Grup titik

240pxBauhinia blakeana bunga di bendera wilayah Hong Kong memiliki simetri C 5 ; bintang di setiap kelopak memiliki simetri D 5 .160pxYin dan Yang simbol memiliki geometri simetri C 2  dengan warna terbalik

Dalam geometri, grup titik adalah grup geometris simetri (isometri) yang menjaga setidaknya satu titik tetap. Kelompok titik dapat ada dalam ruang Euklides dengan dimensi apa pun, dan setiap kelompok titik dalam dimensi  d  adalah subkelompok dari grup ortogonal O(d). Kelompok titik dapat direalisasikan sebagai himpunan matriks ortogonal  M  yang mengubah titik  x  menjadi titik  y :

 y = Mx

dimana asal adalah titik tetap. Elemen kelompok titik dapat berupa rotasi (determinan dari M = 1) atau yang lain refleksi, atau rotasi tidak tepat (determinan dari  M  = −1).

Kelompok titik diskrit di lebih dari satu dimensi datang dalam keluarga tak berhingga, tetapi dari teorema pembatasan kristalografi dan salah satu teorema Bieberbach. setiap jumlah dimensi hanya memiliki jumlah terbatas dari kelompok titik yang simetris di beberapa kisi atau kisi dengan nomor itu. Ini adalah grup titik kristalografi.

 Grup titik kiral dan akiral, grup refleksi 
Kelompok titik dapat diklasifikasikan ke dalam kelompok  kiral  (atau rotasi murni) dan kelompok  akiral .
Gugus kiral adalah subgrup dari  grup ortogonal khusus SO( d ): grup ini hanya berisi transformasi ortogonal yang mempertahankan orientasi, yaitu, determinan +1. Gugus akiral juga mengandung transformasi determinan −1. Dalam gugus akiral, transformasi yang mempertahankan orientasi membentuk subgrup (kiral) dari indeks 2.

Grup Coxeter Hingga atau  kelompok refleksi  adalah kelompok titik yang dihasilkan murni oleh sekumpulan cermin pantul yang melewati titik yang sama. Grup peringkat  n  Coxeter memiliki mirror  n  dan diwakili oleh diagram Coxeter-Dynkin. Notasi Coxeter menawarkan notasi tanda kurung yang setara dengan diagram Coxeter, dengan simbol markup untuk rotasi dan grup titik subsimetri lainnya. Kelompok refleksi harus akiral (kecuali untuk grup trivial yang hanya mengandung elemen identitas).

 Daftar grup titik 

 Satu dimensi 
Hanya ada dua grup titik satu dimensi, yaitu grup identitas dan kelompok refleksi.

GroupCoxeter Diagram Coxeter UrutanDeskripsiC1[ ]+ 1IdentityD1[ ]2Reflection group

 Dua dimensi 
Grup titik dalam dua dimensi, terkadang disebut grup roset.

Mereka datang dalam dua keluarga yang tidak terbatas:
 Grup siklik Cn of n  - grup rotasi lipat
 Grup dihedral Dn of n-kelompok rotasi dan refleksi lipat
Menerapkan teorema pembatasan kristalografi membatasi  n  ke nilai 1, 2, 3, 4, dan 6 untuk kedua famili, menghasilkan 10 kelompok.

 Grup Intl Orbifold Coxeter Urutan Deskripsi Cn n n• [n]+ nSiklik:  n  - rotasi lipat. Grup abstrak Z  n  , grup bilangan bulat di bawah penambahan modulo  n . Dn nm *n• [n] 2nDihedral: siklik dengan refleksi. Grup abstrak Dihn, grup dihedral.
Himpunan bagian dari grup titik pantulan murni, yang ditentukan oleh 1 atau 2 mirror, juga dapat diberikan oleh grup Coxeter dan poligon terkait. Ini termasuk 5 kelompok kristalografi. Simetri kelompok pantulan dapat digandakan dengan isomorphism, memetakan kedua cermin satu sama lain dengan cermin membagi dua, menggandakan simetri.

