 3810: 3809: 3808: 3805: 3802: 3799: 3790: 3789: 3786: 3779: 3774: 3771: 3768: 3766: 3763: 3760: 3757: 3753: 3752: 3749: 3742: 3737: 3734: 3731: 3729: 3726: 3723: 3720: 3716: 3715: 3708: 3703: 3696: 3691: 3688: 3685: 3683: 3680: 3677: 3674: 3670: 3669: 3666: 3659: 3654: 3651: 3648: 3646: 3643: 3640: 3637: 3634: 3622: 3621: 3614: 3609: 3602: 3597: 3594: 3591: 3589: 3586: 3583: 3580: 3576: 3575: 3572: 3565: 3560: 3557: 3554: 3552: 3549: 3546: 3543: 3540: 3528: 3527: 3520: 3515: 3508: 3503: 3500: 3497: 3495: 3492: 3489: 3486: 3482: 3481: 3478: 3471: 3466: 3463: 3460: 3458: 3455: 3452: 3449: 3446: 3433: 3432: 3425: 3420: 3413: 3408: 3405: 3402: 3400: 3397: 3394: 3391: 3387: 3386: 3379: 3374: 3367: 3362: 3359: 3356: 3354: 3351: 3348: 3345: 3342: 3330: 3329: 3326: 3319: 3314: 3311: 3308: 3306: 3303: 3300: 3297: 3293: 3292: 3289: 3282: 3277: 3274: 3271: 3269: 3266: 3263: 3260: 3257: 3245: 3244: 3239: 3234: 3229: 3224: 3218: 3217: 3214: 3209: 3203: 3198:Crystal system 3184: 3181: 3178: 3177: 3174: 3161: 3156: 3152: 3141: 3138: 3125: 3121: 3110: 3107: 3103: 3102: 3100: 3098: 3095: 3082: 3078: 3067: 3065: 3061: 3060: 3058: 3056: 3053: 3040: 3036: 3025: 3023: 3019: 3018: 3016: 3014: 3011: 2998: 2994: 2983: 2981: 2977: 2976: 2973: 2960: 2955: 2951: 2940: 2937: 2924: 2920: 2909: 2906: 2903: 2899: 2898: 2895: 2882: 2877: 2873: 2862: 2859: 2846: 2842: 2831: 2826: 2822: 2821: 2818: 2805: 2800: 2796: 2785: 2782: 2769: 2765: 2754: 2749: 2745: 2744: 2741: 2728: 2723: 2719: 2708: 2705: 2692: 2688: 2677: 2672: 2668: 2667: 2664: 2651: 2646: 2642: 2631: 2628: 2615: 2611: 2600: 2595: 2592: 2588: 2587: 2584: 2571: 2566: 2562: 2551: 2548: 2535: 2531: 2520: 2515: 2511: 2510: 2507: 2494: 2489: 2485: 2474: 2471: 2458: 2454: 2443: 2438: 2434: 2433: 2430: 2417: 2412: 2408: 2397: 2394: 2381: 2377: 2366: 2361: 2358: 2354: 2353: 2350: 2337: 2332: 2328: 2317: 2314: 2301: 2297: 2286: 2281: 2277: 2276: 2273: 2260: 2255: 2251: 2240: 2237: 2224: 2220: 2209: 2204: 2201: 2197: 2196: 2193: 2180: 2175: 2171: 2160: 2157: 2144: 2140: 2129: 2126: 2123: 2119: 2118: 2115: 2112: 2109: 2106: 2102: 2101: 2098: 2095: 2090: 2026:Main article: 2023: 2020: 2017: 2016: 1988: 1972: 1968:227 + 44 = 271 1947: 1926: 1922: 1907: 1884: 1867: 1857: 1856: 1853: 1852: 1849: 1848:28927915 (+?) 1846: 1843: 1840: 1837: 1834: 1831: 1828: 1824: 1823: 1820: 1817: 1814: 1811: 1808: 1805: 1802: 1799: 1795: 1794: 1791: 1788: 1785: 1782: 1779: 1776: 1773: 1770: 1766: 1765: 1762: 1759: 1756: 1753: 1750: 1747: 1744: 1741: 1737: 1736: 1733: 1730: 1727: 1724: 1721: 1718: 1715: 1712: 1708: 1707: 1704: 1701: 1698: 1695: 1692: 1689: 1686: 1683: 1679: 1678: 1675: 1672: 1669: 1666: 1663: 1660: 1657: 1654: 1650: 1649: 1640: 1631: 1622: 1613: 1604: 1595: 1586: 1577: 1567: 1564: 1490: 1487: 1485: 1482: 1449: 1448: 1443: 1442: 1435: 1434: 1426: 1418: 1415: 1410: 1409: 1403: 1396: 1395: 1386: 1381: 1373: 1365: 1360: 1359: 1356: 1348: 1347: 1346: 1345: 1341: 1337: 1334: 1330: 1326: 1322: 1319: 1315: 1311: 1307: 1304: 1300: 1296: 1292: 1288: 1284: 1252: 1247: 1246: 1242: 1241: 1236: 1235: 1231: 1230: 1218: 1211:Sohncke groups 1206: 1202: 1199:Symmetry group 1178: 1177: 1166: 1163: 1162: 1161: 1154: 1149: 1146:Coxeter groups 1142: 1136: 1125: 1119: 1117: 1111: 1109: 