Polytope of Type {18,4,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,4,12}*1728
Also Known As : {{18,4|2},{4,12|2}}. if this polytope has another name.
Group : SmallGroup(1728,14460)
Rank : 4
Schlafli Type : {18,4,12}
Number of vertices, edges, etc : 18, 36, 24, 12
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {18,2,12}*864, {18,4,6}*864
3-fold quotients : {18,4,4}*576, {6,4,12}*576
4-fold quotients : {9,2,12}*432, {18,2,6}*432
6-fold quotients : {18,2,4}*288, {18,4,2}*288a, {6,2,12}*288, {6,4,6}*288
8-fold quotients : {9,2,6}*216, {18,2,3}*216
9-fold quotients : {2,4,12}*192a, {6,4,4}*192
12-fold quotients : {9,2,4}*144, {18,2,2}*144, {3,2,12}*144, {6,2,6}*144
16-fold quotients : {9,2,3}*108
18-fold quotients : {2,2,12}*96, {2,4,6}*96a, {6,2,4}*96, {6,4,2}*96a
24-fold quotients : {9,2,2}*72, {3,2,6}*72, {6,2,3}*72
27-fold quotients : {2,4,4}*64
36-fold quotients : {3,2,4}*48, {2,2,6}*48, {6,2,2}*48
48-fold quotients : {3,2,3}*36
54-fold quotients : {2,2,4}*32, {2,4,2}*32
72-fold quotients : {2,2,3}*24, {3,2,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)( 10, 21)( 11, 20)( 12, 19)( 13, 24)( 14, 23)
( 15, 22)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 32, 33)( 35, 36)( 37, 48)
( 38, 47)( 39, 46)( 40, 51)( 41, 50)( 42, 49)( 43, 54)( 44, 53)( 45, 52)
( 56, 57)( 59, 60)( 62, 63)( 64, 75)( 65, 74)( 66, 73)( 67, 78)( 68, 77)
( 69, 76)( 70, 81)( 71, 80)( 72, 79)( 83, 84)( 86, 87)( 89, 90)( 91,102)
( 92,101)( 93,100)( 94,105)( 95,104)( 96,103)( 97,108)( 98,107)( 99,106)
(110,111)(113,114)(116,117)(118,129)(119,128)(120,127)(121,132)(122,131)
(123,130)(124,135)(125,134)(126,133)(137,138)(140,141)(143,144)(145,156)
(146,155)(147,154)(148,159)(149,158)(150,157)(151,162)(152,161)(153,160)
(164,165)(167,168)(170,171)(172,183)(173,182)(174,181)(175,186)(176,185)
(177,184)(178,189)(179,188)(180,187)(191,192)(194,195)(197,198)(199,210)
(200,209)(201,208)(202,213)(203,212)(204,211)(205,216)(206,215)(207,214);;
s1 := ( 1, 10)( 2, 12)( 3, 11)( 4, 13)( 5, 15)( 6, 14)( 7, 16)( 8, 18)
( 9, 17)( 19, 21)( 22, 24)( 25, 27)( 28, 37)( 29, 39)( 30, 38)( 31, 40)
( 32, 42)( 33, 41)( 34, 43)( 35, 45)( 36, 44)( 46, 48)( 49, 51)( 52, 54)
( 55, 64)( 56, 66)( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)( 62, 72)
( 63, 71)( 73, 75)( 76, 78)( 79, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 94)
( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(100,102)(103,105)(106,108)
(109,172)(110,174)(111,173)(112,175)(113,177)(114,176)(115,178)(116,180)
(117,179)(118,163)(119,165)(120,164)(121,166)(122,168)(123,167)(124,169)
(125,171)(126,170)(127,183)(128,182)(129,181)(130,186)(131,185)(132,184)