 Reflektif Rotasi Terkait poligon Group Grup Coxeter Diagram Coxeter Urutan Subgrup Coxeter Urutan D1 A1[ ]2C1[]+1 Digon D2 A12[2]4C2[2]+2 Persegi panjang D3 A2[3]6C3[3]+3 Segitiga sama sisi D4 BC2[4]8C4[4]+4 Persegi D5 H2[5]10C5[5]+5 Segi lima biasa D6 G2[6]12C6[6]+6 Segi enam biasa Dn I2(n)[n]2nCn[n]+n Poligon beraturan D2×2 A12×2 = [4] = 8 D3×2 A2×2 = [6] = 12 D4×2 BC2×2 = [8] = 16 D5×2 H2×2 = [10] = 20 D6×2 G2×2 = [12] = 24 Dn×2 I2(n)×2 = [2n] = 4n

 Tiga dimensi 

Grup titik dalam tiga dimensi, kadang disebut grup titik molekul setelah digunakan secara luas dalam mempelajari kesimetrian molekul kecil.

Mereka datang dalam 7 kelompok tak terbatas dari kelompok aksial atau prismatik, dan 7 kelompok polihedral atau Platonis tambahan. Dalam Notasi Schönflies, *
 Grup aksial: Cn,  S2n, Cnh,  Cnv, Dn, Dnd, Dnh
 Grup polihedral: T, Td, Th, O, Oh, I, Ih
Menerapkan teorema pembatasan kristalografi ke grup ini menghasilkan 32 grup titik kristalografi.

+ Domain fundamental berwarna genap/ganjil dari grup reflektifC1vUrutan 2C2vUrutan 4C3vUrutan 6C4vUrutan 8C5vUrutan 10C6vUrutan 12...80px80px80px80px80px80pxD1hUrutan 4D2hUrutan 8D3hUrutan 12D4hUrutan 16D5hUrutan 20D6hUrutan 24...80px80px80px80px80px80pxTdUrutan 24OhUrutan 48IhUrutan 12080px80px80px

Intl]* GeoGrup Ruang Kristalografi dalam aljabar geometris, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) PDF   Orbifold Schönflies Conway Coxeter Urutan 1  1 C1 C1 [ ]+ 1   ×1 Ci = S2 CC2 [2+,2+] 2  = m 1 *1 Cs = C1v = C1h ±C1 = CD2 [ ] 2 23456n  2233445566nn C2C3C4C5C6Cn C2C3C4C5C6Cn [2]+[3]+[4]+[5]+[6]+[n]+ 23456n mm23m4mm5m6mmnmmnm 23456n *22*33*44*55*66*nn C2vC3vC4vC5vC6vCnv CD4CD6CD8CD10CD12CD2n [2][3][4][5][6][n] 46810122n 2/m4/m6/mn/m  2 2 2 2 2 2 2*3*4*5*6*n* C2hC3hC4hC5hC6hCnh ±C2CC6±C4CC10±C6±Cn / CC2n [2,2+][2,3+][2,4+][2,5+][2,6+][2,n+] 46810122n   2×3×4×5×6×n× S4S6S8S10S12S2n CC4±C3CC8±C5CC12CC2n / ±Cn [2+,4+][2+,6+][2+,8+][2+,10+][2+,12+][2+,2n+] 46810122n Intl Geo Intl Geo Orbifold Schönflies Conway Coxeter Urutan 2223242252622n22n2        22222322422522622n D2D3D4D5D6Dn D4D6D8D10D12D2n [2,2]+[2,3]+[2,4]+[2,5]+[2,6]+[2,n]+ 46810122n mmmm24/mmmm26/mmmn/mmmm2 2 23 24 25 26 2n 2 *222*223*224*225*226*22n D2hD3hD4hD5hD6hDnh ±D4DD12±D8DD20±D12±D2n / DD4n [2,2][2,3][2,4][2,5][2,6][2,n] 8121620244n 2mm2mm2m2mm 4 6 8 10 12 n  2*22*32*42*52*62*n D2dD3dD4dD5dD6dDnd ±D4±D6DD16±D10DD24DD4n / ±D2n [2+,4][2+,6][2+,8][2+,10][2+,12][2+,2n] 8121620244n 23   332 T T [3,3]+ 12 m 4  3*2 Th ±T [3+,4] 24 3m 3 3 *332 Td TO [3,3] 24 432   432 O O [3,4]+ 24 mm 4 3 *432 Oh ±O [3,4] 48 532   532 I I [3,5]+ 60 m 5 3 *532 Ih ±I [3,5] 120(*) Ketika entri Intl digandakan, yang pertama untuk genap  n , yang kedua untuk ganjil  n .

