3701:
3534:
3515:
3689:
8837:
8694:
4982:
4889:
4871:
4762:
4744:
4964:
4880:
4753:
4973:
8844:
3986:, which is similar to the cube described, with each rectangle replaced by a pentagon with one symmetry axis and 4 equal sides and 1 different side (the one corresponding to the line segment dividing the cube's face); i.e., the cube's faces bulge out at the dividing line and become narrower there. It is a subgroup (but not a normal subgroup) of the full icosahedral symmetry group (as isometry group, not just as abstract group), with 4 of the 10 3-fold axes. It is a normal subgroup of
89:
53:
237:
191:
145:
27:
8526:
8491:
8324:
8291:
8240:
8731:
8373:
8036:
7993:
7957:
7891:
7426:
8653:
7855:
7806:
7766:
7375:
7346:
7310:
7281:
7224:
7195:
7155:
7124:
7075:
8131:
3898:
9468:
1809:
1711:
1518:
1439:
864:
9179:. There is thus no notion of a "binary polyhedron" that covers a 3-dimensional polyhedron. Binary polyhedral groups are discrete subgroups of a Spin group, and under a representation of the spin group act on a vector space, and may stabilize a polyhedron in this representation β under the map Spin(3) β SO(3) they act on the same polyhedron that the underlying (non-binary) group acts on, while under
1610:
1337:
1237:
293:
8873:. An object with a given symmetry in a given orientation is characterized by the fundamental domain. If the object is a surface it is characterized by a surface in the fundamental domain continuing to its radial bordal faces or surface. If the copies of the surface do not fit, radial faces or surfaces can be added. They fit anyway if the fundamental domain is bounded by reflection planes.
1978:
896:) having infinite rotational symmetry must also have mirror symmetry for every plane through the axis. Physical objects having infinite rotational symmetry will also have the symmetry of mirror planes through the axis, but vector fields may not, for instance the velocity vectors of a cone rotating about its axis, or the magnetic field surrounding a wire.
3045:), which is isomorphic to the integers. The following table gives the five continuous axial rotation groups. They are limits of the finite groups only in the sense that they arise when the main rotation is replaced by rotation by an arbitrary angle, so not necessarily a rational number of degrees as with the finite groups. Physical objects can only have
3982:. It is the symmetry of a cube with on each face a line segment dividing the face into two equal rectangles, such that the line segments of adjacent faces do not meet at the edge. The symmetries correspond to the even permutations of the body diagonals and the same combined with inversion. It is also the symmetry of a
789:
number as when there is only one mirror or axis.) The conjugacy definition would also allow a mirror image of the structure, but this is not needed, the structure itself is achiral. For example, if a symmetry group contains a 3-fold axis of rotation, it contains rotations in two opposite directions. (The structure
8884:. Adjusting the orientation of the plane gives various possibilities of combining two or more adjacent faces to one, giving various other polyhedra with the same symmetry. The polyhedron is convex if the surface fits to its copies and the radial line perpendicular to the plane is in the fundamental domain.
911:
who was the first to investigate them. The seven infinite series of axial groups lead to five limiting groups (two of them are duplicates), and the seven remaining point groups produce two more continuous groups. In international notation, the list is β, β2, β/m, βmm, β/mm, ββ, and ββm. Not all of
788:
In the case of multiple mirror planes and/or axes of rotation, two symmetry groups are of the same symmetry type if and only if there is a rotation mapping the whole structure of the first symmetry group to that of the second. (In fact there will be more than one such rotation, but not an infinite
887:
about an axis through the origin, and those with additionally reflection in the planes through the axis, and/or reflection in the plane through the origin, perpendicular to the axis. Those with reflection in the planes through the axis, with or without reflection in the plane through the origin
8907:
between subgroups of Spin(3) and subgroups of SO(3) (rotational point groups): the image of a subgroup of Spin(3) is a rotational point group, and the preimage of a point group is a subgroup of Spin(3). (Note that Spin(3) has alternative descriptions as the special unitary group
721:
When comparing the symmetry type of two objects, the origin is chosen for each separately, i.e. they need not have the same center. Moreover, two objects are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of O(3) (two subgroups
2452:"Equal" is meant here as the same up to conjugacy in space. This is stronger than "up to algebraic isomorphism". For example, there are three different groups of order two in the first sense, but there is only one in the second sense. Similarly, e.g.
2153:), which has vertical mirror planes containing the main rotation axis, but instead of having a horizontal mirror plane, it has an isometry that combines a reflection in the horizontal plane and a rotation by an angle 180Β°/
1892:
The terms horizontal (h) and vertical (v), and the corresponding subscripts, refer to the additional mirror plane, that can be parallel to the rotation axis (vertical) or perpendicular to the rotation axis (horizontal).
6876:. This total number is one of the characteristics helping to distinguish the various abstract group types, while their isometry type helps to distinguish the various isometry groups of the same abstract group.
630:, also called rotation-reflection: rotation about an axis by an angle ΞΈ, combined with reflection in the plane through the origin perpendicular to the axis. Rotation-reflection by ΞΈ = 360Β°/
5004:
offers a bracketed notation equivalent to the
Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups. In Schoenflies notation, the reflective point groups in 3D are
6385:
5658:
etc. are the rotation groups of plane regular polygons embedded in three-dimensional space, and such a figure may be considered as a degenerate regular prism. Therefore, it is also called a
9213:
514:
944:, which have multiple 3-or-more-fold rotation axes; these groups can also be characterized as point groups having multiple 3-fold rotation axes. The possible combinations are:
6073:
Thus there is a 1-to-1 correspondence between all direct isometries and all indirect isometries, through inversion. Also there is a 1-to-1 correspondence between all groups
2121:
includes reflections, which can also be viewed as flipping over flat objects without distinction of frontside and backside; but in 3D, the two operations are distinguished:
1099:. The latter three are not only conveniently related to its properties, but also to the order of the group. The orbifold notation is a unified notation, also applicable for
4347:
4337:
4327:
4036:
4026:
4016:
3861:
3851:
3489:
3479:
3469:
3310:
3300:
3290:
3185:
3175:
3122:
3112:
1685:
1675:
1585:
1575:
1565:
1400:
1390:
1311:
1301:
1286:
1211:
1201:
275:
265:
255:
229:
219:
209:
183:
173:
163:
127:
117:
107:
81:
71:
45:
6447:
4166:
3368:
3305:
3244:
3180:
3117:
5555:
5519:
5509:
5479:
5469:
5430:
5420:
5410:
5374:
5364:
5354:
5324:
5314:
5276:
5266:
5256:
5226:
5216:
5206:
5176:
5166:
5156:
4954:
4944:
4930:
4920:
4906:
4861:
4851:
4841:
4823:
4813:
4803:
4789:
4779:
4734:
4724:
4714:
4700:
4690:
4680:
4666:
4656:
4646:
4496:
4486:
4476:
4207:
4197:
4187:
3871:
3663:
3653:
3643:
3373:
3363:
3353:
3249:
3239:
3165:
1780:
1770:
1760:
1660:
1491:
1481:
1410:
6710:-fold rotations about that axis is a normal subgroup of the group of all rotations about that axis. Since any subgroup of index two is normal, the group of rotations (
3295:
1680:
1665:
1580:
1306:
1291:
5514:
5474:
5425:
5415:
5369:
5359:
5319:
5271:
5261:
5221:
5211:
5171:
5161:
4949:
4925:
4856:
4846:
4818:
4808:
4784:
4729:
4719:
4695:
4685:
4661:
4651:
4491:
4481:
4342:
4332:
4202:
4192:
4031:
4021:
3866:
3856:
3658:
3648:
3484:
3474:
3358:
3170:
1775:
1765:
1670:
1570:
1486:
1405:
1395:
1296:
1206:
270:
260:
224:
214:
178:
168:
122:
112:
76:
7660:
9131:
9098:
9065:
9021:
8981:
920:
Symmetries in 3D that leave the origin fixed are fully characterized by symmetries on a sphere centered at the origin. For finite 3D point groups, see also
6879:
Within the possibilities of isometry groups in 3D, there are infinitely many abstract group types with 0, 1 and 3 elements of order 2, there are two with 4
912:
these are possible for physical objects, for example objects with ββ symmetry also have ββm symmetry. See below for other designations and more details.
6961:(Schoenflies notation) generated by a rotation by an angle 180Β°/n about an axis, combined with a reflection in the plane perpendicular to the axis. For
9641:
9485:
9532:
9504:
31:
9511:
812:
9656:
6034:
The rotation group SO(3) is a subgroup of O(3), the full point rotation group of the 3D Euclidean space. Correspondingly, O(3) is the
9518:
9291:
6786:. The two groups are obtained from it by changing 2-fold rotational symmetry to 4-fold, and adding 5-fold symmetry, respectively.
527:
6789:
There are two crystallographic point groups with the property that no crystallographic point group has it as proper subgroup:
485:
The point groups that are generated purely by a finite set of reflection mirror planes passing through the same point are the
9500:
9367:
5295:
5076:
4073:
axes, and additionally there are two-fold rotation axes through the midpoints of the edges of the cube, giving rise to three
1615:
93:
3749:
subgroups. This group has six mirror planes, each containing two edges of the cube or one edge of the tetrahedron, a single
741:
9662:
7023:
6325:
1052:=1 covers the cases of no rotational symmetry at all. There are four series with no other axes of rotational symmetry (see
1009:
9392:
7454:
includes reflections, which can also be viewed as flipping over flat objects without distinction of front- and backside.
5450:
57:
2360:
All symmetry groups in the 7 infinite series are different, except for the following four pairs of mutually equal ones:
474:, the full rotation group of the 3D space. The rotation group of a bounded object is equal to its full symmetry group
9592:
9551:
9342:
9208:
921:
367:
4308:
1111:
restricted to 1, 2, 3, 4, and 6; removing crystallographic restriction allows any positive integer. The series are:
3828:
1985:= 6 for each of the 7 infinite families of point groups. The symmetry group of each pattern is the indicated group.
845:. There are also non-abelian groups generated by rotations around different axes. These are usually (generically)
9223:
6855:
The column "# of order 2 elements" in the following tables shows the total number of isometry subgroups of types
3890:
899:
There are seven continuous groups which are all in a sense limits of the finite isometry groups. These so called
3452:(full, and abbreviated if different) and the order (number of elements) of the symmetry group. The groups are:
457:
453:
9567:
9489:
9218:
8876:
For a polyhedron this surface in the fundamental domain can be part of an arbitrary plane. For example, in the
4110:
3803:
2432:: group of order 4 with a reflection in a plane and a 180Β° rotation through a line perpendicular to that plane.
2308:
is generated by the combination of a reflection in the horizontal plane and a rotation by an angle 360Β°/n. For
1061:
9525:
6765:
There are two discrete point groups with the property that no discrete point group has it as proper subgroup:
3612:, and the octahedral symmetries. The elements of the group correspond 1-to-2 to the rotations given by the 24
9683:
9360:
Modern
Crystallography, Vol. 1. Fundamentals of Crystals. Symmetry, and Methods of Structural Crystallography
7000:-fold rotation axis, and a perpendicular plane of reflection. It is generated by a rotation by an angle 360Β°/
2235:
are noteworthy in that there is no special rotation axis. Rather, there are three perpendicular 2-fold axes.
2109:, which still has the 2-fold rotation axes perpendicular to the primary rotation axis, but no mirror planes.
938:, or equivalently, those on a finite cylinder. They are sometimes called the axial or prismatic point groups.
9651:
5051:
mirror planes. Coxeter groups having fewer than 3 generators have degenerate spherical triangle domains, as
934:, which have at most one more-than-2-fold rotation axis; they are the finite symmetry groups on an infinite
3449:
3065:
1117:
1076:
1017:
429:
of geometric objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible
3575:
axes, through the centers of the cube's faces, or the midpoints of the tetrahedron's edges. This group is
1969:
349:
5824:. In other words, the chiral objects are those with their symmetry group in the list of rotation groups.
4410:
9292:"Sur la symétrie dans les phénomènes physiques, symétrie d'un champ électrique et d'un champ magnétique"
8899:
in 3 dimensions. (This is the only connected cover of SO(3), since Spin(3) is simply connected.) By the
3700:
437:(finite) 3D object have one or more common fixed points. We follow the usual convention by choosing the
9238:
9198:), and thus a tessellation of projective space or lens space yields a distinct notion of polyhedron.
8877:
4094:
because its elements are in 1-to-1 correspondence to the 24 permutations of the 3-fold axes, as with
842:
661:
592:
463:
303:
9668:
9233:
9134:
9068:
8947:
6035:
3620:
7545:. However, there are three more infinite series of symmetry groups with this abstract group type:
6426:
2803:
around a direction in the plane perpendicular to the axis. Its elements are the elements of group
9478:
9101:
7520:
5821:
5820:
The rotation group of an object is equal to its full symmetry group if and only if the object is
4147:
1016:: 27 from the 7 infinite series, and 5 of the 7 others. Together, these make up the 32 so-called
537:
479:
9294:[On symmetry in physical phenomena, symmetry of an electric field and a magnetic field]
6941:. However, there are two more infinite series of symmetry groups with this abstract group type:
2416:: group of order 4 with a reflection in a plane and a 180Β° rotation through a line in that plane
9156:
For point groups that reverse orientation, the situation is more complicated, as there are two
8605:
The remaining seven are, with bolding of the 5 crystallographic point groups (see also above):
6123:
4997:
3533:
3514:
2010:. In addition to this, one may add a mirror plane perpendicular to the axis, giving the group
1013:
841:
numbers of degrees around the circle illustrates a point group requiring an infinite number of
5828:
4117:
corresponds to the set of permutations of these four objects. It is the rotation group of the
2723:
in a direction in the plane perpendicular to the axis. Its elements are the elements of group
2079:
axes of rotation through 180Β°, so the group is no longer uniaxial. This new group of order 4
1084:
9688:
9669:
The
Geometry Center: 10.1 Formulas for Symmetries in Cartesian Coordinates (three dimensions)
9575:
9024:
8934:, represented as β¨l,n,mβ©, and is called by the same name as its point group, with the prefix
5845:
3721:
3035:
543:
that leave the origin fixed, forming the group O(3). These operations can be categorized as:
395:
in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a
7645:
6706:
a normal subgroup of O(2) and SO(2). Accordingly, in 3D, for every axis the cyclic group of
9258:
9187:
9109:
9076:
9043:
8999:
8959:
8881:
8852:
7484:
6454:
6319:
2, then there is a corresponding group that contains indirect isometries but no inversion:
6050:
5072:
5064:
4595:
4360:
4141:
3502:
3070:
2293:
1919:
1142:
1122:
989:
948:
889:
648:
579:
486:
471:
438:
241:
149:
8:
9180:
8984:
7022:
Thus we have, with bolding of the 10 cyclic crystallographic point groups, for which the
6316:
5068:
4379:
4049:
3688:
1963:
1935:
1041:
972:
935:
408:
195:
9384:
561:
Rotation about an axis through the origin by an angle ΞΈ. Rotation by ΞΈ = 360Β°/
9623:
9603:
9409:
9334:
9311:
9263:
9150:
9142:
8866:
7567:
5589:
5052:
5040:
4451:
4137:
4134:
3616:
3613:
3418:
symmetry because they have more than one rotation axis of order greater than 2. Here,
2290:
2047:
1523:
1088:
883:
is the corresponding rotation group. The other infinite isometry groups consist of all
876:
533:
8836:
7523:
by e.g. an identical chiral marking on every face, or some modification in the shape.