(133,189)(134,188)(135,187)(136,199)(137,201)(138,200)(139,202)(140,204)
(141,203)(142,205)(143,207)(144,206)(145,190)(146,192)(147,191)(148,193)
(149,195)(150,194)(151,196)(152,198)(153,197)(154,210)(155,209)(156,208)
(157,213)(158,212)(159,211)(160,216)(161,215)(162,214);;
s2 := ( 1,109)( 2,110)( 3,111)( 4,115)( 5,116)( 6,117)( 7,112)( 8,113)
( 9,114)( 10,118)( 11,119)( 12,120)( 13,124)( 14,125)( 15,126)( 16,121)
( 17,122)( 18,123)( 19,127)( 20,128)( 21,129)( 22,133)( 23,134)( 24,135)
( 25,130)( 26,131)( 27,132)( 28,136)( 29,137)( 30,138)( 31,142)( 32,143)
( 33,144)( 34,139)( 35,140)( 36,141)( 37,145)( 38,146)( 39,147)( 40,151)
( 41,152)( 42,153)( 43,148)( 44,149)( 45,150)( 46,154)( 47,155)( 48,156)
( 49,160)( 50,161)( 51,162)( 52,157)( 53,158)( 54,159)( 55,163)( 56,164)
( 57,165)( 58,169)( 59,170)( 60,171)( 61,166)( 62,167)( 63,168)( 64,172)
( 65,173)( 66,174)( 67,178)( 68,179)( 69,180)( 70,175)( 71,176)( 72,177)
( 73,181)( 74,182)( 75,183)( 76,187)( 77,188)( 78,189)( 79,184)( 80,185)
( 81,186)( 82,190)( 83,191)( 84,192)( 85,196)( 86,197)( 87,198)( 88,193)
( 89,194)( 90,195)( 91,199)( 92,200)( 93,201)( 94,205)( 95,206)( 96,207)
( 97,202)( 98,203)( 99,204)(100,208)(101,209)(102,210)(103,214)(104,215)
(105,216)(106,211)(107,212)(108,213);;
s3 := ( 1, 4)( 2, 5)( 3, 6)( 10, 13)( 11, 14)( 12, 15)( 19, 22)( 20, 23)
( 21, 24)( 28, 31)( 29, 32)( 30, 33)( 37, 40)( 38, 41)( 39, 42)( 46, 49)
( 47, 50)( 48, 51)( 55, 58)( 56, 59)( 57, 60)( 64, 67)( 65, 68)( 66, 69)
( 73, 76)( 74, 77)( 75, 78)( 82, 85)( 83, 86)( 84, 87)( 91, 94)( 92, 95)
( 93, 96)(100,103)(101,104)(102,105)(109,139)(110,140)(111,141)(112,136)
(113,137)(114,138)(115,142)(116,143)(117,144)(118,148)(119,149)(120,150)
(121,145)(122,146)(123,147)(124,151)(125,152)(126,153)(127,157)(128,158)
(129,159)(130,154)(131,155)(132,156)(133,160)(134,161)(135,162)(163,193)
(164,194)(165,195)(166,190)(167,191)(168,192)(169,196)(170,197)(171,198)
(172,202)(173,203)(174,204)(175,199)(176,200)(177,201)(178,205)(179,206)
(180,207)(181,211)(182,212)(183,213)(184,208)(185,209)(186,210)(187,214)
(188,215)(189,216);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 2, 3)( 5, 6)( 8, 9)( 10, 21)( 11, 20)( 12, 19)( 13, 24)
( 14, 23)( 15, 22)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 32, 33)( 35, 36)
( 37, 48)( 38, 47)( 39, 46)( 40, 51)( 41, 50)( 42, 49)( 43, 54)( 44, 53)
( 45, 52)( 56, 57)( 59, 60)( 62, 63)( 64, 75)( 65, 74)( 66, 73)( 67, 78)
( 68, 77)( 69, 76)( 70, 81)( 71, 80)( 72, 79)( 83, 84)( 86, 87)( 89, 90)
( 91,102)( 92,101)( 93,100)( 94,105)( 95,104)( 96,103)( 97,108)( 98,107)
( 99,106)(110,111)(113,114)(116,117)(118,129)(119,128)(120,127)(121,132)