1105: 1101: 1097: 1093: 1089: 1061: 1057: 1049: 1034: 1029: 1027: 1021: 1019: 1013: 1011: 1010: 1004: 992: 980: 970: 963: 958: 954: 950: 947: 946: 944: 942: 940: 938: 935: 933: 930: 926: 925: 923: 920: 917: 914: 911: 909: 906: 902: 901: 898: 895: 892: 889: 886: 883: 880: 876: 875: 872: 869: 866: 863: 860: 857: 854: 841: 836: 832: 825:crystal system 820: 777: 772: 770: 767: 765: 764: 760: 746: 743: 737: 734: 730:enantiomorphic 717: 714: 713: 712: 709: 703: 693: 687: 677: 613:. In general, 595: 594: 572: 569: 564: 552: 549: 546: 545: 542: 539: 536: 533: 530: 526: 525: 522: 519: 516: 513: 510: 506: 505: 502: 499: 496: 493: 490: 436: 416: 396: 376: 356: 340: 337: 316: 313: 300: 297: 293:NaCl structure 237: 234: 225: 217: 197: 178:Evgraf Fedorov 157: 154: 84:symmetry group 43: 37: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 7246: 7235: 7232: 7230: 7227: 7225: 7224:Finite groups 7222: 7220: 7217: 7215: 7212: 7211: 7209: 7194: 7191: 7189: 7186: 7184: 7181: 7180: 7177: 7170: 7167: 7164: 7162: 7161:Quantum group 7159: 7157: 7154: 7152: 7149: 7147: 7144: 7143: 7141: 7137: 7131: 7128: 7126: 7123: 7121: 7120:Lorentz group 7118: 7116: 7113: 7112: 7109: 7103: 7101: 7095: 7093: 7087: 7085: 7079: 7077: 7071: 7069: 7066: 7065: 7061: 7058: 7055: 7052: 7049: 7048:Unitary group 7046: 7043: 7040: 7037: 7034: 7031: 7028: 7025: 7022: 7021: 7019: 7017: 7013: 7007: 7004: 7001: 6997: 6994: 6991: 6988: 6984: 6981: 6978: 6975: 6974: 6970: 6969:Monster group 6967: 6964: 6961: 6955: 6954:Fischer group 6952: 6950: 6943: 6936: 6929: 6923:Janko groups 6922: 6916: 6913: 6903: 6902:Mathieu group 6900: 6898: 6895: 6894: 6887: 6884: 6878: 6875: 6873: 6870: 6869: 6867: 6865: 6861: 6855: 6854:Trivial group 6852: 6850: 6847: 6845: 6842: 6840: 6837: 6835: 6832: 6830: 6827: 6825: 6824:Simple groups 6822: 6820: 6817: 6815: 6814:Cyclic groups 6812: 6810: 6807: 6805: 6804:Finite groups 6802: 6801: 6799: 6795: 6789: 6786: 6784: 6780: 6776: 6774: 6771: 6769: 6766: 6764: 6761: 6759: 6756: 6754: 6751: 6750: 6748: 6746: 6745:Basic notions 6742: 6738: 6731: 6726: 6724: 6719: 6717: 6712: 6711: 6708: 6702: 6699: 6697: 6694: 6688: 6687: 6681: 6679: 6675: 6672: 6669: 6667: 6664: 6662: 6659: 6657: 6654: 6652: 6649: 6647: 6644: 6642: 6640: 6636: 6633: 6630: 6628: 6625: 6624: 6620: 6614: 6610: 6606: 6602: 6598: 6594: 6590: 6586: 6582: 6578: 6574: 6570: 6566: 6563: 6559: 6558: 6553: 6548: 6545: 6541: 6536: 6531: 6527: 6523: 6519: 6515: 6511: 6507: 6502: 6499: 6495: 6491: 6487: 6483: 6479: 6478: 6470: 6465: 6462: 6458: 6454: 6451: 6447: 6443: 6439: 6435: 6431: 6427: 6423: 6419: 6415: 6411: 6406: 6402: 6397: 6393: 6389: 6384: 6381: 6377: 6373: 6369: 6365: 6361: 6357: 6353: 6352: 6346: 6342: 6337: 6333: 6329: 6325: 6321: 6316: 6313: 6309: 6304: 6299: 6296:(Pt 3): 301, 6295: 6291: 6286: 6283: 6279: 6275: 6271: 6267: 6263: 6259: 6255: 6248: 6243: 6240: 6236: 6232: 6228: 6224: 6220: 6216: 6212: 6207: 6204: 6200: 6196: 6192: 6188: 6182: 6178: 6174: 6170: 6166: 6161: 6158: 6154: 6149: 6144: 6140: 6136: 6132: 6128: 6124: 6121: 6117: 6113: 6109: 6105: 6101: 6096: 6093: 6087: 6083: 6079: 6075: 6071: 6070: 6064: 6060: 6056: 6052: 6048: 6043: 6037: 6031: 6030: 6027: 6023: 6019: 6015: 6010: 6007: 6003: 5999: 5995: 5991: 5987: 5983: 5979: 5975: 5971: 5968: 5964: 5960: 5956: 5952: 5948: 5944: 5940: 5936: 5932: 5931: 5925: 5922: 5918: 5914: 5912: 5906: 5901: 5898: 5894: 5890: 5884: 5880: 5876: 5871: 5868: 5864: 5860: 5856: 5852: 5848: 5844: 5840: 5839: 5834: 5830: 5825: 5822: 5818: 5814: 5810: 5806: 5802: 5798: 5794: 5793: 5788: 5783: 5780: 5776: 5772: 5768: 5764: 5760: 5756: 5751: 5750: 5735: 5729: 5726: 5721: 5717: 5713: 5709: 5705: 5701: 5697: 5690: 5687: 5682: 5678: 5674: 5670: 5667:(2): 023514. 