4434:
on 5 letters, since its elements correspond 1-to-1 with even permutations of the five
4405:(rotational symmetries like prisms and antiprisms). It also contains five versions of
2075:
If both horizontal and vertical reflection planes are added, their intersections give
9588:
9363:
9338:
8943:
8904:
6950:
5858:
5853:
5836:
4431:
3902:
3587:
3437:
3075:
2354:
1127:
1092:
1057:
1053:
857:
853:
830:
627:
547:
The direct (orientation-preserving) symmetry operations, which form the group SO(3):
505:
416:
9627:
9413:
9615:
9401:
9330:
9307:
9191:
9172:
8939:
8895:
The map Spin(3) β SO(3) is the double cover of the rotation group by the
7492:
7457:
However, in 3D the two operations are distinguished: the symmetry group denoted by
6833:
6039:
5832:
5602:
5138:
5108:
5060:
5001:
4609:
3441:
3436:
denotes an axis of improper rotation through the same. On successive lines are the
3080:
2172:
1814:
1132:
1096:
490:
404:
135:
4981:
8900:
8693:
8679:
6465:
that contain indirect isometries but no inversion we can obtain a rotation group
4888:
4761:
4743:
4375:
4300:
4118:
3973:
3778:
3595:
3445:
1147:
1100:
1080:
1064:
extended with an axial coordinate and reflections in it. They are related to the
838:
420:
309:
9443:
9160:, so there are two possible binary groups corresponding to a given point group.
7677:
Thus we have, with bolding of the 12 crystallographic point groups, and writing
5579:
The rotation groups, i.e. the finite subgroups of SO(3), are: the cyclic groups
4963:
4752:
4440:
symmetries (or the five tetrahedra just mentioned). Representing rotations with
2006:(also applicable in 2D), which are generated by a single rotation of angle 360Β°/
837:
by adding more rotations around the same axis. The set of points on a circle at
9606:; Huson, Daniel H. (2002), "The Orbifold Notation for Two-Dimensional Groups",
9248:
9176:
7527:
7443:
6902:
6845:
5120:
5088:
4972:
4870:
3781:
on 4 letters, because there is a 1-to-1 correspondence between the elements of
3023:
Groups with continuous axial rotations are designated by putting β in place of
2570:
is a reflection in the direction of the axis. Its elements are the elements of
2279:
2094:
1500:
1347:
475:
426:
392:
9619:
9405:
8887:
Also the surface in the fundamental domain may be composed of multiple faces.
7519:. The group is also the full symmetry group of such objects after making them
4879:
4525:
This is the symmetry group of the icosahedron and the dodecahedron. The group
784:
if both have 3-fold rotational symmetry, but with respect to a different axis.
9677:
8623:
8460:
8205:
7709:
7516:
7044:
4993:
2206:
834:
826:
509:
8843:
8442:
Thus we have, with bolding of the 3 dihedral crystallographic point groups:
7437:
3933:
axes, and there is inversion symmetry. The two-fold axes give rise to three
88:
52:
9287:
8856:
6923:
6683:
5638:
5129:
4371:
3983:
3897:
2275:
1990:
1896:
The simplest nontrivial axial groups are equivalent to the abstract group Z
1220:
1137:
1104:
1065:
908:
825:" (meaning that it is generated by one element β not to be confused with a
822:
236:
190:
144:
26:
8187:
Thus we have, with bolding of the 2 cyclic crystallographic point groups:
6896:
6024:
781:
if both have mirror symmetry, but with respect to a different mirror plane
9253:
9243:
8913:
8870:
6302:
5634:
5622:
5115:
4367:
3565:
1044:
about an axis, i.e. symmetry with respect to a rotation by an angle 360Β°/
798:
434:
386:
9385:"Three-dimensional finite point groups and the symmetry of beaded beads"
8730:
8372:
888:
perpendicular to the axis, are the symmetry groups for the two types of
813:
Rotational symmetry Β§ Rotational symmetry with respect to any angle
9195:
8896:
8525:
8490:
8323:
8290:
8239:
5804:) has as its rotation group the corresponding one without a subscript:
5626:
5101:
4441:
4420:
4304:
4126:
4122:
3576:
3415:
846:
692:
528:
Euclidean group Β§ Overview of isometries in up to three dimensions
8130:
8035:
7992:
7956:
7890:
7425:
6825:
Below the groups explained above are arranged by abstract group type.
688:), so these operations are often considered to be improper rotations.
9157:
8917:
8652:
7854:
7805:
7765:
7590:
7508:
7500:
7374:
7345:
7309:
7280:
7223:
7194:
7154:
7123:
7074:
6891:β₯ 8 ). There is never a positive even number of elements of order 2.
5606:
4105:
symmetry under one of the 3-fold axes gives rise under the action of
3795:
symmetry under one of the 3-fold axes gives rise under the action of
2198:
2180:
2069:
1716:
1012:, only a limited number of point groups are compatible with discrete
497:
9648:=5) of the 7 infinite series and 5 of the 7 separate 3D point groups
9467:
9325:
Shubnikov, A.V. (1988). "On the Works of Pierre Curie on
Symmetry".
5000:
and represent a set of mirrors that intersect at one central point.
863:
9228:
8921:
6910:
6691:
5660:
2271:
884:
849:. They will be infinite unless the rotations are specially chosen.
501:
430:
412:
400:
381:
674:
Inversion is a special case of rotation-reflection (i =
9425:
9214:
List of character tables for chemically important 3D point groups
4454:
9582:
6803:. Their maximal common subgroups, depending on orientation, are
4992:
The reflective point groups in three dimensions are also called
2312:
odd this is equal to the group generated by the two separately,
2242:
is a subgroup of all the polyhedral symmetries (see below), and
2218:
is the symmetry group of a partially rotated ("twisted") prism.
927:
Up to conjugacy, the set of finite 3D point groups consists of:
860:
of O(3). We now discuss topologically closed subgroups of O(3).
695:
is sometimes added to the symbol to indicate an operator, as in
466:. For a bounded object, the proper symmetry group is called its
5056:
4235:, but with mirror planes, comprising both the mirror planes of
1808:
1710:
1517:
1438:
1028:
The infinite series of axial or prismatic groups have an index
868:
396:
3813:
corresponds to the set of permutations of these four objects.
3788:
and the 24 permutations of the four 3-fold axes. An object of
1609:
1336:
1236:
1075:
The following table lists several notations for point groups:
931:
415:
that leave the origin fixed, or correspondingly, the group of
9362:(2nd enlarged ed.). Springer-Verlag Berlin. p. 93.
9183:
or other representations they may stabilize other polyhedra.
8909:
6820:
4559:
are both normal subgroups. The group contains 10 versions of
3742:
axes, whereas the four three-fold axes now give rise to four
2289:
are in 1-to-2 correspondence with the rotations given by the
880:
444:
The symmetry group of an object is sometimes also called its
1068:; they can be interpreted as frieze-group patterns repeated
1023:
521:
5630:
3561:
1977:
3414:
The remaining point groups are said to be of very high or
9585:
Generators and
Relations for Discrete Groups, 4th edition
7438:
Symmetry groups in 3D that are dihedral as abstract group
3564:(red cube in images), or through one vertex of a regular
1989:
The second of these is the first of the uniaxial groups (
1056:) and three with additional axes of 2-fold symmetry (see
777:
For example, two 3D objects have the same symmetry type:
496:
The point groups in three dimensions are heavily used in
470:. It is the intersection of its full symmetry group with
9642:
Graphic overview of the 32 crystallographic point groups
4620:, any physical object having K symmetry will also have K
4299:
are normal subgroups), and is the symmetry group of the
6897:
Symmetry groups in 3D that are cyclic as abstract group
6738:) reflection planes through its axis and in the group (
6025:
Correspondence between rotation groups and other groups
5664:(Greek: solid with two faces), which explains the name
6303:
Groups containing indirect isometries but no inversion
9112:
9079:
9046:
9002:
8962:
7648:
6429:
6380:{\displaystyle M=L\cup ((H\setminus L)\times \{-I\})}
6328:
4256:
subgroups. The three perpendicular four-fold axes of
4150:
4080:
subgroups. The three-fold axes now give rise to four
3919:, with mirror planes parallel to the cube faces. The
3560:
axes, each through two vertices of a circumscribing
1981:
Patterns on a cylindrical band illustrating the case
941:
8859:, with right spherical triangle fundamental domains
6852:(of order 12), and 10 of the 14 groups of order 16.
5079:. The number of mirrors for an irreducible group is
2497:
around the axis. Its elements are E (the identity),
2028:
mirror planes containing the axis, giving the group
624:
Reflection in a plane through the origin, denoted Ο.
9492:. Unsourced material may be challenged and removed.
6883:+ 1 elements of order 2, and there are three with 4
4586:The continuous groups related to these groups are:
3409:
833:of turns about an axis. We may create non-cyclical
452:, the intersection of its full symmetry group with
9657:Simplest Canonical Polyhedra of Each Symmetry Type
9125:
9092:
9059:
9015:
8975:
7654:
7018:is used; it is generated by reflection in a plane.
7004:about the axis, combined with the reflection. For
6441:
6379:
4160:
9432:, Β§12.6 The number of reflections, equation 12.61
8930:The preimage of a finite point group is called a
3059:symmetry, but vector fields can have the others.
2442:is the group of order 2 with a single inversion (
2042:. The latter is the symmetry group for a regular
852:All the infinite groups mentioned so far are not
607:The indirect (orientation-reversing) operations:
9675:
9583:Coxeter, H. S. M. & Moser, W. O. J. (1980).
9149:by the action of a binary polyhedral group is a
6757:) a reflection plane perpendicular to its axis.
4573:(symmetries like antiprisms), and 5 versions of
4267:subgroups, while the six two-fold axes give six
2357:without containing the corresponding rotations.
1032:, which can be any integer; in each series, the
932:Β§ The seven infinite series of axial groups
9327:Crystal Symmetries: Shubnikov Centennial papers
8409:this is already covered above, so we have here
3427:denotes an axis of rotation through 360Β°/n and
2133:There is one more group in this family, called
318:<math>... \smallsetminus ...</math>
6081:of isometries in O(3) that contain inversion:
6029:
4457:. As before, this is a 1-to-2 correspondence.
4168:. As before, this is a 1-to-2 correspondence.
2493:, which corresponds to a rotation by angle 2Ο/
2400:: group of order 2 with a single 180Β° rotation
6077:of direct isometries in SO(3) and all groups
4617:
2377:: group of order 2 with a single reflection (
602:, is a special case of the rotation operator.
500:, especially to describe the symmetries of a
9644:β form the first parts (apart from skipping
9599:6.5 The binary polyhedral groups, p. 68
9382:
9282:
9280:
6371:
6362:
4627:
4144:of squared norm 2 normalized by dividing by
4087:subgroups. This group is also isomorphic to
9602:
9357:
8942:(l,m,n). For instance, the preimage of the
8890:
5043:domains on the surface of a sphere. A rank
3972:are both normal subgroups), and not to the
3724:. This group has the same rotation axes as
2130:contains "flipping over", not reflections.
806:
512:, and in this context they are also called
9570:(1974), "7 The Binary Polyhedral Groups",
8828:
7503:with regular base, and also of a regular,
6821:The groups arranged by abstract group type
6472:For finite groups, the correspondence is:
6133:For finite groups, the correspondence is:
2472:The groups may be constructed as follows:
314:<math>... \setminus ...</math>
284:Groups of point isometries in 3 dimensions
9552:Learn how and when to remove this message
9324:
9277:
8880:one full face is a fundamental domain of
4633:Fundamental domains of 3D Coxeter groups
4612:, all possible rotations and reflections.
4231:This group has the same rotation axes as
3915:This group has the same rotation axes as
3829:the isometries of the regular tetrahedron
2981:. Its elements are the elements of group
2869:. Its elements are the elements of group
2646:. Its elements are the elements of group
2524:, corresponding to rotation angles 0, 2Ο/
1024:The seven infinite series of axial groups
915:
522:3D isometries that leave the origin fixed
368:Learn how and when to remove this message
5764:An object having a polyhedral symmetry (
4249:. The three-fold axes give rise to four
3896:
3457:
1976:
971:Four 3-fold axes and three 4-fold axes (
875:The whole O(3) is the symmetry group of
862:
595:). The identity operation, also written
9574:, Cambridge University Press, pp.
9566:
8938:, with double the order of the related
6469:by inverting the indirect isometries.
4274:subgroups. This group is isomorphic to
2249:is a subgroup of the polyhedral groups
2050:. A typical object with symmetry group
942:Β§ The seven remaining point groups
9676:
9652:Overview of properties of point groups
6975:is used; it is generated by inversion.