 Grup refleksi 
Kelompok titik refleksi, ditentukan oleh 1 sampai 3 bidang cermin, juga dapat diberikan oleh grup Coxeter dan polihedra terkait. Grup [3,3] dapat digandakan, ditulis sebagai , memetakan cermin pertama dan terakhir satu sama lain, menggandakan simetri menjadi 48, dan isomorfik ke grup [4,3].
 Schönflies grup Coxeter Diagram Coxeter Urutan Terkait reguler dan Td A3[3,3]24TetrahedronTd×Dih1 = Oh A3×2 = BC3 = [4,3]= 48Oktahedron bintangOh BC3[4,3]48kubus, segi delapanIhH3[5,3]120Icosahedron, dodecahedronD3h A2×A1[3,2]12Prisma segitigaD3h×Dih1 = D6h A2×A1×2[[3],2]= 24Prisma segi enamD4h BC2×A1[4,2]16Prisma persegiD4h×Dih1 = D8h BC2×A1×2[[4],2] = [8,2]= 32Octagonal prismD5h H2×A1[5,2]20Prisma segi limaD6h G2×A1[6,2]24Prisma segi enamDnh I2(n)×A1[n,2]4nn-gonal prismaDnh×Dih1 = D2nh I2(n)×A1×2[[n],2]= 8nD2h A13[2,2]8BalokD2h×Dih1 A13×2[[2],2] = [4,2]= 16D2h×Dih3 = Oh A13×6[3[2,2]] = [4,3]= 48C3v A2[1,3]6HosohedronC4v BC2[1,4]8C5v H2[1,5]10C6v G2[1,6]12Cnv I2(n)[1,n]2nCnv×Dih1 = C2nv I2(n)×2[1,[n]] = [1,2n]= 4nC2v A12[1,2]4C2v×Dih1 A12×2[1,[2]]= 8Cs A1[1,1]2

 Four dimensions 

The  four-dimensional point groups (chiral as well as achiral) are listed in Conway and Smith, Section 4, Tables 4.1-4.3.

The following list gives the four-dimensional reflection groups (excluding those that leave a subspace fixed and that are therefore lower-dimensional reflection groups). Each group is specified as a Coxeter group, and like the polyhedral groups of 3D, it can be named by its related convex regular 4-polytope. Related pure rotational groups exist for each with half the order, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3]+ has three 3-fold gyration points and symmetry order 60. Front-back symmetric groups like [3,3,3] and [3,4,3] can be doubled, shown as double brackets in Coxeter's notation, for example  with its order doubled to 240.

Coxeter group/notationCoxeter diagramOrderRelated polytopesA4[3,3,3]1205-cellA4×2  2405-cell dual compound BC4[4,3,3]38416-cell/TesseractD4[31,1,1]192DemitesseracticD4×2 = BC4<[3,31,1]> = [4,3,3]= 384D4×6 = F4[3[31,1,1]] = [3,4,3]= 1152 F4[3,4,3]115224-cell F4×2230424-cell dual compound H4[5,3,3]14400120-cell/600-cell A3×A1[3,3,2]48Tetrahedral prism A3×A1×2[[3,3],2] = [4,3,2]= 96Octahedral prism BC3×A1[4,3,2]96 H3×A1[5,3,2]240Icosahedral prism A2×A2[3,2,3]36Duoprism A2×BC2[3,2,4]48 A2×H2[3,2,5]60 A2×G2[3,2,6]72 BC2×BC2[4,2,4]64 BC22×2128 BC2×H2[4,2,5]80 BC2×G2[4,2,6]96 H2×H2[5,2,5]100 H2×G2[5,2,6]120 G2×G2[6,2,6]144 I2(p)×I2(q)[p,2,q]4pq I2(2p)×I2(q)[[p],2,q] = [2p,2,q]= 8pq I2(2p)×I2(2q),2, = [2p,2,2q]= 16pq I2(p)2×2 8p2 I2(2p)2×2 [[[p],2,[p]]] =  = 32p2 A2×A1×A1[3,2,2]24 BC2×A1×A1[4,2,2]32 H2×A1×A1[5,2,2]40 G2×A1×A1[6,2,2]48 I2(p)×A1×A1[p,2,2]8p I2(2p)×A1×A1×2[[p],2,2] = [2p,2,2]= 16p I2(p)×A12×2[p,2,[2]] = [p,2,4]= 16p I2(2p)×A12×4,2, = [2p,2,4]= 32p A1×A1×A1×A1[2,2,2]164-orthotope A12×A1×A1×2[[2],2,2] = [4,2,2]= 32 A12×A12×4,2, = [4,2,4]= 64 A13×A1×6[3[2,2],2] = [4,3,2]= 96 A14×24[3,3[2,2,2]] = [4,3,3]= 384

 Five dimensions 
The following table gives the five-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups. Related chiral groups exist for each with half the order, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3]+ has four 3-fold gyration points and symmetry order 360.