5666: 5662: 5655: 5648: 5645: 5640: 5636: 5630: 5627: 5622: 5615: 5612: 5607: 5604:(in German). 5603: 5599: 5592: 5589: 5584: 5580: 5573: 5570: 5566: 5561: 5558: 5553: 5549: 5545: 5544: 5536: 5533: 5529: 5524: 5521: 5510:on 2022-09-29 5509: 5505: 5501: 5497: 5493: 5489: 5485: 5484: 5479: 5472: 5469: 5463: 5458: 5454: 5451: 5450: 5446: 5444: 5441: 5436: 5434: 5430: 5424: 5422: 5418: 5413: 5359: 5356: 5354: 5351: 5345: 5338: 5335: 5334: 5304: 5301: 5299: 5296: 5290: 5284: 5281: 5280: 5250: 5247: 5245: 5242: 5239: 5236: 5233: 5232: 5201: 5198: 5196: 5193: 5187: 5181: 5178: 5177: 5164:P23, F23, I23 5163: 5160: 5158: 5155: 5152: 5149: 5146: 5140: 5134: 5127: 5121: 5120: 5108: 5105: 5103: 5100: 5094: 5091: 5088: 5087: 5067: 5064: 5062: 5059: 5053: 5047: 5044: 5043: 5031: 5028: 5026: 5023: 5017: 5014: 5011: 5010: 4986: 4983: 4981: 4978: 4972: 4969: 4966: 4965: 4957: 4954: 4952: 4949: 4943: 4940: 4937: 4936: 4929: 4926: 4924: 4921: 4915: 4910: 4907: 4906: 4883: 4880: 4878: 4875: 4869: 4866: 4863: 4856: 4850: 4849: 4819: 4816: 4814: 4811: 4805: 4799: 4796: 4789: 4786: 4784: 4781: 4775: 4772: 4769: 4768: 4746: 4743: 4741: 4738: 4732: 4729: 4726: 4725: 4714: 4711: 4709: 4706: 4700: 4695: 4692: 4691: 4678: 4675: 4673: 4670: 4664: 4661: 4658: 4653: 4646: 4640: 4593: 4590: 4588: 4585: 4579: 4576: 4573: 4572: 4510: 4507: 4505: 4502: 4496: 4490: 4487: 4486: 4456: 4453: 4451: 4448: 4442: 4439: 4436: 4435: 4385: 4382: 4380: 4377: 4371: 4368: 4365: 4364: 4346: 4343: 4341: 4338: 4332: 4329: 4326: 4325: 4314: 4311: 4309: 4306: 4300: 4295: 4292: 4291: 4272: 4269: 4267: 4264: 4258: 4255: 4252: 4246: 4239: 4233: 4232: 4222: 4219: 4217: 4214: 4208: 4205: 4202: 4201: 4171: 4168: 4166: 4163: 4157: 4154: 4151: 4150: 4107: 4104: 4102: 4099: 4093: 4090: 4087: 4082: 4076: 4071: 4064: 4058: 4057: 4041: 4038: 4036: 4033: 4027: 4024: 4021: 4020: 4014: 4011: 4009: 4006: 4000: 3997: 3994: 3993: 3984: 3981: 3979: 3976: 3970: 3967: 3964: 3959: 3952: 3946: 3945: 3938: 3935: 3933: 3930: 3924: 3919: 3916: 3915: 3911: 3908: 3906: 3903: 3897: 3894: 3891: 3884: 3878: 3877: 3874: 3871: 3869: 3866: 3864: 3861: 3859: 3856: 3854: 3851: 3850: 3844: 3834: 3829: 3823: 3819: 3811: 3806: 3803: 3800: 3797: 3796: 3795: 3784: 3780: 3775: 3772: 3769: 3767: 3764: 3758: 3755: 3747: 3743: 3738: 3735: 3732: 3730: 3727: 3721: 3718: 3713: 3709: 3704: 3701: 3697: 3692: 3689: 3686: 3684: 3681: 3675: 3672: 3664: 3660: 3655: 3652: 3649: 3647: 3644: 3638: 3635: 3632: 3627: 3619: 3615: 3610: 3607: 3603: 3598: 3595: 3592: 3590: 3587: 3581: 3578: 3570: 3566: 3561: 3558: 3555: 3553: 3550: 3544: 3541: 3538: 3533: 3525: 3521: 3516: 3513: 3509: 3504: 3501: 3498: 3496: 3493: 3487: 3484: 3476: 3472: 3467: 3464: 3461: 3459: 3456: 3450: 3447: 3444: 3439: 3430: 3426: 3421: 3418: 3414: 3409: 