6828:The smallest abstract groups that are
4000:, it does not apply to a tetrahedron.
3720:This group is the symmetry group of a
3706:A four-fold rotation-reflection axis (
2484:. Generated by an element also called
2193:is the symmetry group for a "regular"
2167:is the symmetry group for a "regular"
9286:
8824:
8424:, which is of abstract group type Dih
6760:
4113:consisting of four such objects, and
3806:consisting of four such objects, and
988:Ten 3-fold axes and six 5-fold axes (
9614:(3), Springer Netherlands: 247β257,
9490:adding citations to reliable sources
9461:
9329:. Pergamon Press. pp. 357β364.
4391:. The group contains 10 versions of
3591:
2093:. Its subgroup of rotations is the
1889:are chiral, the others are achiral.
1010:crystallographic restriction theorem
308:In particular, it has problems with
286:
9393:Journal of Mathematics and the Arts
6779:. Their largest common subgroup is
6677:
6461:. Conversely, for all point groups
6049:(where inversion is denoted by its
5644:In particular, the dihedral groups
5588:(the rotation group of a canonical
3594:for a regular tetrahedron. It is a
1107:. The crystallographic groups have
681:), as is reflection (Ο =
550:The identity operation, denoted by
419:. O(3) itself is a subgroup of the
13:
9501:"Point groups in three dimensions"
9441:
9335:10.1016/B978-0-08-037014-9.50007-8
8953:The binary polyhedral groups are:
7585:, the symmetry group of a regular
7562:, the symmetry group of a regular
6887:+ 3 elements of order 2 (for each
6832:any symmetry group in 3D, are the
5574:
4382:2 in the full group of symmetries
4366:This is the rotation group of the
2462:is algebraically isomorphic with Z
2339:even it is distinct, and of order
14:
9700:
9635:
9383:Fisher, G.L.; Mellor, B. (2007),
9209:List of spherical symmetry groups
7470:2-fold axes perpendicular to the
6433:
6350:
5601:(the rotation group of a uniform
5039:The mirror planes bound a set of
5024:, and the full polyhedral groups
9663:Point Groups and Crystal Systems
9466:
9312:10.1051/jphystap:018940030039300
9163:Note that this is a covering of
8842:
8835:
8729:
8692:
8651:
8524:
8489:
8371:
8322:
8289:
8238:
8129:
8034:
7991:
7955:
7889:
7853:
7804:
7764:
7424:
7373:
7344:
7308:
7279:
7222:
7193:
7153:
7122:
7073:
6307:If a group of direct isometries
5723:An object having symmetry group
5672:An object having symmetry group
5553:
5517:
5512:
5507:
5477:
5472:
5467:
5428:
5423:
5418:
5413:
5408:
5372:
5367:
5362:
5357:
5352:
5322:
5317:
5312:
5274:
5269:
5264:
5259:
5254:
5224:
5219:
5214:
5209:
5204:
5174:
5169:
5164:
5159:
5154:
4980:
4971:
4962:
4952:
4947:
4942:
4928:
4923:
4918:
4904:
4887:
4878:
4869:
4859:
4854:
4849:
4844:
4839:
4821:
4816:
4811:
4806:
4801:
4787:
4782:
4777:
4760:
4751:
4742:
4732:
4727:
4722:
4717:
4712:
4698:
4693:
4688:
4683:
4678:
4664:
4659:
4654:
4649:
4644:
4494:
4489:
4484:
4479:
4474:
4345:
4340:
4335:
4330:
4325:
4205:
4200:
4195:
4190:
4185:
4034:
4029:
4024:
4019:
4014:
3869:
3864:
3859:
3854:
3849:
3699:
3694:A mirror plane of a tetrahedron.
3687:
3661:
3656:
3651:
3646:
3641:
3532:
3520:The three-fold rotational axes (
3513:
3487:
3482:
3477:
3472:
3467:
3410:The seven remaining point groups
3371:
3366:
3361:
3356:
3351:
3308:
3303:
3298:
3293:
3288:
3247:
3242:
3237:
3183:
3178:
3173:
3168:
3163:
3120:
3115:
3110:
1807:
1778:
1773:
1768:
1763:
1758:
1709:
1683:
1678:
1673:
1668:
1663:
1658:
1608:
1583:
1578:
1573:
1568:
1563:
1516:
1489:
1484:
1479:
1437:
1408:
1403:
1398:
1393:
1388:
1335:
1309:
1304:
1299:
1294:
1289:
1284:
1235:
1209:
1204:
1199:
829:) generated by a rotation by an
291:
273:
268:
263:
258:
253:
235:
227:
222:
217:
212:
207:
189:
181:
176:
171:
166:
161:
143:
125:
120:
115:
110:
105:
87:
79:
74:
69:
51:
43:
25:
19:point groups in three dimensions
9477:needs additional citations for
9224:Point groups in four dimensions
6719:) is normal both in the group (
6423:by inverting the isometries in
5839:), the rotation subgroups are:
2990:and the additional elements of
2878:and the additional elements of
532:The symmetry group operations (
9435:
9419:
9376:
9351:
9318:
9219:Point groups in two dimensions
6702:is for every positive integer
6374:
6356:
6344:
6341:
4631:
4125:. Representing rotations with
3539:The two-fold rotational axes (
1528:in biology, biradial symmetry
1062:point groups in two dimensions
462:, i.e., isometries preserving
300:This article needs editing to
1:
9665:, by Yi-Shu Wei, pp. 4β6
9587:. New York: Springer-Verlag.
9457:
8851:The planes of reflection for
8397:is of abstract group type Dih
7631:Note the following property:
7474:-fold axis, not reflections.
6922:; its abstract group type is
2326:, and therefore the notation
1060:). They can be understood as
1018:crystallographic point groups
9270:
9141:These are classified by the
8991: + 1)-gon, order 2
8791:
8768:
8736:
8699:
8658:
8628:
8570:
8531:
8496:
8465:
8329:
8296:
8245:
8210:
8180:, already covered above, so
8072:
8041:
7998:
7962:
7896:
7860:
7811:
7771:
7714:
7623:, already covered above, so
7380:
7351:
7315:
7286:
7229:
7200:
7160:
7129:
7080:
7049:
7024:crystallographic restriction
6442:{\displaystyle H\setminus L}
6000:
5972:
5951:
5912:
5863:
5542:
5495:
5455:
5390:
5340:
5300:
5243:
5193:
5143:
4960:
4867:
4740:
4461:
4315:
4172:
4004:
3836:
3628:
3329:
3269:
3218:
3141:
3091:
3034:here is not the same as the
2335:is not needed; however, for
2270:occurs in molecules such as
1722:
1620:
1531:
1446:
1353:
1248:
1175:
1158:
947:Four 3-fold axes (the three
716:
15:
7:
9358:Vainshtein., B. K. (1994).
9201:
8161:is of abstract group type Z
7536:, which is also denoted by
7526:The abstract group type is
6932:, which is also denoted by
6030:Groups containing inversion
5609:), and the rotation groups
4411:Compound of five tetrahedra
4161:{\displaystyle {\sqrt {2}}}
3993:. In spite of being called
3038:(also sometimes designated
2201:, and also for a "regular"
1036:th symmetry group contains
10:
9705:
8830:Disdyakis triacontahedron
6453:is, when considered as an
6154:Group containing inversion
5099:
4309:the isometries of the cube
3590:on 4 elements, and is the
1245:-fold rotational symmetry
1115:
892:. Any 3D shape (subset of
810:
525:
9572:Regular Complex Polytopes
9406:10.1080/17513470701416264
8878:disdyakis triacontahedron
8850:
6748:) obtained by adding to (
6729:) obtained by adding to (
6574:
6563:
6521:
6510:
6204:
6162:
5538:
5449:
5294:
5137:
5107:
4628:Reflective Coxeter groups
4598:, all possible rotations.
4521:full icosahedral symmetry
3678:full tetrahedral symmetry
3079:
2175:and also for a "regular"
1176:
1154:
1151:
1146:
1141:
1131:
1126:
1121:
1116:
1072:times around a cylinder.
922:spherical symmetry groups
634:for any positive integer
617:. The matrix notation is
565:for any positive integer
134:
9234:Euclidean plane isometry
9190:β the sphere does cover
9135:binary icosahedral group
9069:binary tetrahedral group
8948:binary icosahedral group
8891:Binary polyhedral groups
8855:intersect the sphere on
8142:
4618:infinite isometry groups
4227:full octahedral symmetry
3820:is a normal subgroup of
3621:binary tetrahedral group
3450:HermannβMauguin notation
2964:. Generated by elements
2851:. Generated by elements
2353:it contains a number of
1077:HermannβMauguin notation
819:infinite isometry groups
807:Infinite isometry groups
610:Inversion, denoted i or
536:) are the isometries of
456:, which consists of all
423:E(3) of all isometries.
302:comply with Knowledge's
9620:10.1023/A:1015851621002
9186:This is in contrast to
9102:binary octahedral group
8932:binary polyhedral group
7495:with regular base, and
5592:), the dihedral groups
5087:is the Coxeter group's
4616:As noted above for the
2795:and 180Β° rotation U = Ο
2786:. Generated by element
2710:. Generated by element
2633:. Generated by element
2552:. Generated by element
1873:(including the trivial
795:chiral for 11 pairs of
554:or the identity matrix
538:three-dimensional space
9444:"Du Val Singularities"
9145:, and the quotient of
9127:
9094:
9061:
9017:
8977:
8916:. Topologically, this
8869:of a point group is a
7656:
7655:{\displaystyle \cong }
6443:
6381:
5850:Reflection/rotational
5095:is the dimension (3).
4998:Coxeter-Dynkin diagram
4996:and can be given by a
4450:is made up of the 120
4162:
3913:
1986:
1014:translational symmetry
990:icosahedral symmetries
949:tetrahedral symmetries
916:Finite isometry groups
872:
515:molecular point groups
433:. All isometries of a
9128:
9126:{\displaystyle E_{8}}
9095:
9093:{\displaystyle E_{7}}
9062:
9060:{\displaystyle E_{6}}
9025:binary dihedral group
9018:
9016:{\displaystyle D_{n}}
8978:
8976:{\displaystyle A_{n}}
8922:3-dimensional sphere
8620:# of order 2 elements
8457:# of order 2 elements
8202:# of order 2 elements
7706:# of order 2 elements
7657:
7041:# of order 2 elements
6444:
6382:
6114:where the isometry (
4163:
4142:Lipschitz quaternions
4133:is made up of the 24
3900:
3891:pyritohedral symmetry
3036:infinite cyclic group
2294:Lipschitz quaternions
1980:
973:octahedral symmetries
905:Curie limiting groups
901:limiting point groups
866:
858:topological subgroups
487:finite Coxeter groups
450:proper symmetry group
32:Involutional symmetry
9684:Euclidean symmetries
9608:Structural Chemistry
9486:improve this article
9259:List of small groups
9188:projective polyhedra
9181:spin representations
9137:, β¨2,3,5β©, order 120
9110:
9077:
9044:
9000:
8960:
8912:and as the group of
8882:icosahedral symmetry
8853:icosahedral symmetry
7646:
6427:
6326:
5073:icosahedral symmetry
5065:tetrahedral symmetry
4361:icosahedral symmetry
4148:
3503:tetrahedral symmetry
3027:. Note however that
890:cylindrical symmetry
821:; for example, the "
803:with a screw axis.)
649:Schoenflies notation
580:Schoenflies notation
448:, as opposed to its
346:for set subtraction.
242:Icosahedral symmetry
150:Tetrahedral symmetry
9604:Conway, John Horton
9300:Journal de Physique
9104:, β¨2,3,4β©, order 48
9071:, β¨2,3,3β©, order 24
8985:binary cyclic group
8831:
6949:there is the group
6502:indirect isometries
5829:SchΓΆnflies notation
5752:has rotation group
5711:has rotation group
5069:octahedral symmetry
4634:
4398:and 6 versions of
4138:Hurwitz quaternions
4055:This group is like
4050:octahedral symmetry
3722:regular tetrahedron
3713:) of a tetrahedron.
3617:Hurwitz quaternions
3546:) of a tetrahedron.
3527:) of a tetrahedron.
2812:, with elements U,
1964:reflection symmetry
1936:rotational symmetry
1085:SchΓΆnflies notation
1042:rotational symmetry
534:symmetry operations
446:full symmetry group
417:orthogonal matrices
389:in three dimensions
350:improve the content
196:Octahedral symmetry
21:
9264:Molecular symmetry
9175:, and thus has no
9167:not a covering of
9151:Du Val singularity
9143:ADE classification
9123:
9090:
9057:
9013:
8973:
8867:fundamental domain
8829:
8825:Fundamental domain
7684:as the equivalent
7652:
7511:and of a regular,
6844:(of order 9), the
6761:Maximal symmetries
6439:
6377:
5047:Coxeter group has
5041:spherical triangle
4632:
4158:
3914:
2355:improper rotations
2282:. The elements of
1987:
1970:bilateral symmetry
1089:molecular symmetry
1087:(used to describe
877:spherical symmetry
873:
871:has O(3) symmetry.
746:, if there exists
506:molecular orbitals
16:
9568:Coxeter, H. S. M.