 Coxeter group/notation Coxeterdiagrams Order Related regular and prismatic polytopesA5 [3,3,3,3]7205-simplexA5×2  14405-simplex dual compoundBC5[4,3,3,3]38405-cube, 5-orthoplexD5[32,1,1]19205-demicubeD5×2<[3,3,31,1]> = 3840 A4×A1 [3,3,3,2]2405-cell prism A4×A1×2 [[3,3,3],2]480 BC4×A1 [4,3,3,2] 768tesseract prism F4×A1 [3,4,3,2]230424-cell prism F4×A1×2 [[3,4,3],2]4608 H4×A1 [5,3,3,2]28800600-cell or 120-cell prism D4×A1 [31,1,1,2]384Demitesseract prism A3×A2 [3,3,2,3]144Duoprism A3×A2×2 [[3,3],2,3]288 A3×BC2 [3,3,2,4]192 A3×H2 [3,3,2,5] 240 A3×G2 [3,3,2,6] 288 A3×I2(p) [3,3,2,p] 48p BC3×A2 [4,3,2,3] 288 BC3×BC2 [4,3,2,4] 384 BC3×H2 [4,3,2,5] 480 BC3×G2 [4,3,2,6] 576 BC3×I2(p) [4,3,2,p] 96p H3×A2 [5,3,2,3] 720 H3×BC2 [5,3,2,4] 960 H3×H2 [5,3,2,5] 1200 H3×G2 [5,3,2,6] 1440 H3×I2(p) [5,3,2,p] 240p A3×A12 [3,3,2,2] 96 BC3×A12 [4,3,2,2] 192 H3×A12 [5,3,2,2] 480 A22×A1 [3,2,3,2] 72duoprism prism A2×BC2×A1 [3,2,4,2] 96 A2×H2×A1 [3,2,5,2] 120 A2×G2×A1 [3,2,6,2] 144 BC22×A1 [4,2,4,2] 128 BC2×H2×A1 [4,2,5,2] 160 BC2×G2×A1 [4,2,6,2] 192 H22×A1 [5,2,5,2] 200 H2×G2×A1 [5,2,6,2] 240 G22×A1 [6,2,6,2] 288 I2(p)×I2(q)×A1 [p,2,q,2] 8pq A2×A13 [3,2,2,2] 48 BC2×A13 [4,2,2,2] 64 H2×A13 [5,2,2,2] 80 G2×A13 [6,2,2,2] 96 I2(p)×A13 [p,2,2,2] 16p A15 [2,2,2,2] 325-orthotope A15×(2!) [[2],2,2,2]  = 64 A15×(2!×2!) ,2,[2],2]  = 128 A15×(3!) [3[2,2],2,2]  = 192 A15×(3!×2!) [3[2,2],2,  = 384 A15×(4!) [3,3[2,2,2],2]]  = 768 A15×(5!) [3,3,3[2,2,2,2]]  = 3840

 Six dimensions 
The following table gives the six-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups. Related pure rotational groups exist for each with half the order, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3,3]+ has five 3-fold gyration points and symmetry order 2520.