3406: 3403: 3401: 3398: 3392: 3389: 3384: 3380: 3375: 3372: 3368: 3363: 3360: 3357: 3355: 3352: 3346: 3343: 3340: 3335: 3324: 3320: 3315: 3312: 3309: 3307: 3304: 3298: 3295: 3287: 3283: 3278: 3275: 3272: 3270: 3267: 3261: 3258: 3255: 3250: 3243: 3240: 3238: 3235: 3233: 3230: 3228: 3225: 3223: 3220: 3208: 3199: 3195: 3192: 3190: 3187:Table of the 3182: 3175: 3159: 3154: 3150: 3142: 3139: 3123: 3119: 3111: 3108: 3105: 3104: 3101: 3099: 3096: 3080: 3076: 3068: 3066: 3063: 3062: 3059: 3057: 3054: 3038: 3034: 3026: 3024: 3021: 3020: 3017: 3015: 3012: 2996: 2992: 2984: 2982: 2979: 2978: 2974: 2958: 2953: 2949: 2941: 2938: 2922: 2918: 2910: 2907: 2904: 2900: 2896: 2880: 2875: 2871: 2863: 2860: 2844: 2840: 2832: 2830: 2827: 2824: 2823: 2819: 2803: 2798: 2794: 2786: 2783: 2767: 2763: 2755: 2753: 2750: 2747: 2746: 2742: 2726: 2721: 2717: 2709: 2706: 2690: 2686: 2678: 2676: 2673: 2670: 2669: 2665: 2649: 2644: 2640: 2632: 2629: 2613: 2609: 2601: 2599: 2596: 2593: 2589: 2585: 2569: 2564: 2560: 2552: 2549: 2533: 2529: 2521: 2519: 2516: 2513: 2512: 2508: 2492: 2487: 2483: 2475: 2472: 2456: 2452: 2444: 2442: 2441:Frieze groups 2439: 2436: 2435: 2431: 2415: 2410: 2406: 2398: 2395: 2379: 2375: 2367: 2365: 2362: 2359: 2355: 2351: 2335: 2330: 2326: 2318: 2315: 2299: 2295: 2287: 2285: 2282: 2279: 2278: 2274: 2258: 2253: 2249: 2241: 2238: 2222: 2218: 2210: 2208: 2205: 2202: 2198: 2194: 2178: 2173: 2169: 2161: 2158: 2142: 2138: 2130: 2127: 2124: 2121: 2120: 2116: 2113: 2110: 2107: 2104: 2103: 2085: 2082: 2080: 2076: 2072: 2068: 2064: 2060: 2056: 2052: 2048: 2044: 2040: 2035: 2029: 2021: 2013: 2009: 2005: 2001: 1997: 1992: 1989: 1985: 1981: 1976: 1973: 1961: 1957: 1951: 1948: 1944: 1940: 1936: 1935:Barlow (1894) 1932: 1929:12). Usually 1920: 1918: 1911: 1908: 1904: 1900: 1899: 1894: 1888: 1885: 1881: 1877: 1871: 1868: 1865:Trivial group 1862: 1859: 1850: 1847: 1844: 1841: 1838: 1835: 1832: 1829: 1826: 1825: 1821: 1819:222018 (+79) 1818: 1815: 1812: 1809: 1806: 1803: 1800: 1797: 1796: 1792: 1789: 1786: 1783: 1780: 1777: 1774: 1771: 1768: 1767: 1763: 1760: 1757: 1754: 1751: 1748: 1745: 1742: 1739: 1738: 1734: 1731: 1728: 1725: 1722: 1719: 1716: 1713: 1710: 1709: 1705: 1702: 1699: 1696: 1693: 1690: 1687: 1684: 1681: 1680: 1676: 1673: 1670: 1667: 1664: 1661: 1658: 1655: 1652: 1651: 1648: 1641: 1639: 1632: 1630: 1623: 1621: 1614: 1612: 1605: 1603: 1596: 1594: 1587: 1585: 1578: 1575: 1574: 1571: 1565: 1563: 1561: 1555: 1553: 1549: 1545: 1541: 1537: 1533: 1529: 1524: 1520: 1516: 1512: 1508: 1504: 1500: 1496: 1483: 1481: 1479: 1475: 1471: 1467: 1463: 1459: 1455: 1444: 1440: 1436: 1432: 1424: 1416: 1412: 1411: 1407: 1404: 1401: 1398: 1397: 1394: 1392: 1382: 1379: 1371: 1362: 1361: 1357: 1354: 1350: 1349: 1340:: p4m, p4g; D 1335: 1320: 1305: 1282: 1281: 1280: 1278: 1270: 1266: 1262: 1258: 1248: 1243: 1237: 1232: 1228: 1224: 1216: 1215:enantiomorphs 1212: 1200: 1196: 1192: 1188: 1184: 1179: 1174: 1171: 1164: 1159: 1155: 1153: 1150: 1147: 1143: 1141: 1138: 1137: 1134: 1130: 1126: 1122: 1118: 1114: 1110: 1087: 1083: 1079: 1075: 1071: 1067: 1055: 1047: 1043: 1039: 1035: 1032: 1028: 1024: 1020: 1016: 1012: 1000: 988: 976: 969: 964: 962: 959: 955: 953:Hall notation 952: 951: 945: 943: 941: 939: 936: 934: 931: 