9562:
9561:
9554:
9536:
9430:Regular polytopes
9369:978-3-642-08153-8
8944:icosahedral group
8905:Galois connection
8863:
8862:
8822:
8821:
8600:
8599:
8379:
8378:
8137:
8136:
7432:
7431:
6909:-fold rotational
6675:
6674:
6500:Group containing
6419:is obtained from
6300:
6299:
6061:O(3) = SO(3) Γ {
6038:of SO(3) and the
6022:
6021:
5854:Improper rotation
5837:orbifold notation
5572:
5571:
5139:Polyhedral groups
5077:dihedral symmetry
5063:these groups are
4990:
4989:
4584:
4583:
4566:, 6 versions of
4534:is isomorphic to
4432:alternating group
4156:
3947:is isomorphic to
3770:is isomorphic to
3588:alternating group
3438:orbifold notation
3407:
3406:
2732:, with elements Ο
2655:, with elements Ο
2536: β 1)Ο/
2038:, also of order 2
1821:
1820:
1616:Dihedral symmetry
1093:orbifold notation
1058:dihedral symmetry
1054:cyclic symmetries
1008:According to the
831:irrational number
628:Improper rotation
489:, represented by
459:direct isometries
378:
377:
370:
282:
281:
94:Dihedral symmetry
9696:
9630:
9598:
9578:
9557:
9550:
9546:
9543:
9537:
9535:
9494:
9470:
9462:
9451:
9450:
9448:
9439:
9433:
9423:
9417:
9416:
9389:
9380:
9374:
9373:
9355:
9349:
9348:
9322:
9316:
9315:
9297:
9284:
9192:projective space
9173:simply connected
9171:β the sphere is
9132:
9130:
9129:
9124:
9122:
9121:
9099:
9097:
9096:
9091:
9089:
9088:
9066:
9064:
9063:
9058:
9056:
9055:
9022:
9020:
9019:
9014:
9012:
9011:
8982:
8980:
8979:
8974:
8972:
8971:
8940:polyhedral group
8914:unit quaternions
8846:
8839:
8832:
8733:
8696:
8655:
8608:
8607:
8528:
8493:
8445:
8444:
8375:
8326:
8293:
8242:
8190:
8189:
8133:
8038:
7995:
7959:
7893:
7857:
7808:
7768:
7694:
7693:
7661:
7659:
7658:
7653:
7428:
7377:
7348:
7312:
7283:
7226:
7197:
7157:
7126:
7077:
7029:
7028:
6986:is odd, we have
6945:For even order 2
6834:quaternion group
6678:Normal subgroups
6485:Index-2 subgroup
6475:
6474:
6457:, isomorphic to
6448:
6446:
6445:
6440:
6386:
6384:
6383:
6378:
6136:
6135:
5842:
5841:
5833:Coxeter notation
5558:
5557:
5556:
5522:
5521:
5520:
5516:
5515:
5511:
5510:
5482:
5481:
5480:
5476:
5475:
5471:
5470:
5433:
5432:
5431:
5427:
5426:
5422:
5421:
5417:
5416:
5412:
5411:
5377:
5376:
5375:
5371:
5370:
5366:
5365:
5361:
5360:
5356:
5355:
5327:
5326:
5325:
5321:
5320:
5316:
5315:
5279:
5278:
5277:
5273:
5272:
5268:
5267:
5263:
5262:
5258:
5257:
5229:
5228:
5227:
5223:
5222:
5218:
5217:
5213:
5212:
5208:
5207:
5179:
5178:
5177:
5173:
5172:
5168:
5167:
5163:
5162:
5158:
5157:
5098:
5097:
5061:Coxeter notation
5002:Coxeter notation
4984:
4975:
4966:
4957:
4956:
4955:
4951:
4950:
4946:
4945:
4933:
4932:
4931:
4927:
4926:
4922:
4921:
4909:
4908:
4907:
4891:
4882:
4873:
4864:
4863:
4862:
4858:
4857:
4853:
4852:
4848:
4847:
4843:
4842:
4826:
4825:
4824:
4820:
4819:
4815:
4814:
4810:
4809:
4805:
4804:
4792:
4791:
4790:
4786:
4785:
4781:
4780:
4764:
4755:
4746:
4737:
4736:
4735:
4731:
4730:
4726:
4725:
4721:
4720:
4716:
4715:
4703:
4702:
4701:
4697:
4696:
4692:
4691:
4687:
4686:
4682:
4681:
4669:
4668:
4667:
4663:
4662:
4658:
4657:
4653:
4652:
4648:
4647:
4635:
4514:
4511:
4507:
4504:
4499:
4498:
4497:
4493:
4492:
4488:
4487:
4483:
4482:
4478:
4477:
4350:
4349:
4348:
4344:
4343:
4339:
4338:
4334:
4333:
4329:
4328:
4220:
4216:
4210:
4209:
4208:
4204:
4203:
4199:
4198:
4194:
4193:
4189:
4188:
4167:
4165:
4164:
4159:
4157:
4152:
4039:
4038:
4037:
4033:
4032:
4028:
4027:
4023:
4022:
4018:
4017:
3884:
3880:
3874:
3873:
3872:
3868:
3867:
3863:
3862:
3858:
3857:
3853:
3852:
3703:
3691:
3671:
3666:
3665:
3664:
3660:
3659:
3655:
3654:
3650:
3649:
3645:
3644:
3536:
3517:
3492:
3491:
3490:
3486:
3485:
3481:
3480:
3476:
3475:
3471:
3470:
3455:
3454:
3442:Coxeter notation
3376:
3375:
3374:
3370:
3369:
3365:
3364:
3360:
3359:
3355:
3354:
3333:
3313:
3312:
3311:
3307:
3306:
3302:
3301:
3297:
3296:
3292:
3291:
3252:
3251:
3250:
3246:
3245:
3241:
3240:
3188:
3187:
3186:
3182:
3181:
3177:
3176:
3172:
3171:
3167:
3166:
3145:
3125:
3124:
3123:
3119:
3118:
3114:
3113:
3062:
3061:
2898:, with elements
2719:and reflection Ο
2642:and reflection Ο
1811:
1783:
1782:
1781:
1777:
1776:
1772:
1771:
1767:
1766:
1762:
1761:
1735:
1713:
1688:
1687:
1686:
1682:
1681:
1677:
1676:
1672:
1671:
1667:
1666:
1662:
1661:
1635:
1627:
1612:
1588:
1587:
1586:
1582:
1581:
1577:
1576:
1572:
1571:
1567:
1566:
1520:
1494:
1493:
1492:
1488:
1487:
1483:
1482:
1441:
1413:
1412:
1411:
1407:
1406:
1402:
1401:
1397:
1396:
1392:
1391:
1366:
1339:
1314:
1313:
1312:
1308:
1307:
1303:
1302:
1298:
1297:
1293:
1292:
1288:
1287:
1262:
1255:
1239:
1214:
1213:
1212:
1208:
1207:
1203:
1202:
1114:
1113:
1101:wallpaper groups
1097:Coxeter notation
907:are named after
491:Coxeter notation
441:as one of them.
405:orthogonal group
373:
366:
362:
359:
353:
345:
342:
339:
337:
332:
329:
326:
324:
319:
315:
295:
294:
287:
278:
277:
276:
272:
271:
267:
266:
262:
261:
257:
256:
239:
232:
231:
230:
226:
225:
221:
220:
216:
215:
211:
210:
193:
186:
185:
184:
180:
179:
175:
174:
170:
169:
165:
164:
147:
136:Polyhedral group
130:
129:
128:
124:
123:
119:
118:
114:
113:
109:
108:
91:
84:
83:
82:
78:
77:
73:
72:
55:
48:
47:
46:
29:
22:
9704:
9703:
9699:
9698:
9697:
9695:
9694:
9693:
9674:
9673:
9638:
9633:
9595:
9558:
9547:
9541:
9538:
9495:
9493:
9483:
9471:
9460:
9455:
9454:
9446:
9440:
9436:
9424:
9420:
9387:
9381:
9377:
9370:
9356:
9352:
9345:
9323:
9319:
9295:
9285:
9278:
9273:
9268:
9204:
9177:covering spaces
9117:
9113:
9111:
9108:
9107:
9084:
9080:
9078:
9075:
9074:
9051:
9047:
9045:
9042:
9041:
9007:
9003:
9001:
8998:
8997:
8967:
8963:
8961:
8958:
8957:
8946:(2,3,5) is the
8901:lattice theorem
8893:
8827:
8813:
8809:
8801:
8783:
8760:
8756:
8747:
8723:
8719:
8710:
8685:
8669:
8645:
8590:
8586:
8580:
8561:
8557:
8553:
8549:
8542:
8518:
8514:
8507:
8483:
8476:
8434:
8430:
8419:
8404:
8400:
8392:
8365:
8361:
8357:
8353:
8349:
8345:
8339:
8316:
8312:
8306:
8283:
8279:
8275:
8271:
8267:
8263:
8256:
8232:
8228:
8221:
8179:
8171:
8167:
8156:
8145:
8123:
8119:
8112:
8106:
8099:
8092:
8082:
8064:
8058:
8051:
8028:
8022:
8015:
8008:
7985:
7979:
7972:
7949:
7945:
7941:
7934:
7925:
7916:
7907:
7883:
7877:
7870:
7847:
7840:
7831:
7822:
7798:
7791:
7782:
7758:
7754:
7750:
7743:
7734:
7725:
7700:Isometry groups
7690:
7683:
7673:
7669:
7647:
7644:
7643:
7642:
7622:
7603:
7580:
7557:
7544:
7535:
7482:
7465:
7453:
7440:
7418:
7414:
7410:
7404:
7397:
7390:
7367:
7361:
7338:
7332:
7325:
7302:
7296:
7273:
7269:
7265:
7258:
7249:
7240:
7216:
7210:
7187:
7180:
7171:
7147:
7140:
7116:
7109:
7100:
7091:
7067:
7060:
7035:Isometry groups
7017:
7010:
6995:
6978:For any order 2
6974:
6967:
6959:
6940:
6931:
6921:
6899:
6875:
6868:
6861:
6851:
6843:
6839:
6836:(of order 8), Z
6823:
6816:
6809:
6802:
6795:
6785:
6778:
6771:
6763:
6756:
6747:
6737:
6728:
6718:
6701:
6680:
6671:
6648:
6634:
6624:
6612:
6596:
6582:
6572:
6559:
6543:
6529:
6519:
6503:
6501:
6493:
6486:
6479:
6428:
6425:
6424:
6411:
6401:corresponds to
6400:
6327:
6324:
6323:
6311:has a subgroup
6305:
6296:
6278:
6260:
6242:
6226:
6212:
6200:
6184:
6170:
6155:
6147:
6140:
6048:
6032:
6027:
6007:
5988:
5979:
5958:
5947:
5934:
5922:
5908:
5897:
5885:
5873:
5803:
5792:
5781:
5774:
5760:
5751:
5741:
5731:
5719:
5710:
5700:
5690:
5680:
5657:
5650:
5605:, or canonical
5600:
5587:
5577:
5575:Rotation groups
5554:
5552:
5549:
5518:
5513:
5508:
5506:
5502:
5478:
5473:
5468:
5466:
5463:
5429:
5424:
5419:
5414:
5409:
5407:
5404:
5397:
5373:
5368:
5363:
5358:
5353:
5351:
5348:
5323:
5318:
5313:
5311:
5308:
5296:Dihedral groups
5275:
5270:
5265:
5260:
5255:
5253:
5250:
5225:
5220:
5215:
5210:
5205:
5203:
5200:
5175:
5170:
5165:
5160:
5155:
5153:
5150:
5132:
5125:
5122:
5110:
5103:
5023:
5013:
4985:
4976:
4967:
4953:
4948:
4943:
4941:
4939:
4929:
4924:
4919:
4917:
4915:
4905:
4903:
4901:
4892:
4883:
4874:
4860:
4855:
4850:
4845:
4840:
4838:
4836:
4832:
4822:
4817:
4812:
4807:
4802:
4800:
4798:
4788:
4783:
4778:
4776:
4774:
4765:
4756:
4747:
4733:
4728:
4723:
4718:
4713:
4711:
4709:
4699:
4694:
4689:
4684:
4679:
4677:
4675:
4665:
4660:
4655:
4650:
4645:
4643:
4641:
4630:
4623:
4606:
4579:
4572:
4565:
4558:
4547:
4540:
4532:
4516:
4512:
4509:
4505:
4502:
4501:
4495:
4490:
4485:
4480:
4475:
4473:
4471:
4468:
4429:
4404:
4397:
4389:
4376:normal subgroup
4354:
4352:
4346:
4341:
4336:
4331:
4326:
4324:
4322:
4298:
4287:
4280:
4273:
4266:
4255:
4248:
4241:
4222:
4218:
4214:
4212:
4206:
4201:
4196:
4191:
4186:
4184:
4182:
4179:
4151:
4149:
4146:
4145:
4104:
4098:. An object of
4093:
4086:
4079:
4072:
4065:
4043:
4041:
4035:
4030:
4025:
4020:
4015:
4013:
4011:
3999:
3992:
3981:
3974:symmetric group
3971:
3960:
3953:
3946:
3939:
3932:
3925:
3911:
3901:The seams of a
3885:
3882:
3878:
3876:
3870:
3865:
3860:
3855:
3850:
3848:
3846:
3843:
3826:
3819:
3812:
3801:
3794:
3787:
3779:symmetric group
3776:
3769:
3762:
3755:
3748:
3741:
3734:
3718:
3717:
3716:
3715:
3714:
3712:
3704:
3696:
3695:
3692:
3673:
3669:
3668:
3662:
3657:
3652:
3647:
3642:
3640:
3638:
3635:
3611:
3604:
3596:normal subgroup
3585:
3574:
3559:
3553:There are four
3551:
3550:
3549:
3548:
3547:
3545:
3537:
3529:
3528:
3526:
3518:
3496:
3494:
3488:
3483:
3478:
3473:
3468:
3466:
3464:
3446:Coxeter diagram
3435:
3426:
3412:
3403:
3397:
3387:
3372:
3367:
3362:
3357:
3352:
3350:
3342:
3331:
3323:
3309:
3304:
3299:
3294:
3289:
3287:
3279:
3263:
3248:
3243:
3238:
3236:
3228:
3215:
3209:
3199:
3184:
3179:
3174:
3169:
3164:
3162:
3154:
3143:
3135:
3121:
3116:
3111:
3109:
3101:
3088:Abstract group
3058:
3051:
3044:
3033:
3018:
3009:
2999:
2989:
2980:
2976:
2972:
2963:
2951:
2947:
2943:
2933:
2929:
2925:
2915:
2911:
2907:
2897:
2887:
2877:
2868:
2864:
2860:
2850:
2838:
2829:
2820:
2811:
2802:
2798:
2794:
2785:
2774:
2770:
2761:
2757:
2748:
2744:
2735:
2731:
2722:
2718:
2709:
2697:
2693:
2684:
2680:
2671:
2667:
2658:
2654:
2645:
2641:
2632:
2620:
2616:
2606:
2602:
2592:
2588:
2578:
2569:
2565:
2561:
2551:
2523:
2514:
2505:
2492:
2483:
2468:
2461:
2448:
2441:
2431:
2424:
2415:
2408:
2399:
2392:
2383:
2376:
2369:
2352:
2334:
2321:
2307:
2288:
2269:
2262:
2255:
2248:
2241:
2234:
2227:
2217:
2192:
2166:
2152:
2142:
2129:
2120:
2104:
2092:
2067:
2058:
2037:
2019:
2001:
1960:
1953:
1947:(equivalent to
1946:
1931:
1916:
1910:(equivalent to
1909:
1899:
1888:
1879:
1872:
1860:
1856:
1850:
1843:
1839:
1833:
1798:
1794:
1779:
1774:
1769:
1764:
1759:
1757:
1756:
1747:
1730:
1700:
1684:
1679:
1674:
1669:
1664:
1659:
1657:
1656:
1647:
1631:
1622:
1599:
1584:
1579:
1574:
1569:
1564:
1562:
1561:
1552:
1527:
1506:
1490:
1485:
1480:
1478:
1477:
1468:
1428:
1424:
1409:
1404:
1399:
1394:
1389:
1387:
1386:
1377:
1361:
1326:
1310:
1305:
1300:
1295:
1290:
1285:
1283:
1282:
1273:
1258:
1250:
1226:
1210:
1205:
1200:
1198:
1197:
1189:
1081:crystallography
1026:
1001:
984:
967:
960:
918:
817:There are many
815:
809:
770:
760:
735:
728:
719:
712:
703:
687:
680:
659:
646:
616:
601:
590:
577:
530:
524:
427:Symmetry groups
421:Euclidean group
374:
363:
357:
354:
347:
344:REVERSE SOLIDUS
343:
340:
335:
334:
330:
327:
322:
321:
317:
313:
310:MOS:MATHSPECIAL
304:Manual of Style
296:
292:
285:
274:
269:
264:
259:
254:
252:
250:
248:
244:
240:
228:
223:
218:
213:
208:
206:
204:
202:
198:
194:
182:
177:
172:
167:
162:
160:
158:
156:
152:
148:
126:
121:
116:
111:
106:
104:
102:
100:
96:
92:
80:
75:
70:
68:
66:
64:
60:
58:Cyclic symmetry
56:
44:
42:
40:
38:
34:
30:
12:
11:
5:
9702:
9692:
9691:
9686:
9672:
9671:
9666:
9660:
9654:
9649:
9637:
9636:External links
9634:
9632:
9631:
9600:
9593:
9580:
9563:
9560:
9559:
9474:
9472:
9465:
9459:
9456:
9453:
9452:
9442:Burban, Igor.