 Coxeter group Coxeterdiagram Order Related regular and prismatic polytopesA6 [3,3,3,3,3]5040 (7!)6-simplexA6×2 10080 (2×7!)6-simplex dual compoundBC6[4,3,3,3,3]46080 (26×6!)6-cube, 6-orthoplexD6[3,3,3,31,1]23040 (25×6!)6-demicubeE6[3,32,2]51840 (72×6!)122, 221A5×A1 [3,3,3,3,2]1440 (2×6!)5-simplex prismBC5×A1[4,3,3,3,2]7680 (26×5!)5-cube prismD5×A1 [3,3,31,1,2]3840 (25×5!)5-demicube prismA4×I2(p) [3,3,3,2,p]240pDuoprismBC4×I2(p) [4,3,3,2,p]768pF4×I2(p) [3,4,3,2,p]2304pH4×I2(p) [5,3,3,2,p]28800p D4×I2(p) [3,31,1,2,p]384pA4×A12 [3,3,3,2,2]480BC4×A12 [4,3,3,2,2]1536F4×A12 [3,4,3,2,2]4608H4×A12 [5,3,3,2,2]57600 D4×A12 [3,31,1,2,2]768A32 [3,3,2,3,3]576A3×BC3 [3,3,2,4,3]1152A3×H3 [3,3,2,5,3]2880BC32 [4,3,2,4,3]2304BC3×H3 [4,3,2,5,3]5760H32 [5,3,2,5,3]14400A3×I2(p)×A1 [3,3,2,p,2]96pDuoprism prismBC3×I2(p)×A1 [4,3,2,p,2]192pH3×I2(p)×A1 [5,3,2,p,2]480pA3×A13 [3,3,2,2,2]192BC3×A13 [4,3,2,2,2]384H3×A13 [5,3,2,2,2]960I2(p)×I2(q)×I2(r) [p,2,q,2,r]8pqrTriaprismI2(p)×I2(q)×A12 [p,2,q,2,2]16pqI2(p)×A14 [p,2,2,2,2]32pA16 [2,2,2,2,2]646-orthotope

 Seven dimensions 
The following table gives the seven-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups. Related chiral groups exist for each with half the order, defined by an even number of reflections, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3,3,3]+ has six 3-fold gyration points and symmetry order 20160.

Coxeter groupCoxeter diagramOrderRelated polytopesA7 [3,3,3,3,3,3]40320 (8!)7-simplexA7×2 80640 (2×8!)7-simplex dual compoundBC7[4,3,3,3,3,3]645120 (27×7!)7-cube, 7-orthoplexD7[3,3,3,3,31,1]322560 (26×7!)7-demicubeE7[3,3,3,32,1]2903040 (8×9!)321, 231, 132A6×A1 [3,3,3,3,3,2] 10080 (2×7!)BC6×A1 [4,3,3,3,3,2] 92160 (27×6!)D6×A1 [3,3,3,31,1,2] 46080 (26×6!)E6×A1 [3,3,32,1,2]103680 (144×6!)A5×I2(p) [3,3,3,3,2,p] 1440pBC5×I2(p) [4,3,3,3,2,p] 7680pD5×I2(p) [3,3,31,1,2,p] 3840pA5×A12 [3,3,3,3,2,2] 2880BC5×A12 [4,3,3,3,2,2] 15360D5×A12 [3,3,31,1,2,2] 7680A4×A3 [3,3,3,2,3,3] 2880A4×BC3 [3,3,3,2,4,3] 5760A4×H3 [3,3,3,2,5,3] 14400BC4×A3 [4,3,3,2,3,3] 9216BC4×BC3 [4,3,3,2,4,3] 18432BC4×H3 [4,3,3,2,5,3] 46080H4×A3 [5,3,3,2,3,3] 345600H4×BC3 [5,3,3,2,4,3] 691200H4×H3 [5,3,3,2,5,3] 1728000F4×A3 [3,4,3,2,3,3] 27648F4×BC3 [3,4,3,2,4,3] 55296F4×H3 [3,4,3,2,5,3] 138240D4×A3 [31,1,1,2,3,3] 4608D4×BC3 [3,31,1,2,4,3] 9216D4×H3 [3,31,1,2,5,3] 23040A4×I2(p)×A1 [3,3,3,2,p,2]480pBC4×I2(p)×A1 [4,3,3,2,p,2] 1536pD4×I2(p)×A1 [3,31,1,2,p,2] 768pF4×I2(p)×A1 [3,4,3,2,p,2] 4608pH4×I2(p)×A1 [5,3,3,2,p,2] 57600pA4×A13 [3,3,3,2,2,2] 960BC4×A13 [4,3,3,2,2,2] 3072F4×A13 [3,4,3,2,2,2] 9216H4×A13 [5,3,3,2,2,2] 115200D4×A13 [3,31,1,2,2,2] 1536A32×A1 [3,3,2,3,3,2] 1152A3×BC3×A1 [3,3,2,4,3,2] 2304A3×H3×A1 [3,3,2,5,3,2] 5760BC32×A1 [4,3,2,4,3,2] 4608BC3×H3×A1 [4,3,2,5,3,2]11520H32×A1 [5,3,2,5,3,2] 28800A3×I2(p)×I2(q) [3,3,2,p,2,q] 96pqBC3×I2(p)×I2(q) [4,3,2,p,2,q] 192pqH3×I2(p)×I2(q) [5,3,2,p,2,q] 480pqA3×I2(p)×A12 [3,3,2,p,2,2] 192pBC3×I2(p)×A12 [4,3,2,p,2,2] 384pH3×I2(p)×A12 [5,3,2,p,2,2] 960pA3×A14 [3,3,2,2,2,2] 384BC3×A14 [4,3,2,2,2,2] 768H3×A14 [5,3,2,2,2,2] 1920I2(p)×I2(q)×I2(r)×A1 [p,2,q,2,r,2] 16pqrI2(p)×I2(q)×A13 [p,2,q,2,2,2] 32pqI2(p)×A15 [p,2,2,2,2,2] 64pA17 [2,2,2,2,2,2] 128