928: 927: 924: 921: 918: 915: 912: 910: 907: 904: 903: 899: 896: 893: 890: 887: 884: 881: 878: 877: 873: 870: 867: 864: 862:Orthorhombic 861: 858: 855: 852: 851: 848: 830: 826: 818: 814: 810: 806: 802: 798: 794: 790: 786: 782: 778: 775: 771: 766: 761: 758: 757: 756: 752: 744: 742: 735: 733: 731: 726: 723: 715: 710: 708: 704: 702: 698: 694: 692: 688: 686: 685:frieze groups 682: 678: 676: 672: 671: 670: 667: 665: 661: 660:diamond cubic 656: 652: 648: 644: 640: 636: 632: 628: 624: 620: 616: 612: 608: 604: 600: 593: 589: 585: 581: 578: 577: 576: 570: 568: 562: 558: 550: 543: 540: 537: 534: 531: 528: 527: 523: 520: 517: 514: 511: 508: 507: 503: 500: 497: 494: 491: 488: 487: 484: 482: 478: 474: 470: 466: 462: 458: 454: 450: 434: 414: 394: 374: 354: 346: 338: 336: 334: 330: 326: 322: 314: 312: 310: 306: 298: 296: 294: 286: 281: 279: 275: 271: 267: 263: 259: 255: 251: 247: 243: 235: 233: 231: 207: 203: 183: 179: 175: 171: 166: 163: 155: 153: 151: 148: 144: 140: 138: 133: 129: 124: 122: 121: 116: 112: 108: 105: 101: 97: 93: 89: 85: 81: 77: 73: 69: 61: 57: 53: 49: 41: 32: 19: 7188:Applications 7115:Circle group 6999: 6995: 6986: 6982: 6915:Conway group 6877:Cyclic group 6838: 6685: 6580: 6576: 6555: 6509: 6505: 6481: 6475: 6460: 6417: 6413: 6391: 6387: 6355: 6349: 6323: 6319: 6293: 6289: 6257: 6253: 6214: 6210: 6164: 6138: 6134: 6130: 6103: 6099: 6068: 6058: 6054: 6050: 6035: 6025: 6021: 6017: 5989: 5985: 5934: 5928: 5908: 5904: 5878: 5842: 5836: 5796: 5790: 5762: 5758: 5737:. Retrieved 5728: 5703: 5699: 5689: 5664: 5660: 5647: 5638: 5629: 5614: 5605: 5601: 5591: 5582: 5578: 5572: 5560: 5552:B.G. Teubner 5547: 5542: 5535: 5523: 5512:. Retrieved 5508:the original 5487: 5481: 5471: 5437: 5425: 5416: 5411: 5409: 4351:/m, P4/n, P4 4184:, Pba2, Pna2 4180:, Pnc2, Pmn2 4063:Orthorhombic 3793: 3186: 2752:Layer groups 2031: 1991: 1975: 1950: 1930: 1915: 1910: 1903:plane groups 1902: 1896: 1892: 1887: 1870: 1861: 1790:4783 (+111) 1569: 1559: 1556: 1551: 1547: 1544:affine space 1535: 1522: 1514: 1502: 1494: 1492: 1478:cubic groups 1452: 1430: 1422: 1390: 1384: 1377: 1369: 1351:(geometric) 1273: 1264: 1260: 1256: 1168: 1085: 1081: 1077: 1073: 1069: 1065: 1064:compounds, ( 1053: 1045: 1041: 1037: 998: 986: 974: 967: 846: 828: 813:point groups 808: 804: 800: 796: 792: 788: 780: 754: 739: 736:Combinations 727: 719: 707:Layer groups 668: 650: 646: 642: 638: 634: 630: 626: 618: 614: 610: 606: 602: 598: 596: 591: 587: 583: 579: 574: 560: 554: 480: 476: 472: 468: 464: 460: 456: 452: 342: 339:Glide planes 333:point groups 318: 315:Translations 307:, including 302: 282: 244:with the 14 239: 167: 159: 146: 135: 131: 125: 118: 79: 65: 59: 55: 51: 50:. The first 47: 6844:Point group 6839:Space group 6583:: 117–141, 6388:Math. Comp. 6127:Janssen, T. 5421:Hahn (2002) 3843:Point group 3438:rectangular 3334:Rectangular 3211:Arithmetic 3207:point group 2071:Litvin 2005 2067:Litvin 2008 2006:. See also 1931:space group 1845:85308 (+?) 1576:Dimensions 1310:: pm, pg; D 1195:orientation 1052:compounds, 865:Tetragonal 859:Monoclinic 697:line groups 681:line groups 675:line groups 655:point group 609:into point 529:Old Symbol 509:New symbol 345:glide plane 278:glide plane 256:(including 150:Hahn (2002) 80:space group 68:mathematics 36:hexagonal H 7208:Categories 7156:Loop group 7016:Lie groups 6788:direct sum 6535:2066/35218 6061:: 345–390. 