9434:
9418:
9375:
9368:
9350:
9343:
9317:
9306:(1): 393β415.
9275:
9274:
9272:
9269:
9267:
9266:
9261:
9256:
9251:
9249:Crystal system
9246:
9241:
9236:
9231:
9226:
9221:
9216:
9211:
9205:
9203:
9200:
9139:
9138:
9120:
9116:
9105:
9087:
9083:
9072:
9054:
9050:
9039:
9010:
9006:
8995:
8970:
8966:
8892:
8889:
8861:
8860:
8848:
8847:
8840:
8826:
8823:
8820:
8819:
8817:
8814:
8811:
8807:
8802:
8799:
8794:
8790:
8789:
8787:
8784:
8781:
8776:
8771:
8767:
8766:
8764:
8761:
8758:
8754:
8749:
8745:
8739:
8735:
8734:
8727:
8724:
8721:
8717:
8712:
8708:
8702:
8698:
8697:
8690:
8687:
8683:
8677:
8667:
8661:
8657:
8656:
8649:
8646:
8643:
8638:
8631:
8627:
8626:
8621:
8618:
8617:Abstract group
8615:
8614:Isometry group
8612:
8598:
8597:
8594:
8591:
8588:
8584:
8581:
8578:
8573:
8569:
8568:
8565:
8562:
8559:
8555:
8551:
8547:
8544:
8540:
8534:
8530:
8529:
8522:
8519:
8516:
8512:
8509:
8505:
8499:
8495:
8494:
8487:
8484:
8481:
8478:
8474:
8468:
8464:
8463:
8458:
8455:
8454:Abstract group
8452:
8451:Isometry group
8449:
8432:
8425:
8413:
8402:
8398:
8387:
8377:
8376:
8369:
8366:
8363:
8359:
8355:
8351:
8347:
8343:
8340:
8337:
8332:
8328:
8327:
8320:
8317:
8314:
8310:
8307:
8304:
8299:
8295:
8294:
8287:
8284:
8281:
8277:
8273:
8269:
8265:
8261:
8258:
8254:
8248:
8244:
8243:
8236:
8233:
8230:
8226:
8223:
8219:
8213:
8209:
8208:
8203:
8200:
8199:Abstract group
8197:
8196:Isometry group
8194:
8177:
8176:= 1 we get Dih
8169:
8162:
8150:
8144:
8141:
8135:
8134:
8127:
8124:
8121:
8117:
8110:
8107:
8104:
8097:
8087:
8080:
8075:
8071:
8070:
8068:
8065:
8062:
8059:
8056:
8049:
8044:
8040:
8039:
8032:
8029:
8026:
8023:
8020:
8013:
8006:
8001:
7997:
7996:
7989:
7986:
7983:
7980:
7977:
7970:
7965:
7961:
7960:
7953:
7950:
7947:
7943:
7939:
7936:
7932:
7923:
7914:
7905:
7899:
7895:
7894:
7887:
7884:
7881:
7878:
7875:
7868:
7863:
7859:
7858:
7851:
7848:
7845:
7842:
7838:
7829:
7820:
7814:
7810:
7809:
7802:
7799:
7796:
7793:
7789:
7780:
7774:
7770:
7769:
7762:
7759:
7756:
7752:
7748:
7745:
7741:
7732:
7723:
7717:
7713:
7712:
7707:
7704:
7703:Abstract group
7701:
7698:
7688:
7681:
7675:
7674:
7671:
7663:
7651:
7636:
7629:
7628:
7620:
7598:
7593:
7575:
7570:
7552:
7540:
7531:
7528:dihedral group
7485:rotation group
7478:
7461:
7449:
7444:dihedral group
7439:
7436:
7430:
7429:
7422:
7419:
7416:
7412:
7408:
7405:
7402:
7395:
7388:
7383:
7379:
7378:
7371:
7368:
7365:
7362:
7359:
7354:
7350:
7349:
7342:
7339:
7336:
7333:
7330:
7323:
7318:
7314:
7313:
7306:
7303:
7300:
7297:
7294:
7289:
7285:
7284:
7277:
7274:
7271:
7267:
7263:
7260:
7256:
7247:
7238:
7232:
7228:
7227:
7220:
7217:
7214:
7211:
7208:
7203:
7199:
7198:
7191:
7188:
7185:
7182:
7178:
7169:
7163:
7159:
7158:
7151:
7148:
7145:
7142:
7138:
7132:
7128:
7127:
7120:
7117:
7114:
7111:
7107:
7098:
7089:
7083:
7079:
7078:
7071:
7068:
7065:
7062:
7058:
7052:
7048:
7047:
7042:
7039:
7038:Abstract group
7036:
7033:
7020:
7019:
7015:
7008:
6990:
6976:
6972:
6965:
6954:
6936:
6927:
6917:
6903:symmetry group
6898:
6895:
6873:
6866:
6859:
6849:
6846:dicyclic group
6841:
6837:
6822:
6819:
6814:
6807:
6800:
6793:
6783:
6776:
6769:
6762:
6759:
6752:
6742:
6733:
6723:
6714:
6697:
6679:
6676:
6673:
6672:
6669:
6664:
6661:
6656:
6650:
6649:
6643:
6638:
6635:
6630:
6625:
6620:
6614:
6613:
6607:
6602:
6598:
6597:
6591:
6586:
6583:
6578:
6573:
6567:
6561:
6560:
6554:
6549:
6545:
6544:
6538:
6533:
6530:
6525:
6520:
6514:
6508:
6507:
6498:
6490:
6483:
6478:Rotation group
6455:abstract group
6438:
6435:
6432:
6409:
6398:
6388:
6387:
6376:
6373:
6370:
6367:
6364:
6361:
6358:
6355:
6352:
6349:
6346:
6343:
6340:
6337:
6334:
6331:
6304:
6301:
6298:
6297:
6294:
6289:
6286:
6280:
6279:
6276:
6271:
6268:
6262:
6261:
6258:
6253:
6250:
6244:
6243:
6237:
6232:
6228:
6227:
6221:
6216:
6213:
6208:
6202:
6201:
6195:
6190:
6186:
6185:
6179:
6174:
6171:
6166:
6160:
6159:
6152:
6144:
6139:Rotation group
6112:
6111:
6101:
6071:
6070:
6046:
6036:direct product
6031:
6028:
6026:
6023:
6020:
6019:
6013:
6011:
6009:
6005:
5999:
5998:
5992:
5990:
5986:
5981:
5977:
5971:
5970:
5964:
5962:
5960:
5956:
5950:
5949:
5943:
5938:
5936:
5929:
5924:
5917:
5911:
5910:
5904:
5899:
5892:
5887:
5880:
5875:
5868:
5862:
5861:
5856:
5851:
5848:
5818:
5817:
5801:
5790:
5779:
5772:
5762:
5756:
5746:
5736:
5727:
5721:
5715:
5705:
5695:
5685:
5676:
5666:dihedral group
5655:
5648:
5596:
5583:
5576:
5573:
5570:
5569:
5566:
5564:
5561:
5559:
5550:
5547:
5541:
5540:
5539:Single mirror
5536:
5535:
5532:
5530:
5525:
5523:
5504:
5500:
5494:
5493:
5490:
5488:
5485:
5483:
5464:
5461:
5454:
5453:
5447:
5446:
5443:
5441:
5436:
5434:
5405:
5402:
5395:
5389:
5388:
5385:
5383:
5380:
5378:
5349:
5346:
5339:
5338:
5335:
5333:
5330:
5328:
5309:
5306:
5299:
5298:
5292:
5291:
5288:
5285:
5282:
5280:
5251:
5248:
5242:
5241:
5238:
5235:
5232:
5230:
5201:
5198:
5192:
5191:
5188:
5185:
5182:
5180:
5151:
5148:
5142:
5141:
5135:
5134:
5127:
5118:
5113:
5106:
5089:Coxeter number
5018:
5008:
4994:Coxeter groups
4988:
4987:
4978:
4969:
4959:
4958:
4937:
4934:
4913:
4910:
4899:
4895:
4894:
4885:
4876:
4866:
4865:
4834:
4830:
4827:
4796:
4793:
4772:
4768:
4767:
4758:
4749:
4739:
4738:
4707:
4704:
4673:
4670:
4639:
4629:
4626:
4621:
4614:
4613:
4604:
4599:
4582:
4581:
4577:
4570:
4563:
4556:
4545:
4538:
4530:
4523:
4518:
4466:
4459:
4458:
4427:
4402:
4395:
4387:
4364:
4356:
4313:
4312:
4296:
4288:(because both
4285:
4278:
4271:
4264:
4253:
4246:
4239:
4229:
4224:
4177:
4170:
4169:
4155:
4102:
4091:
4084:
4077:
4070:
4063:
4053:
4045:
4002:
4001:
3997:
3990:
3979:
3969:
3958:
3951:
3944:
3937:
3930:
3923:
3909:
3894:
3887:
3841:
3834:
3833:
3824:
3817:
3810:
3799:
3792:
3785:
3774:
3767:
3760:
3756:axis, and two
3753:
3746:
3739:
3732:
3710:
3705:
3698:
3697:
3693:
3686:
3685:
3684:
3683:
3682:
3680:
3675:
3633:
3626:
3625:
3609:
3602:
3592:rotation group
3583:
3572:
3557:
3543:
3538:
3531:
3530:
3524:
3519:
3512:
3511:
3510:
3509:
3508:
3506:
3498:
3431:
3422:
3411:
3408:
3405:
3404:
3401:
3398:
3392:
3382:
3377:
3348:
3346:
3343:
3340:
3335:
3328:
3327:
3324:
3319:
3314:
3285:
3283:
3280:
3277:
3272:
3268:
3267:
3264:
3258:
3253:
3234:
3232:
3229:
3226:
3221:
3217:
3216:
3213:
3210:
3204:
3194:
3189:
3160:
3158:
3155:
3152:
3147:
3140:
3139:
3136:
3131:
3126:
3107:
3105:
3102:
3099:
3094:
3090:
3089:
3086:
3083:
3078:
3073:
3068:
3056:
3049:
3042:
3031:
3021:
3020:
3014:
3004:
2994:
2985:
2978:
2974:
2968:
2958:
2953:
2949:
2945:
2938:
2931:
2927:
2920:
2913:
2909:
2902:
2892:
2882:
2873:
2866:
2862:
2855:
2845:
2840:
2834:
2825:
2816:
2807:
2800:
2796:
2790:
2781:
2776:
2772:
2766:
2759:
2753:
2746:
2740:
2733:
2727:
2720:
2714:
2704:
2699:
2695:
2689:
2682:
2676:
2669:
2663:
2656:
2650:
2643:
2637:
2627:
2622:
2618:
2611:
2604:
2597:
2590:
2583:
2574:
2567:
2563:
2556:
2546:
2541:
2519:
2510:
2501:
2488:
2479:
2463:
2456:
2446:
2439:
2434:
2433:
2429:
2422:
2417:
2413:
2406:
2401:
2397:
2390:
2385:
2381:
2374:
2367:
2347:
2330:
2316:
2303:
2286:
2280:Concanavalin A
2267:
2260:
2253:
2246:
2239:
2232:
2225:
2213:
2187:
2161:
2147:
2137:
2125:
2116:
2100:
2095:dihedral group
2087:
2063:
2054:
2032:
2024:, or a set of
2014:
1997:
1975:
1974:
1967:, also called
1958:
1951:
1944:
1939:
1929:
1924:
1914:
1907:
1897:
1884:
1877:
1868:
1858:
1852:
1845:
1841:
1835:
1828:
1819:
1818:
1812:
1805:
1799:
1796:
1790:
1787:
1784:
1754:
1748:
1742:
1737:
1728:
1721:
1720:
1714:
1707:
1701:
1695:
1692:
1689:
1654:
1648:
1642:
1637:
1629:
1619:
1618:
1613:
1606:
1600:
1595:
1592:
1589:
1559:
1553:
1548:
1543:
1537:
1530:
1529:
1521:
1514:
1508:
1502:
1498:
1495:
1475:
1469:
1463:
1458:
1452:
1445:
1444:
1442:
1435:
1429:
1426:
1420:
1417:
1414:
1384:
1378:
1372:
1367:
1359:
1352:
1351:
1348:rotoreflection
1340:
1333:
1327:
1321:
1318:
1315:
1280:
1274:
1268:
1263:
1256:
1247:
1246:
1240:
1233:
1228:
1222:
1218:
1215:
1195:
1190:
1185:
1180:
1174:
1173:
1170:
1164:
1157:
1156:
1153:
1150:
1145:
1140:
1135:
1130:
1125:
1120:
1025:
1022:
1006:
1005:
1004:
1003:
999:
986:
982:
969:
965:
958:
939:
917:
914:
835:abelian groups
808:
805:
786:
785:
782:
768:
758:
733:
726:
718:
715:
708:
699:
685:
678:
672:
671:
670:
669:
655:
651:for the group
642:
625:
622:
614:
605:
604:
603:
599:
586:
582:for the group
573:
559:
523:
520:
510:covalent bonds
478:the object is
476:if and only if
468:rotation group
393:isometry group
376:
375:
299:
297:
290:
283:
280:
279:
246:
233:
200:
187:
154:
140:
139:
132:
131:
98:
85:
62:
49:
36:
9:
6:
4:
3:
2:
9701:
9690:
9687:
9685:
9682:
9681:
9679:
9670:
9667:
9664:
9661:
9658:
9655:
9653:
9650:
9647:
9643:
9640:
9639:
9629:
9625:
9621:
9617:
9613:
9609:
9605:
9601:
9596:
9594:0-387-09212-9
9590:
9586:
9581:
9577:
9573:
9569:
9565:
9564:
9556:
9553:
9545:
9534:
9531:
9527:
9524:
9520:
9517:
9513:
9510:
9506:
9503: β
9502:
9498:
9497:Find sources:
9491:
9487:
9481:
9480:
9475:This article
9473:
9469:
9464:
9463:
9445:
9438:
9431:
9427:
9422:
9415:
9411:
9407:
9403:
9399:
9395:
9394:
9386:
9379:
9371:
9365:
9361:
9354:
9346:
9344:0-08-037014-4
9340:
9336:
9332:
9328:
9321:
9313:
9309:
9305:
9302:(in French).