 Eight dimensions 
The following table gives the eight-dimensional reflection groups (excluding those that are lower-dimensional reflection groups), by listing them as Coxeter groups. Related chiral groups exist for each with half the order, defined by an even number of reflections, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3,3,3,3]+ has seven 3-fold gyration points and symmetry order 181440.

Coxeter groupCoxeter diagramOrderRelated polytopesA8 [3,3,3,3,3,3,3]362880 (9!)8-simplexA8×2   725760 (2×9!)8-simplex dual compoundBC8[4,3,3,3,3,3,3]10321920 (288!)8-cube,8-orthoplexD8[3,3,3,3,3,31,1]5160960 (278!)8-demicubeE8[3,3,3,3,32,1]696729600 (192×10!)421, 241, 142A7×A1 [3,3,3,3,3,3,2]806407-simplex prismBC7×A1[4,3,3,3,3,3,2]6451207-cube prismD7×A1[3,3,3,3,31,1,2]3225607-demicube prismE7×A1[3,3,3,32,1,2]5806080321 prism, 231 prism, 142 prismA6×I2(p) [3,3,3,3,3,2,p] 10080pduoprismBC6×I2(p) [4,3,3,3,3,2,p] 92160pD6×I2(p) [3,3,3,31,1,2,p] 46080pE6×I2(p) [3,3,32,1,2,p] 103680pA6×A12 [3,3,3,3,3,2,2] 20160BC6×A12 [4,3,3,3,3,2,2] 184320D6×A12 [33,1,1,2,2] 92160E6×A12 [3,3,32,1,2,2] 207360A5×A3 [3,3,3,3,2,3,3]17280BC5×A3[4,3,3,3,2,3,3]92160D5×A3[32,1,1,2,3,3]46080A5×BC3 [3,3,3,3,2,4,3]34560BC5×BC3[4,3,3,3,2,4,3]184320D5×BC3[32,1,1,2,4,3]92160A5×H3 [3,3,3,3,2,5,3]BC5×H3[4,3,3,3,2,5,3]D5×H3[32,1,1,2,5,3]A5×I2(p)×A1 [3,3,3,3,2,p,2] BC5×I2(p)×A1 [4,3,3,3,2,p,2] D5×I2(p)×A1 [32,1,1,2,p,2] A5×A13 [3,3,3,3,2,2,2] BC5×A13 [4,3,3,3,2,2,2] D5×A13 [32,1,1,2,2,2] A4×A4[3,3,3,2,3,3,3]BC4×A4[4,3,3,2,3,3,3]D4×A4[31,1,1,2,3,3,3]F4×A4[3,4,3,2,3,3,3]H4×A4[5,3,3,2,3,3,3]BC4×BC4[4,3,3,2,4,3,3]D4×BC4[31,1,1,2,4,3,3]F4×BC4[3,4,3,2,4,3,3]H4×BC4[5,3,3,2,4,3,3]D4×D4[31,1,1,2,31,1,1]F4×D4[3,4,3,2,31,1,1]H4×D4[5,3,3,2,31,1,1]F4×F4[3,4,3,2,3,4,3]H4×F4[5,3,3,2,3,4,3]H4×H4[5,3,3,2,5,3,3]A4×A3×A1 [3,3,3,2,3,3,2] duoprism prismsA4×BC3×A1 [3,3,3,2,4,3,2] A4×H3×A1 [3,3,3,2,5,3,2] BC4×A3×A1 [4,3,3,2,3,3,2] BC4×BC3×A1 [4,3,3,2,4,3,2] BC4×H3×A1 [4,3,3,2,5,3,2] H4×A3×A1 [5,3,3,2,3,3,2] H4×BC3×A1 [5,3,3,2,4,3,2] H4×H3×A1 [5,3,3,2,5,3,2] F4×A3×A1 [3,4,3,2,3,3,2] F4×BC3×A1 [3,4,3,2,4,3,2] F4×H3×A1 [3,4,2,3,5,3,2] D4×A3×A1 [31,1,1,2,3,3,2] D4×BC3×A1 [31,1,1,2,4,3,2] D4×H3×A1 [31,1,1,2,5,3,2] A4×I2(p)×I2(q) [3,3,3,2,p,2,q]triaprismBC4×I2(p)×I2(q) [4,3,3,2,p,2,q]F4×I2(p)×I2(q) [3,4,3,2,p,2,q]H4×I2(p)×I2(q) [5,3,3,2,p,2,q]D4×I2(p)×I2(q) [31,1,1,2,p,2,q]A4×I2(p)×A12 [3,3,3,2,p,2,2]BC4×I2(p)×A12 [4,3,3,2,p,2,2]F4×I2(p)×A12 [3,4,3,2,p,2,2]H4×I2(p)×A12 [5,3,3,2,p,2,2]D4×I2(p)×A12 [31,1,1,2,p,2,2]A4×A14 [3,3,3,2,2,2,2]BC4×A14 [4,3,3,2,2,2,2]F4×A14 [3,4,3,2,2,2,2]H4×A14 [5,3,3,2,2,2,2]D4×A14 [31,1,1,2,2,2,2]A3×A3×I2(p)[3,3,2,3,3,2,p]BC3×A3×I2(p)[4,3,2,3,3,2,p]H3×A3×I2(p)[5,3,2,3,3,2,p]BC3×BC3×I2(p)[4,3,2,4,3,2,p]H3×BC3×I2(p)[5,3,2,4,3,2,p]H3×H3×I2(p)[5,3,2,5,3,2,p] A3×A3×A12[3,3,2,3,3,2,2] BC3×A3×A12[4,3,2,3,3,2,2]H3×A3×A12[5,3,2,3,3,2,2] BC3×BC3×A12[4,3,2,4,3,2,2]H3×BC3×A12[5,3,2,4,3,2,2]H3×H3×A12[5,3,2,5,3,2,2]A3×I2(p)×I2(q)×A1 [3,3,2,p,2,q,2] BC3×I2(p)×I2(q)×A1 [4,3,2,p,2,q,2] H3×I2(p)×I2(q)×A1 [5,3,2,p,2,q,2] A3×I2(p)×A13 [3,3,2,p,2,2,2] BC3×I2(p)×A13 [4,3,2,p,2,2,2] H3×I2(p)×A13 [5,3,2,p,2,2,2] A3×A15 [3,3,2,2,2,2,2] BC3×A15 [4,3,2,2,2,2,2] H3×A15 [5,3,2,2,2,2,2] I2(p)×I2(q)×I2(r)×I2(s) [p,2,q,2,r,2,s] 16pqrsI2(p)×I2(q)×I2(r)×A12 [p,2,q,2,r,2,2] 32pqrI2(p)×I2(q)×A14 [p,2,q,2,2,2,2] 64pqI2(p)×A16 [p,2,2,2,2,2,2] 128pA18 [2,2,2,2,2,2,2] 256

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 Referensi 

 H. S. M. Coxeter: Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,   
 (Paper 23) H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
 H. S. M. Coxeter and W. O. J. Moser. Generators and Relations for Discrete Groups 4th ed, Springer-Verlag. New York. 1980
 N. W. Johnson: Geometries and Transformations, (2018) Chapter 11: Finite symmetry groups

 Pranala luar 
Web-based point group tutorial  (needs Java and Flash)
Subgroup enumeration  (needs Java)
 The Geometry Center: 2.1 Formulas for Symmetries in Cartesian Coordinates (two dimensions) 
 The Geometry Center: 10.1 Formulas for Symmetries in Cartesian Coordinates (three dimensions) 

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