6028:(2): 1–146 5909:Groups of 5514:2015-01-31 5464:References 4238:Tetragonal 4227:Fmmm, Fddd 4196:Fmm2, Fdd2 4190:Cmm2, Cmc2 4172:Pmm2, Pmc2 4108:P222, P222 3951:Monoclinic 2675:Rod groups 2094:dimension 2089:dimension 1787:710 (+70) 1784:227 (+44) 1761:219 (+11) 1499:Bieberbach 1321:Between: D 1261:symmorphic 1170:one down. 871:Hexagonal 856:Triclinic 701:rod groups 557:screw axis 551:Screw axes 274:screw axis 262:reflection 111:isometries 94:) are the 6613:120651709 6597:0010-2571 6562:EMS Press 6512:: 77–82, 6434:1058-6458 6203:117849701 5998:0138-4821 5967:121994079 5951:0003-9519 5867:119472023 5859:0025-5831 5821:124429194 5813:0025-5831 5779:102301331 5456:rotation) 5410:Note: An 4855:Hexagonal 4792:R3m, R3c 4386:P422, P42 3883:Triclinic 3837:(count), 3626:Hexagonal 3436:Centered 2034:Shubnikov 1925:12 and P3 1919:character 1778:64 (+10) 1466:fibrifold 1160:notation. 1023:Shubnikov 868:Trigonal 716:Chirality 666:applies. 633:), where 254:unit cell 174:chirality 104:cocompact 76:chemistry 7214:Symmetry 6753:Subgroup 6674:Archived 6635:Archived 6571:(1948), 6544:99946564 6498:12714771 6450:40588234 6380:95680790 6312:11961294 6282:15846043 6239:18421131 6157:12388880 5980:(2001), 5877:(1978), 5765:: 1–63, 5608:: 25–75. 5336:221–230 5282:215–220 5257:F432, F4 5251:P432, P4 5234:207–214 5179:200–206 5122:195–199 5113:/mcm, P6 5089:191–194 5045:187–190 5012:183–186 4987:P622, P6 4967:177–182 4958:P6/m, P6 4938:175–176 4851:168–173 4797:162–167 4770:156–161 4727:149–155 4693:147–148 4645:Trigonal 4641:143–146 4632:/amd, I4 4622:/nmc, P4 4618:/mnm, P4 4614:/mbc, P4 4610:/nnm, P4 4606:/nbc, P4 4602:/mcm, P4 4598:/mmc, P4 4574:123–142 4488:111–122 4428:I422, I4 4357:I4/m, I4 4347:P4/m, P4 4285:, I4, I4 4042:P2/m, P2 3863:Orbifold 3690:Between 3502:Between 3465:Between 3232:Orbifold 2092:Lattice 2087:Overall 2059:Kim 1999 1921:(e.g. P3 1775:33 (+7) 1772:23 (+6) 1470:orbifold 1329:: cmm; D 1306:Along: D 1133:Thurston 1080:Alloys, 1017:notation 845:in use. 763:numbers. 745:Notation 329:quotient 266:rotation 236:Elements 220:d, and P 143:symmetry 7183:History 6605:0024424 6514:Bibcode 6442:1795312 6360:Bibcode 6328:Bibcode 6262:Bibcode 6219:Bibcode 6195:1713786 6108:Bibcode 6006:1865535 5959:0220837 5921:0020553 5897:0484179 5708:Bibcode 5669:Bibcode 5504:2322930 4437:99–110 4273:P4, P4 4017:Cm, Cc 3788: 3751: 3668: 3574: 3480: 3412:(*2222) 3328: 3291: 3249:Oblique 2114:Symbol 2108:Symbol 1822:222097 1519:abelian 1336:Both: D 1325:: cm; D 1299:: p4; C 1295:: p3; C 1291:: p2; C 1287:: p1; C 1283:None: C 1104:, and B 1096:, and C 1072:, ..., 1015:Fedorov 732:pairs. 705:(3,2): 689:(2,2): 653:form a 623:lattice 449:diamond 291:m, the 204: ( 156:History 137:Fedorov 82:is the 72:physics 6958:22..24 6910:22..24 6906:11..12 6737:Groups 6611: 6603: 6595: 6542: 6496: 6448: 6440: 6432: 6378: 6310: 6280: 6237: 6201: 6193: 6183: 6155: 6088: 6004: 5996: 5965: 5957: 5949: 5919: 5895: 5885: 5865: 5857: 5819: 5811: 5777: 5739:11 May 5502: 5273:32, I4 5269:32, P4 5036:cm, P6 5003:22, P6 4999:22, P6 4995:22, P6 4991:22, P6 4884:P6, P6 4759:12, P3 4755:21, P3 4751:12, P3 4679:P3, P3 4479:md, I4 4469:mc, P4 4461:cm, P4 4418:22, P4 4406:22, P4 4394:22, P4 4366:89–98 4327:83–88 4293:81–82 4234:75–80 4203:47–74 4188:, Pnn2 4152:25–46 4132:, C222 4059:16–24 4022:10–15 4015:Pm, Pc 3985:P2, P2 3858:Schön. 