9301:
9293:
9289:
9288:Curie, Pierre
9283:
9281:
9276:
9265:
9262:
9260:
9257:
9255:
9252:
9250:
9247:
9245:
9242:
9240:
9237:
9235:
9232:
9230:
9227:
9225:
9222:
9220:
9217:
9215:
9212:
9210:
9207:
9206:
9199:
9197:
9193:
9189:
9184:
9182:
9178:
9174:
9170:
9166:
9161:
9159:
9154:
9152:
9148:
9144:
9136:
9118:
9114:
9106:
9103:
9085:
9081:
9073:
9070:
9052:
9048:
9040:
9038:
9034:
9030:
9026:
9008:
9004:
8996:
8994:
8990:
8986:
8968:
8964:
8956:
8955:
8954:
8951:
8949:
8945:
8941:
8937:
8933:
8928:
8926:
8925:
8919:
8915:
8911:
8906:
8903:, there is a
8902:
8898:
8888:
8885:
8883:
8879:
8874:
8872:
8868:
8858:
8857:great circles
8854:
8849:
8845:
8841:
8838:
8834:
8833:
8818:
8815:
8806:
8803:
8798:
8795:
8792:
8788:
8785:
8780:
8777:
8775:
8772:
8769:
8765:
8762:
8753:
8750:
8748:
8744:
8740:
8737:
8732:
8728:
8725:
8716:
8713:
8711:
8707:
8703:
8700:
8695:
8691:
8688:
8686:
8682:
8678:
8676:
8675:
8670:
8666:
8662:
8659:
8654:
8650:
8647:
8642:
8639:
8637:
8636:
8632:
8629:
8625:
8624:Cycle diagram
8622:
8619:
8616:
8613:
8610:
8609:
8606:
8603:
8595:
8592:
8582:
8577:
8574:
8571:
8566:
8563:
8545:
8543:
8539:
8535:
8532:
8527:
8523:
8520:
8510:
8508:
8504:
8500:
8497:
8492:
8488:
8485:
8479:
8477:
8473:
8469:
8466:
8462:
8461:Cycle diagram
8459:
8456:
8453:
8450:
8447:
8446:
8443:
8440:
8438:
8429:
8423:
8417:
8412:
8408:
8396:
8390:
8386:
8382:
8374:
8370:
8367:
8341:
8336:
8333:
8330:
8325:
8321:
8318:
8308:
8303:
8300:
8297:
8292:
8288:
8285:
8259:
8257:
8253:
8249:
8246:
8241:
8237:
8234:
8224:
8222:
8218:
8214:
8211:
8207:
8206:Cycle diagram
8204:
8201:
8198:
8195:
8192:
8191:
8188:
8185:
8183:
8175:
8166:
8160:
8154:
8149:
8140:
8132:
8128:
8125:
8116:
8108:
8103:
8096:
8091:
8086:
8079:
8076:
8073:
8069:
8066:
8060:
8055:
8048:
8045:
8042:
8037:
8033:
8030:
8024:
8019:
8012:
8005:
8002:
7999:
7994:
7990:
7987:
7981:
7976:
7969:
7966:
7963:
7958:
7954:
7951:
7937:
7935:
7931:
7926:
7922:
7917:
7913:
7908:
7904:
7900:
7897:
7892:
7888:
7885:
7879:
7874:
7867:
7864:
7861:
7856:
7852:
7849:
7843:
7841:
7837:
7832:
7828:
7823:
7819:
7815:
7812:
7807:
7803:
7800:
7794:
7792:
7788:
7783:
7779:
7775:
7772:
7767:
7763:
7760:
7746:
7744:
7740:
7735:
7731:
7726:
7722:
7718:
7715:
7711:
7710:Cycle diagram
7708:
7705:
7702:
7699:
7696:
7695:
7692:
7687:
7680:
7667:
7649:
7640:
7634:
7633:
7632:
7626:
7619:
7615:
7611:
7607:
7601:
7597:
7594:
7592:
7588:
7584:
7578:
7574:
7571:
7569:
7565:
7561:
7555:
7551:
7548:
7547:
7546:
7543:
7539:
7534:
7529:
7524:
7522:
7518:
7517:trapezohedron
7514:
7510:
7506:
7502:
7498:
7494:
7490:
7486:
7481:
7477:
7473:
7469:
7464:
7460:
7455:
7452:
7448:
7445:
7435:
7427:
7423:
7420:
7406:
7401:
7394:
7387:
7384:
7381:
7376:
7372:
7369:
7363:
7358:
7355:
7352:
7347:
7343:
7340:
7334:
7329:
7322:
7319:
7316:
7311:
7307:
7304:
7298:
7293:
7290:
7287:
7282:
7278:
7275:
7261:
7259:
7255:
7250:
7246:
7241:
7237:
7233:
7230:
7225:
7221:
7218:
7212:
7207:
7204:
7201:
7196:
7192:
7189:
7183:
7181:
7177:
7172:
7168:
7164:
7161:
7156:
7152:
7149:
7143:
7141:
7137:
7133:
7130:
7125:
7121:
7118:
7112:
7110:
7106:
7101:
7097:
7092:
7088:
7084:
7081:
7076:
7072:
7069:
7063:
7061:
7057:
7053:
7050:
7046:
7045:Cycle diagram
7043:
7040:
7037:
7034:
7031:
7030:
7027:
7025:
7014:
7011:the notation
7007:
7003:
6999:
6993:
6989:
6985:
6981:
6977:
6971:
6968:the notation
6964:
6960:
6958:
6953:
6948:
6944:
6943:
6942:
6939:
6935:
6930:
6925:
6920:
6916:
6912:
6908:
6904:
6894:
6892:
6890:
6886:
6882:
6877:
6872:
6865:
6858:
6853:
6847:
6835:
6831:
6826:
6818:
6813:
6806:
6799:
6792:
6787:
6782:
6775:
6768:
6758:
6755:
6751:
6745:
6741:
6736:
6732:
6726:
6722:
6717:
6713:
6709:
6705:
6700:
6696:
6693:
6689:
6685:
6668:
6665:
6662:
6660:
6657:
6655:
6652:
6651:
6646:
6642:
6639:
6636:
6633:
6629:
6626:
6623:
6619:
6616:
6615:
6610:
6606:
6603:
6600:
6599:
6594:
6590:
6587:
6584:
6581:
6577:
6571:
6566:
6562:
6557:
6553:
6550:
6547:
6546:
6542:
6537:
6534:
6531:
6528:
6524:
6518:
6513:
6509:
6506:
6499:
6497:
6491:
6489:
6484:
6482:
6477:
6476:
6473:
6470:
6468:
6464:
6460:
6456:
6452:
6449:. This group
6436:
6430:
6422:
6418:
6413:
6408:
6404:
6397:
6393:
6390:For example,
6368:
6365:
6359:
6353:
6347:
6338:
6335:
6332:
6329:
6322:
6321:
6320:
6318:
6314:
6310:
6293:
6290:
6287:
6285:
6282:
6281:
6275:
6272:
6269:
6267:
6264:
6263:
6257:
6254:
6251:
6249:
6246:
6245:
6240:
6236:
6233:
6230:
6229:
6224:
6220:
6217:
6214:
6211:
6207:
6203:
6199:
6194:
6191:
6188:
6187:
6182:
6178:
6175:
6172:
6169:
6165:
6161:
6158:
6153:
6151:
6145:
6143:
6138:
6137:
6134:
6131:
6129:
6125:
6121:
6117:
6109:
6105:
6102:
6099:
6095:
6091:
6087:
6084:
6083:
6082:
6080:
6076:
6068:
6064:
6060:
6059:
6058:
6056:
6052:
6045:
6041:
6037:
6017:
6014:
6012:
6010:
6004:
6001:
5996:
5993:
5991:
5985:
5982:
5976:
5973:
5968:
5965:
5963:
5961:
5955:
5952:
5946:
5942:
5939:
5937:
5932:
5928:
5925:
5920:
5916:
5913:
5907:
5903:
5900:
5896:
5891:
5888:
5883:
5879:
5876:
5871:
5867:
5864:
5860:
5857:
5855:
5852:
5849:
5847:
5844:
5843:
5840:
5838:
5834:
5830:
5825:
5823:
5815:
5811:
5807:
5800:
5796:
5789:
5785:
5778:
5771:
5767:
5763:
5759:
5755:
5749:
5745:
5739:
5735:
5730:
5726:
5722:
5718:
5714:
5709:
5704:
5698:
5694:
5688:
5684:
5679:
5675:
5671:
5670:
5669:
5667:
5663:
5662:
5654:
5647:
5642:
5640:
5636:
5632:
5628:
5624:
5621:of a regular
5620:
5616:
5612:
5608:
5604:
5599:
5595:
5591:
5586:
5582:
5567:
5565:
5562:
5560:
5551:
5546:
5543:
5537:
5533:
5531:
5529:
5526:
5524:
5505:
5499:
5496:
5491:
5489:
5486:
5484:
5465:
5460:
5456:
5452:
5451:Cyclic groups
5448:
5444:
5442:
5440:
5437:
5435:
5406:
5401:
5394:
5391:
5386:
5384:
5381:
5379:
5350:
5345:
5341:
5336:
5334:
5331:
5329:
5310:
5305:
5301:
5297:
5293:
5289:
5286:
5283:
5281:
5252:
5247:
5244:
5239:
5236:
5233:
5231:
5202:
5197:
5194:
5189:
5186:
5183:
5181:
5152:
5147:
5144:
5140:
5136:
5131:
5128:
5124:
5119:
5117:
5114:
5112:
5105:
5100:
5096:
5094:
5090:
5086:
5082:
5078:
5074:
5070:
5066:
5062:
5058:
5054:
5050:
5046:
5042:
5037:
5035:
5031:
5027:
5021:
5017:
5011:
5007:
5003:
4999:
4995:
4983:
4979:
4974:
4970:
4965:
4961:
4935:
4911:
4897:
4896:
4890:
4886:
4881:
4877:
4872:
4868:
4828:
4794:
4770:
4769:
4763:
4759:
4754:
4750:
4745:
4741:
4705:
4671:
4637:
4636:
4625:
4619:
4611:
4607:
4600:
4597:
4593:
4589:
4588:
4587:
4576:
4569:
4562:
4555:
4551:
4544:
4537:
4533:
4529:
4524:
4522:
4519:
4469:
4465:
4460:
4456:
4453:
4449:
4448:
4443:
4439:
4438:
4433:
4426:
4422:
4418:
4417:
4413:). The group
4412:
4408:
4401:
4394:
4390:
4386:
4381:
4377:
4373:
4369:
4365:
4363:
4362:
4357:
4320:
4319:
4314:
4310:
4306:
4302:
4295:
4291:
4284:
4277:
4270:
4263:
4259:
4252:
4245:
4238:
4234:
4230:
4228:
4225:
4180:
4176:
4171:
4153:
4143:
4139:
4136:
4132:
4128:
4124:
4120:
4116:
4112:
4108:
4101:
4097:
4090:
4083:
4076:
4069:
4066:axes are now
4062:
4058:
4054:
4052:
4051:
4046:
4009:
4008:
4003:
3996:
3989:
3985:
3978:
3975:
3968:
3964:
3957:
3950:
3943:
3936:
3929:
3922:
3918:
3908:
3904:
3899:
3895:
3893:
3892:
3888:
3844:
3840:
3835:
3832:
3830:
3823:
3816:
3809:
3805:
3798:
3791:
3784:
3780:
3773:
3766:
3759:
3752:
3745:
3738:
3735:axes are now
3731:
3727:
3723:
3709:
3702:
3690:
3681:
3679:
3676:
3636:
3632:
3627:
3624:
3622:
3618:
3615:
3608:
3601:
3597:
3593:
3589:
3582:
3578:
3571:
3567:
3563:
3556:
3542:
3535:
3523:
3516:
3507:
3505:
3504:
3499:
3462:
3461:
3456:
3453:
3451:
3447:
3443:
3439:
3434:
3430:
3425:
3421:
3417:
3399:
3395:
3391:
3385:
3381:
3378:
3349:
3347:
3344:
3339:
3336:
3330:
3325:
3322:
3318:
3315:
3286:
3284:
3281:
3276:
3273:
3270:
3265:
3261:
3257:
3254:
3235:
3233:
3230:
3225:
3222:
3219:
3211:
3208:
3203:
3197:
3193:
3190:
3161:
3159:
3156:
3151:
3148:
3142:
3137:
3134:
3130:
3127:
3108:
3106:
3103:
3098:
3095:
3092:
3087:
3084:
3082:
3077:
3074:
3072:
3069:
3067:
3064:
3063:
3060:
3055:
3048:
3041:
3037:
3030:
3026:
3017:
3013:
3007:
3003:
2997:
2993:
2988:
2984:
2971:
2967:
2961:
2957:
2954:
2942:
2937:
2924:
2919:
2906:
2901:
2895:
2891:
2886:
2881:
2876:
2872:
2859:
2854:
2848:
2844:
2841:
2837:
2833:
2828:
2824:
2819:
2815:
2810:
2806:
2793:
2789:
2784:
2780:
2777:
2769:
2765:
2756:
2752:
2743:
2739:
2730:
2726:
2717:
2713:
2707:
2703:
2700:
2692:
2688:
2679:
2675:
2666:
2662:
2653:
2649:
2640:
2636:
2630:
2626:
2623:
2615:
2610:
2601:
2596:
2587:
2582:
2577:
2573:
2560:
2555:
2550:
2545:
2542:
2539:
2535:
2531:
2527:
2522:
2518:
2513:
2509:
2504:
2500:
2496:
2491:
2487:
2482:
2478:
2475:
2474:
2473:
2470:
2467:
2460:
2455:
2450:
2445:
2438:
2428:
2421:
2418:
2412:
2405:
2402:
2396:
2389:
2386:
2380:
2373:
2366:
2363:
2362:
2361:
2358:
2356:
2350:
2346:
2342:
2338:
2333:
2329:
2325:
2319:
2315:
2311:
2306:
2302:
2297:
2295:
2292:
2285:
2281:
2277:
2276:homotetramers
2273:
2266:
2259:
2252:
2245:
2238:
2231:
2224:
2219:
2216:
2212:
2208:
2207:trapezohedron
2204:
2200:
2196:
2190:
2186:
2182:
2178:
2174:
2170:
2164:
2160:
2156:
2150:
2146:
2140:
2136:
2131:
2128:
2124:
2119:
2115:
2112:Note: in 2D,
2110:
2108:
2103:
2099:
2096:
2090:
2086:
2082:
2078:
2073:
2071:
2066:
2062:
2057:
2053:
2049:
2045:
2041:
2035:
2031:
2027:
2023:
2017:
2013:
2009:
2005:
2000:
1996:
1992:
1991:cyclic groups
1984:
1979:
1972:
1971:
1966:
1965:
1957:
1950:
1943:
1940:
1938:
1937:
1928:
1925:
1923:
1921:
1913:
1906:
1903:
1902:
1901:
1894:
1890:
1887:
1883:
1876:
1871:
1867:
1862:
1855:
1849:
1838:
1832:
1826:
1816:
1813:
1810:
1806:
1804:
1800:
1793:
1788:
1785:
1755:
1753:
1749:
1745:
1741:
1738:
1734:
1729:
1726:
1723:
1718:
1717:Antiprismatic
1715:
1712:
1708:
1706:
1702:
1699:
1693:
1690:
1655:
1653:
1649:
1645:
1641:
1638:
1634:
1630:
1626:
1621:
1617:
1614:
1611:
1607:
1605:
1601:
1598:
1593:
1590:
1560:
1558:
1554:
1551:
1547:
1544:
1541:
1538:
1535:
1532:
1525:
1522:
1519:
1515:
1513:
1509:
1507:
1505:
1499:
1496:
1476:
1474:
1470:
1466:
1462:
1459:
1456:
1453:
1450:
1447:
1443:
1440:
1436:
1434:
1430:
1423:
1418:
1415:
1385:
1382:
1379:
1375:
1371:
1368:
1365:
1360:
1357:
1354:
1349:
1345:
1341:
1338:
1334:
1332:
1328:
1325:
1319:
1316:
1281:
1278:
1275:
1272:
1267:
1264:
1261:
1257:
1254:
1249:
1244:
1241:
1238:
1234:
1232:
1229:
1227:
1225:
1219:
1216:
1196:
1194:
1191:
1188:
1184:
1181:
1179:
1171:
1169:
1165:
1163:
1159:
1149:
1144:
1139:
1136:
1134:
1129:
1124:
1119:
1112:
1110:
1106:
1105:frieze groups
1102:
1098:
1094:
1090:
1086:
1082:
1078:
1073:
1071:
1067:
1066:frieze groups
1063:
1059:
1055:
1051:
1047:
1043:
1039:
1035:
1031:
1021:
1019:
1015:
1011:
998:
994:
991:
987:
981:
977:
974:
970:
964:
957:
953:
950:
946:
945:
943:
940:
937:
933:
930:
929:
928:
925:
923:
913:
910:
906:
902:
897:
895:
891:
886:
882:
878:
870:
865:
861:
859:
855:
850:
848:
844:
840:
836:
832:
828:
827:torsion group
824:
820:
814:
804:
802:
801:
800:
794:
793:
783:
780:
779:
778:
775:
773:
767:
764:
757:
753:
749:
745:
744:
739:
732:
725:
714:
711:
707:
702:
698:
694:
689:
684:
677:
667:
663:
658:
654:
650:
645:
641:
637:
633:
629:
626:
623:
620:
613:
609:
608:
606:
598:
594:
589:
585:
581:
576:
572:
568:
564:
560:
557:
553:
549:
548:
546:
545:
544:
542:
539:
535:
529:
519:
517:
516:
511:
507:
503:
499:
494:
492:
488:
483:
481:
477:
473:
469:
465:
461:
460:
455:
451:
447:
442:
440:
436:
432:
428:
424:
422:
418:
414:
410:
406:
402:
398:
394:
390:
388:
383:
372:
369:
361:
358:February 2024
351:
311:
307:
305:
298:
289:
288:
243:
238:
234:
197:
192:
188:
151:
146:
142:
141:
137:
133:
95:
90:
86:
59:
54:
50:
33:
28:
24:
23:
20:
9689:Group theory
9645:
9611:
9607:
9584:
9571:
9548:
9539:
9529:
9522:
9515:
9508:
9496:
9484:Please help
9479:verification
9476:
9437:
9429:
9421:
9400:(2): 85β96,
9397:
9391:
9378:
9359:
9353:
9326:
9320:
9303:
9299:
9239:Group action
9185:
9168:
9164:
9162:
9155:
9146:
9140:
9036:
9032:
9028:
8992:
8988:
8952:
8935:
8931:
8929:
8923:
8894:
8886:
8875:
8864:
8804:
8796:
8778:
8773:
8751:
8742:
8741:
8714:
8705:
8704:
8680:
8673:
8672:
8664:
8663:
8640:
8634:
8633:
8604:
8601:
8575:
8537:
8536:
8502:
8501:
8471:
8470:
8441:
8436:
8427:
8421:
8415:
8410:
8406:
8394:
8388:
8384:
8383:
8380:
8334:
8301:
8251:
8250:
8216:
8215:
8186:
8181:
8173:
8164:
8158:
8152:
8147:
8146:
8138:
8114:
8101:
8094:
8089:
8084:
8077:
8053:
8046:
8017:
8010:
8003:
7974:
7967:
7929:
7928:
7920:
7919:
7911:
7910:
7902:
7901:
7872:
7865:
7835:
7834:
7826:
7825:
7817:
7816:
7786:
7785:
7777:
7776:
7738:
7737:
7729:
7728:
7720:
7719:
7685:
7678:
7676:
7665:
7638:
7630:
7624:
7617:
7613:
7609:
7605:
7599:
7595:
7586:
7582:
7576:
7572:
7563:
7559:
7553:
7549:
7541:
7537:
7532:
7525:
7512:
7504:
7496:
7488:
7479:
7475:
7471:
7467:
7462:
7458:
7456:
7450:
7446:
7441:
7433:
7399:
7392:
7385:
7356:
7327:
7320:
7291:
7253:
7252:
7244:
7243:
7235:
7234:
7205:
7175:
7174:
7166:
7165:
7135:
7134:
7104:
7103:
7095:
7094:
7086:
7085:
7055:
7054:
7021:
7012:
7005:
7001:
6997:
6996:; it has an
6991:
6987:
6983:
6979:
6969:
6962:
6956:
6951:
6946:
6937:
6933:
6928:
6924:cyclic group
6918:
6914:
6906:
6900:
6893:
6888:
6884:
6880:
6878:
6870:
6863:
6856:
6854:
6829:
6827:
6824:
6811:
6804:
6797:
6790:
6788:
6780:
6773:
6766:
6764:
6753:
6749:
6743:
6739:
6734:
6730:
6724:
6720:
6715:
6711:
6707:
6703:
6698:
6694:
6687:
6684:cyclic group
6681:
6666:
6658:
6653:
6644:
6640:
6631:
6627:
6621:
6617:
6608:
6604:
6592:
6588:
6579:
6575:
6569:
6564:
6555:
6551:
6540:
6535:
6526:
6522:
6516:
6511:
6504:
6495:
6487:
6480:
6471:
6466:
6462:
6458:
6450:
6420:
6416:
6414:
6406:
6402:
6395:
6391:
6389:
6312:
6308:
6306:
6291:
6283:
6273:
6265:
6255:
6247:
6238:
6234:
6222:
6218:
6209:
6205:
6197:
6192:
6180:
6176:
6167:
6163:
6156:
6149:
6141:
6132:
6127:
6119:
6115:
6113:
6107:
6103:
6097:
6093:
6089:
6085:
6078:
6074:
6072:
6066:
6062:
6054:
6043:
6033:
6015:
6002:
5994:
5983:
5974:
5966:
5953:
5944:
5940:
5930:
5926:
5918:
5914:
5905:
5901:
5894:
5889:
5881:
5877:
5869:
5865:
5826:
5819:
5813:
5809:
5805:
5798:
5794:
5787:
5783:
5776:
5769:
5765:
5757:
5753:
5747:
5743:
5737:
5733:
5728:
5724:
5716:
5712:
5707:
5702:
5696:
5692:
5686:
5682:
5677:
5673:
5665:
5659:
5652:
5645:
5643:
5639:dodecahedron
5618:
5614:
5610:
5597:
5593:
5584:
5580:
5578:
5544:
5527:
5497:
5458:
5438:
5399:
5392:
5343:
5303:
5245:
5195:
5145:
5092:
5084:
5080:
5048:
5044:
5038:
5033:
5029:
5025:
5019:
5015:
5009:
5005:
4991:
4757:3+6 mirrors
4615:
4602:
4591:
4585:
4574:
4567:
4560:
4553:
4549:
4542:
4535:
4527:
4526:
4520:
4463:
4462:
4446:
4445:
4436:
4435:
4424:
4415:
4414:
4406:
4399:
4392:
4384:
4383:
4372:dodecahedron
4358:
4317:
4316:
4293:
4289:
4282:
4275:
4268:
4261:
4257:
4250:
4243:
4236:
4232:
4226:
4174:
4173:
4130:
4114:
4106:
4099:
4095:
4088:
4081:
4074:
4067:
4060:
4056:
4047:
4006:
4005:
3994:
3987:
3984:pyritohedron
3976:
3966:
3962:
3955:
3948:
3941:
3934:
3927:
3926:axes become
3920:
3916:
3906:
3889:
3838:
3837:
3821:
3814:
3807:
3796:
3789:
3782:
3771:
3764:
3757:
3750:
3743:
3736:
3729:
3725:
3719:
3707:
3677:
3630:
3629:
3606:
3599:
3580:
3569:
3568:, and three
3554:
3552:
3540:
3521:
3500:
3459:
3458:
3432:
3428:
3423:
3419:
3413:
3393:
3389:
3383:
3379:
3337:
3320:
3316:
3274:
3259:
3255:
3223:
3206:
3201:
3195:
3191:
3149:
3132:
3128:
3096:
3053:
3046:
3039:
3028:
3024:
3022:
3015:
3011:
3005:
3001:
2995:
2991:
2986:
2982:
2969:
2965:
2959:
2955:
2940:
2935:
2922:
2917:
2904:
2899:
2893:
2889:
2884:
2879:
2874:
2870:
2857:
2852:
2846:
2842:
2835:
2831:
2826:
2822:
2817:
2813:
2808:
2804:
2791:
2787:
2782:
2778:
2767:
2763:
2754:
2750:
2741:
2737:
2728:
2724:
2715:
2711:
2705:
2701:
2690:
2686:
2677:
2673:
2664:
2660:
2651:
2647:
2638:
2634:
2628:
2624:
2613:
2608:
2599:
2594:
2585:
2580:
2575:
2571:
2558:
2553:
2548:
2543:
2537:
2533:
2529:
2525:
2520:
2516:
2511:
2507:
2502:
2498:
2494:
2489:
2485:
2480:
2476:
2471:
2465:
2458:
2453:
2451:
2443:
2436:
2435:
2426:
2419:
2410:
2403:
2394:
2387:
2378:
2371:
2364:
2359:
2348:
2344:
2340:
2336:
2331:
2327:
2323:
2317:
2313:
2309:
2304:
2300:
2298:
2283:
2264:
2257:
2250:
2243:
2236:
2229:
2222:
2220:
2214:
2210:
2202:
2194:
2188:
2184:
2176:
2168:
2162:
2158:
2154:
2148:
2144:
2138:
2134:
2132:
2126:
2122:
2117:
2113:
2111:
2106:
2101:
2097:
2088:
2084:
2080:
2076:
2074:
2064:
2060:
2055:
2051:
2043:
2039:
2033:
2029:
2025:
2021:
2015:
2011:
2007:
2003:
1998:
1994:
1988:
1982:
1968:
1962:
1955:
1948:
1941:
1933:
1926:
1918:
1911:
1904:
1895:
1891:
1885:
1881:
1874:
1869:
1865:
1863:
1853:
1847:
1836:
1830:
1824:
1822:
1802:
1791:
1751:
1743:
1739:
1732:
1724:
1704:
1697:
1651:
1643:
1639:
1632:
1624:
1603:
1596:
1556:
1549:
1545:
1539:
1533:
1511:
1503:
1472:
1464:
1460:
1454:
1448:
1432:
1421:
1380:
1373:
1369:
1363:
1355:
1343:
1330:
1323:
1276:
1270:
1265:
1259:
1252:
1242:
1230:
1223:
1192:
1186:
1182:
1177:
1167:
1161:
1108:
1074:
1069:
1049:
1045:
1037:
1033:
1029:
1027:
1007:
996:
992:
979:
975:
962:
955:
951:
926:
919:
909:Pierre Curie
904:
900:
898:
893:
874:
867:An unmarked
851:
823:cyclic group
818:
816:
799:space groups
797:
796:
791:
790:
787:
776:
771:
765:
762:
755:
751:
747:
742:
737:
730:
723:
720:
709:
705:
700:
696:
690:
682:
675:
673:
665:
656:
652:
643:
639:
635:
631:
618:
611:
596:
587:
583:
574:
570:
566:
562:
555:
551:
540:
531:
513:
495:
484:
467:
458:
449:
445:
443:
425:
385:
379:
364:
355:
348:Please help
301:
18:
9659:(uses Java)
9254:Space group
9244:Point group
9196:lens spaces
9031:-gon, β¨2,2,
8950:, β¨2,3,5β©.