3778:(*632) 3695:(*333) 3601:(*442) 3532:Square 3507:(2*22) 3407:Along 3361:Along 3318:(2222) 3227:Schön. 3213:class 3176:62227 2117:Count 2111:Count 2047:ferri- 2043:ferro- 1891:These 1878:; see 1542:is an 1454:Conway 1333:: p3m1 1318:: p31m 1227:quartz 1221:3 for 1205:and P4 1129:Conway 1048:for AB 1025:symbol 874:Cubic 817:quartz 759:Number 722:chiral 597:where 139:groups 107:groups 100:chiral 7171:Sp(∞) 7168:SU(∞) 7062:Sp(n) 7056:SU(n) 7044:SO(n) 7032:SL(n) 7026:GL(n) 6779:Semi- 6690:(PDF) 6609:S2CID 6540:S2CID 6472:(PDF) 6446:S2CID 6376:S2CID 6250:(PDF) 6199:S2CID 6049:[ 6016:[ 5963:S2CID 5907:[ 5863:S2CID 5817:S2CID 5775:S2CID 5657:(PDF) 5581:[ 5546:[ 5500:JSTOR 5400:m, Ia 5390:m, Fd 5386:c, Fd 5382:m, Fm 5372:n, Pn 5368:n, Pm 5364:m, Pn 5327:3c, I 5323:3n, F 5313:3m, I 5309:3m, F 5170:3, I2 5126:Cubic 5117:/mmc 5092:6/mmm 5080:2m, P 5076:c2, P 5072:m2, P 4832:m1, P 4828:1c, P 4824:1m, P 4636:/acd 4577:4/mmm 4565:2m, I 4561:c2, I 4557:m2, I 4547:b2, P 4543:c2, P 4539:m2, P 4519:2c, P 4515:2m, P 4414:2, P4 4402:2, P4 4390:2, P4 4120:2, P2 4054:C2/c 3853:Int'l 3773:Both 3765:(*66) 3741:(632) 3736:None 3707:(3*3) 3682:(*33) 3658:(333) 3653:None 3613:(4*2) 3596:Both 3588:(*44) 3564:(442) 3559:None 3519:(22×) 3494:(*22) 3424:(22*) 3399:(*22) 3313:None 3276:None 3222:Int'l 3140:4894 3097:1594 3055:1091 2975:1202 2897:1651 2105:Name 1842:7103 1839:1594 1816:6079 1793:4894 1344:: p6m 1265:split 1187:up to 1056:for A 819:is P3 544:Ccca 524:Ccce 387:, or 212:2d, P 42:is P6 40:O ice 7165:O(∞) 7050:U(n) 7038:O(n) 6919:1..3 6593:ISSN 6494:PMID 6430:ISSN 6308:PMID 6278:PMID 6235:PMID 6181:ISBN 6153:PMID 6086:ISBN 5994:ISSN 5947:ISSN 5883:ISBN 5855:ISSN 5809:ISSN 5741:2015 5438:The 5352:*432 5297:*332 5263:I432 5226:, Ia 5222:, Pa 5218:, Im 5214:, Fd 5210:, Fm 5206:, Pn 5129:(36) 5101:*226 5060:*223 4908:174 4900:, P6 4896:, P6 4892:, P6 4888:, P6 4858:(27) 4842:m, R 4765:R32 4683:, P3 4648:(25) 4626:/ncm 4586:*224 4535:c, P 4527:m, P 4281:, P4 4277:, P4 4241:(68) 4215:*222 4112:, P2 4066:(59) 3995:6–9 3954:(13) 3947:3–5 3873:Ord. 3868:Cox. 3728:(66) 3705:p31m 3693:p3m1 3645:(33) 3551:(44) 3470:(*×) 3378:(××) 3366:(**) 3305:(22) 3242:Ord. 3237:Cox. 3013:343 2939:271 2861:230 2820:528 2743:394 2666:122 2069:), ( 1956:1978 1941:and 1836:841 1833:251 1813:955 1810:239 1807:189 1781:118 1764:230 1644:OEIS 1635:OEIS 1626:OEIS 1617:OEIS 1608:OEIS 1599:OEIS 1590:OEIS 1581:OEIS 1511:1912 1507:1911 1458:2001 1303:: p6 1263:(or 1223:FeSi 1131:and 1123:(3D) 1115:(2D) 625:) + 541:Cmma 538:Cmca 535:Aba2 532:Abm2 521:Cmme 518:Cmce 515:Aea2 512:Aem2 453:i.e. 321:rank 276:and 268:and 206:1894 78:, a 74:and 6585:doi 6530:hdl 6522:doi 6510:221 6486:doi 6422:doi 6396:doi 6368:doi 6336:doi 6298:doi 6270:doi 6227:doi 6173:doi 6143:doi 6116:doi 6078:doi 5939:doi 5847:doi 5801:doi 5767:doi 5716:doi 5677:doi 5492:doi 5357:48 5331:3d 5302:24 5277:32 5248:24 5243:432 5237:432 5199:24 5194:3*2 5161:12 5156:332 5106:24 5084:2c 5065:12 5040:mc 5029:12 5024:*66 5015:6mm 5007:22 4984:12 4979:226 4970:622 4962:/m 4955:12 4941:6/m 4817:12 4812:2*3 4782:*33 4739:223 4719:, R 4688:R3 4591:16 4569:2d 4503:2*2 4483:cd 4449:*44 4440:4mm 4432:22 4378:224 4369:422 4361:/a 4330:4/m 4319:, I 4206:mmm 4164:*22 4155:mm2 4100:222 4091:222 4025:2/m 4007:*11 3990:C2 3912:P1 3886:(2) 3776:p6m 3770:12 3756:6mm 3611:p4g 3599:p4m 3579:4mm 3517:pgg 3505:cmm 3485:2mm 3457:(*) 3422:pmg 3410:pmm 3390:2mm 3353:(*) 3281:(1) 3268:(1) 2784:80 2707:75 2630:32 2586:80 2550:17 2509:31 2432:31 2396:10 2065:, ( 2049:or 1901:or 1830:91 1804:59 1801:32 1758:73 1755:32 1752:18 1749:14 1735:17 1732:17 1729:13 1726:10 1493:In 1380:). 1189:an 1092:, B 807:or 504:68 475:or 467:or 459:or 134:or 126:In 109:of 66:In 48:mmc 7210:: 6944:, 6937:, 6930:, 6917:Co 6908:,M 6781:) 6607:, 6601:MR 6599:, 6591:, 6581:21 6579:, 6560:, 6554:, 6538:, 6528:, 6520:, 6508:, 6492:, 6482:59 6480:, 6474:, 6444:, 6438:MR 6436:, 6428:, 6416:, 6412:, 6392:43 6390:, 6374:, 6366:, 6356:57 6354:, 6334:, 6324:54 6322:, 6306:, 6294:58 6292:, 6276:, 6268:, 6258:61 6256:, 6252:, 6233:, 6225:, 6215:64 6213:, 6197:, 6191:MR 6189:, 6179:, 6171:, 6167:, 6151:, 6139:58 6137:, 6114:, 6104:37 6102:, 6084:, 6076:, 6059:28 6026:28 6002:MR 6000:, 5990:42 5988:, 5984:, 5961:, 5955:MR 5953:, 5945:, 5933:, 5917:MR 5893:MR 5891:, 5861:, 5853:, 5843:72 5841:, 5815:, 5807:, 5797:70 5795:, 5773:, 5763:23 5761:, 5714:. 5704:50 5702:. 5698:. 5675:. 5665:48 5663:. 5659:. 5637:. 5606:20 5498:. 5488:93 5486:. 5480:. 5423:. 5404:d 5396:Im 5378:Fm 5360:Pm 5317:3m 5288:3m 5265:P4 5261:32 5255:32 5202:Pm 5174:3 5166:P2 5150:23 5097:6h 5056:3h 5051:m2 5020:6v 4950:6* 4946:6h 4927:6 4922:3* 4918:3h 4881:6 4876:66 4846:c 4836:c1 4808:3d 4787:6 4778:3v 4773:3m 4763:21 4744:6 4730:32 4712:6 4707:3× 4676:3 4671:33 4582:4h 4551:n2 4508:8 4499:2d 4494:2m 4473:bc 4454:8 4445:4v 4383:8 4355:/n 4344:8 4339:4* 4335:4h 4312:4 4307:2× 4270:4 4265:44 4220:8 4211:2h 4169:4 4160:2v 4105:4 4052:/c 4046:/m 4039:4 4034:2* 4030:2h 4012:2 3982:2 3977:22 3936:2 3931:1× 3917:2 3909:1 3904:11 3879:1 3835:, 3739:p6 3733:6 3687:6 3673:3m 3656:p3 3650:3 3593:8 3562:p4 3556:4 3499:4 3468:cm 3462:2 3404:4 3376:pg 3364:pm 3358:2 3316:p2 3310:2 3279:p1 3273:1 3200:, 3106:4 3081:43 3064:3 3039:42 3022:2 2997:41 2980:1 2954:40 2923:40 2905:0 2902:4 2825:3 2799:32 2768:32 2748:2 2722:31 2691:31 2671:1 2645:30 2614:30 2594:0 2591:3 2514:2 2488:21 2473:7 2457:21 2437:1 2411:20 2380:20 2360:0 2357:2 2352:7 2316:2 2280:1 2275:5 2254:10 2239:2 2223:10 2203:0 2200:1 2195:2 2159:1 2125:0 2122:0 2081:) 2077:)( 2045:, 2010:. 1958:) 1937:, 1851:? 1827:6 1798:5 1769:4 1746:7 1743:6 1740:3 1723:9 1720:5 1717:4 1714:4 1711:2 1706:2 1703:2 1700:2 1697:2 1694:2 1691:1 1688:1 1685:1 1682:1 1677:1 1674:1 1671:1 1668:1 1665:1 1662:1 1659:1 1656:1 1653:0 1562:. 1530:. 1509:, 1364:GL 1251:GL 1156:A 1068:, 1058:m 997:, 985:, 937:c 932:— 929:3 922:a 919:a 916:a 913:b 908:— 905:2 900:a 897:c 894:c 891:c 888:a 885:b 882:— 879:1 840:P3 803:, 799:, 795:, 791:, 683:: 617:= 590:+ 582:= 555:A 501:67 498:64 495:41 492:39 471:, 463:, 367:, 343:A 335:. 311:. 295:. 264:, 152:. 123:. 70:, 7107:8 7105:E 7099:7 7097:E 7091:6 7089:E 7083:4 7081:F 7075:2 7073:G 7000:n 6996:D 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