8871:conic solid
7616:= 1 we get
6682:In 2D, the
6008:, , (*532)
5980:, , (*432)
5959:, , (*332)
5923:, , (*n22)
5635:icosahedron
5623:tetrahedron
4766:15 mirrors
4442:quaternions
4368:icosahedron
4307:. See also
4140:and the 24
4127:quaternions
3940:subgroups.
3827:. See also
3566:tetrahedron
2221:The groups
1864:The groups
1172:(cylinder)
1123:Schoenflies
847:free groups
736:of a group
638:is denoted
569:is denoted
464:orientation
387:point group
320:instead of
138:, , (*n32)
9678:Categories
9512:newspapers
9458:References
9194:(and also
9158:pin groups
9035:β©, order 4
8897:spin group
8420:of order 8
8405:. For odd
8393:of order 4
8157:of order 4
7604:of order 4
7581:of order 4
7558:of order 2
6124:identified
6018:, , (532)
5997:, , (432)
5989:, , (3*2)
5969:, , (332)
5948:, , (n22)
5935:, , (2*n)
5874:, , (*nn)
5846:Reflection
5627:octahedron
5057:hemisphere
4986:3 mirrors
4977:2 mirrors
4893:4 mirrors
4884:3 mirrors
4875:2 mirrors
4748:6 mirrors
4624:symmetry.
4421:isomorphic
4374:. It is a
4305:octahedron
4123:octahedron
4059:, but the
3903:volleyball
3728:, and the
3577:isomorphic
3448:, and the
3416:polyhedral
3071:SchΓΆnflies
2322:of order 2
2299:The group
2105:of order 2
2083:is called
2020:of order 2
843:generators
811:See also:
754:such that
693:circumflex
647:(from the
578:(from the
526:See also:
431:symmetries
413:isometries
407:O(3), the
399:. It is a
9271:Footnotes
8918:Lie group
7650:≅
7591:antiprism
7509:antiprism
7501:bipyramid
7466:contains
7026:applies:
6692:rotations
6434:∖
6366:−
6360:×
6351:∖
6339:∪
6040:inversion
5909:, , (nn)
5898:, , (nΓ)
5886:, , (n*)
5827:Given in
5607:bipyramid
4968:1 mirror
4517:order 120
4260:now give
3912:symmetry.
3085:Limit of
2566:, where Ο
2532:, ..., 2(
2199:antiprism
2181:bipyramid
2070:propeller
2002:of order
1920:inversion
1827:we have Z
1817:symmetry
1815:Prismatic
1719:symmetry
1526:symmetry;
1524:Pyramidal
1350:symmetry
1155:Comments
1079:(used in
885:rotations
743:conjugate
717:Conjugacy
668:is even).
662:generates
593:generates
498:chemistry
331:SET MINUS
17:Selected
9628:33947139
9542:May 2010
9414:40755219
9290:(1894).
9229:Symmetry
9202:See also
7608:for odd
6911:symmetry
5859:Rotation
5661:dihedron
5111:notation
5083:, where
4548:because
4470:, (*532)
4455:icosians
4370:and the
4355:order 60
4223:order 48
4181:, (*432)
4044:order 24
3886:order 24
3674:order 24
3637:, (*332)
3497:order 12
3076:Orbifold
2839:U added.
2830:U, ...,
2278:such as
2272:twistane
1922:symmetry
1823:For odd
1152:Example
1128:Orbifold
936:cylinder
839:rational
660:that it
591:that it
508:forming
502:molecule
401:subgroup
382:geometry
328:∖
249:, (*532)
203:, (*432)
157:, (*332)
101:, (*n22)
9526:scholar
9426:Coxeter
9165:groups,
8987:of an (
8920:is the
8596:
8567:
7589:-sided
7568:pyramid
7566:-sided
7515:-sided
7507:-sided
7499:-sided
7491:-sided
7487:of the
7483:is the
6110:β© SO(3)
5590:pyramid
5130:Mirrors
5121:Coxeter
5109:Coxeter
4359:chiral
4321:, (532)
4048:chiral
4010:, (432)
3961:(since
3845:, (3*2)
3501:chiral
3463:, (332)
3334:m, β/mm
3332:∞
3212:SO(2)ΓZ
3144:∞
3081:Coxeter
2977:, and Ο
2934:, ...,
2762:, ...,
2685:, ...,
2607:, ...,
2515:, ...,
2343:. Like
2274:and in
2205:-gonal
2197:-gonal
2179:-gonal
2171:-gonal
2048:pyramid
2046:-sided
1934:2-fold
1844:and Dih
1143:Struct.
1133:Coxeter
504:and of
435:bounded
411:of all
403:of the
65:, (*nn)
9626:
9591:
9528:
9521:
9514:
9507:
9499:
9412:
9366:
9341:
9169:spaces
9027:of an
8936:binary
8554:= Dih
8172:. For
7612:. For
7521:chiral
7442:In 2D
6982:where
6690:-fold
6663:
6492:Parity
6288:
6270:
6252:
6146:Parity
6051:matrix
6042:group
5822:chiral
5123:number
5075:, and
5032:, and
4430:, the
4217:2/m, m
4109:to an
3802:to an
3777:, the
3763:axes.
3619:(the "
3586:, the
3440:, the
3400:O(2)ΓZ
3138:SO(2)
3010:, and
2952:added.
2775:added.
2698:added.
2621:added.
2068:is a
1880:) and
1346:-fold
1138:Frieze
1095:, and
1040:-fold
961:, and
869:sphere
854:closed
480:chiral
439:origin
397:sphere
391:is an
341:\
338:
336:U+005C
325:
323:U+2216
312:- Use
9624:S2CID
9576:73β82
9533:JSTOR
9519:books
9447:(PDF)
9410:S2CID
9388:(PDF)
9296:(PDF)
8910:SU(2)
8611:Order
8602:etc.
8448:Order
8439:β₯1).
8381:etc.
8362:Γ Dih
8280:Γ Dih
8193:Order
8184:β₯ 2.
8143:Other
8139:etc.
7942:= Dih
7697:Order
7493:prism
7434:etc.
7032:Order
6415:Thus
6317:index
6126:with
6122:) is
5742:, or
5603:prism
5116:Order
5104:group
5059:. In
5055:or a
5053:lunes
4608:, or
4601:ββm,
4596:SO(3)
4594:, or
4508:2/m,
4409:(see
4380:index
4111:orbit
3905:have
3804:orbit
3326:O(2)
3266:O(2)
3146:, β/m
2865:and Ο
2579:with
2528:, 4Ο/
2173:prism
1851:= Dih
1160:Even
1148:Order
881:SO(3)
472:SO(3)
409:group
39:, (*)
9589:ISBN
9505:news
9364:ISBN
9339:ISBN
8927:.)
8865:The
7627:β₯ 3.
6905:for
6901:The
6810:and
6796:and
6772:and
6585:even
6532:even
6215:even
6173:even
6092:Γ {
5701:or
5633:and
5631:cube
5617:and
5445:p+1
5387:2+1
5337:1+1
5240:3+6
5133:(m)
5126:(h)
5102:Weyl
5081:nh/2
4940:, ,
4916:, ,
4902:, ,
4837:, ,
4799:, ,
4775:, ,
4710:, ,
4676:, ,
4642:, ,
4610:O(3)
4590:ββ,
4552:and
4452:unit
4303:and
4301:cube
4292:and
4242:and
4135:unit
4121:and
4119:cube
3965:and
3623:").
3614:unit
3562:cube
3444:and
3345:*22β
2888:and
2425:and
2409:and
2393:and
2370:and
2291:unit
2256:and
2228:and
2143:(or
1961:) β
1954:and
1917:) β
1786:p2mm
1727:/mmm
1691:p2mg
1591:p211
1497:p1m1
1416:p11m
1317:p11g
1166:Odd
1118:Intl
1103:and
995:and
978:and
740:are
704:and
454:E(3)
384:, a
9616:doi
9488:by
9402:doi
9331:doi
9308:doi
8810:Γ Z
8793:120
8757:Γ Z
8720:Γ Z
8587:Γ Z
8583:Dih
8558:Γ Z
8550:Γ Z
8546:Dih
8515:Γ Z
8511:Dih
8431:Γ Z
8401:Γ Z
8358:= Z
8354:Γ Z
8350:= Z
8346:Γ Z
8338:10h
8313:Γ Z
8276:= Z
8272:Γ Z
8268:= Z
8264:Γ Z
8229:Γ Z
8168:Γ Z
8120:Γ Z
8109:Dih
8061:Dih
8025:Dih
7982:Dih
7946:Γ Z
7938:Dih
7880:Dih
7844:Dih
7795:Dih
7755:Γ Z
7751:= Z
7747:Dih
7670:Γ Z
7662:Dih
7635:Dih
7530:Dih
7415:Γ Z
7411:= Z
7270:Γ Z
7266:= Z
6913:is
6848:Dic
6840:Γ Z
6830:not
6686:of
6637:any
6601:odd
6548:odd
6494:of
6315:of
6231:odd
6189:odd
6148:of
6096:, β
6065:, β
6057:):
5835:, (
5812:or
5797:or
5503:(p)
5398:(p)
5290:15
5284:120
4423:to
4419:is
4378:of
4353:532
4213:4/m
4042:432
3881:, m
3877:2/m
3598:of
3579:to
3282:22β
3231:*ββ
3066:HβM
3052:or
2973:, Ο
2821:U,
2449:).
2157:.
2059:or
1857:Γ Z
1840:Γ Z
1834:= Z
1789:Dih
1750:*22
1694:Dih
1594:Dih
1501:Dih
1091:),
1083:),
903:or
856:as
774:).
664:if
493:.
380:In
333:or
316:or
9680::
9622:,
9612:13
9610:,
9428:,
9408:,
9396:,
9390:,
9337:.
9298:.
9279:^
9153:.
9133::
9100::
9067::
9023::
8983::
8816:31
8786:15
8770:60
8763:19
8738:48
8701:24
8671:,
8660:24
8630:12
8593:19
8579:8h
8572:32
8564:15
8541:6h
8533:24
8521:11
8506:4h
8498:16
8475:2h
8344:10
8331:20
8305:8h
8298:16
8255:6h
8247:12
8220:4h
8155:,h
8126:11
8113:=
8111:10
8105:5d
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1362:2
1356:n
1344:n
1342:2
1331:n
1329:2
1324:n
1322:2
1320:Z
1279:Γ
1277:n
1271:n
1269:2
1266:S
1260:n
1253:n
1251:2
1243:n
1231:n
1224:n
1221:Z
1187:n
1183:C
1178:n
1168:n
1162:n
1109:n
1070:n
1050:n
1046:n
1038:n
1034:n
1030:n
1002:)
1000:h
997:I
993:I
985:)
983:h
980:O
976:O
968:)
966:d
963:T
959:h
956:T
952:T
894:R
772:g
769:2
766:H
763:g
759:1
756:H
752:G
748:g
738:G
734:2
731:H
727:1
724:H
710:n
706:Ε
701:n
697:Δ
686:1
683:S
679:2
676:S
666:n
657:n
653:S
644:n
640:S
636:n
632:n
621:.
615:i
612:C
600:1
597:C
588:n
584:C
575:n
571:C
567:n
563:n
558:.
556:I
552:E
541:R
371:)
365:(
360:)
356:(
352:.
306:.
247:h
245:I
201:h
199:O
155:d
153:T
97:D
61:C
37:s
35:C
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