(122,131)(123,130)(124,135)(125,134)(126,133)(137,138)(140,141)(143,144)
(145,156)(146,155)(147,154)(148,159)(149,158)(150,157)(151,162)(152,161)
(153,160)(164,165)(167,168)(170,171)(172,183)(173,182)(174,181)(175,186)
(176,185)(177,184)(178,189)(179,188)(180,187)(191,192)(194,195)(197,198)
(199,210)(200,209)(201,208)(202,213)(203,212)(204,211)(205,216)(206,215)
(207,214);
s1 := Sym(216)!( 1, 10)( 2, 12)( 3, 11)( 4, 13)( 5, 15)( 6, 14)( 7, 16)
( 8, 18)( 9, 17)( 19, 21)( 22, 24)( 25, 27)( 28, 37)( 29, 39)( 30, 38)
( 31, 40)( 32, 42)( 33, 41)( 34, 43)( 35, 45)( 36, 44)( 46, 48)( 49, 51)
( 52, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)
( 62, 72)( 63, 71)( 73, 75)( 76, 78)( 79, 81)( 82, 91)( 83, 93)( 84, 92)
( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(100,102)(103,105)
(106,108)(109,172)(110,174)(111,173)(112,175)(113,177)(114,176)(115,178)
(116,180)(117,179)(118,163)(119,165)(120,164)(121,166)(122,168)(123,167)
(124,169)(125,171)(126,170)(127,183)(128,182)(129,181)(130,186)(131,185)
(132,184)(133,189)(134,188)(135,187)(136,199)(137,201)(138,200)(139,202)
(140,204)(141,203)(142,205)(143,207)(144,206)(145,190)(146,192)(147,191)
(148,193)(149,195)(150,194)(151,196)(152,198)(153,197)(154,210)(155,209)
(156,208)(157,213)(158,212)(159,211)(160,216)(161,215)(162,214);
s2 := Sym(216)!( 1,109)( 2,110)( 3,111)( 4,115)( 5,116)( 6,117)( 7,112)
( 8,113)( 9,114)( 10,118)( 11,119)( 12,120)( 13,124)( 14,125)( 15,126)
( 16,121)( 17,122)( 18,123)( 19,127)( 20,128)( 21,129)( 22,133)( 23,134)
( 24,135)( 25,130)( 26,131)( 27,132)( 28,136)( 29,137)( 30,138)( 31,142)
( 32,143)( 33,144)( 34,139)( 35,140)( 36,141)( 37,145)( 38,146)( 39,147)
( 40,151)( 41,152)( 42,153)( 43,148)( 44,149)( 45,150)( 46,154)( 47,155)
( 48,156)( 49,160)( 50,161)( 51,162)( 52,157)( 53,158)( 54,159)( 55,163)
( 56,164)( 57,165)( 58,169)( 59,170)( 60,171)( 61,166)( 62,167)( 63,168)
( 64,172)( 65,173)( 66,174)( 67,178)( 68,179)( 69,180)( 70,175)( 71,176)
( 72,177)( 73,181)( 74,182)( 75,183)( 76,187)( 77,188)( 78,189)( 79,184)
( 80,185)( 81,186)( 82,190)( 83,191)( 84,192)( 85,196)( 86,197)( 87,198)
( 88,193)( 89,194)( 90,195)( 91,199)( 92,200)( 93,201)( 94,205)( 95,206)
( 96,207)( 97,202)( 98,203)( 99,204)(100,208)(101,209)(102,210)(103,214)
(104,215)(105,216)(106,211)(107,212)(108,213);
s3 := Sym(216)!( 1, 4)( 2, 5)( 3, 6)( 10, 13)( 11, 14)( 12, 15)( 19, 22)
( 20, 23)( 21, 24)( 28, 31)( 29, 32)( 30, 33)( 37, 40)( 38, 41)( 39, 42)
( 46, 49)( 47, 50)( 48, 51)( 55, 58)( 56, 59)( 57, 60)( 64, 67)( 65, 68)
( 66, 69)( 73, 76)( 74, 77)( 75, 78)( 82, 85)( 83, 86)( 84, 87)( 91, 94)
( 92, 95)( 93, 96)(100,103)(101,104)(102,105)(109,139)(110,140)(111,141)
(112,136)(113,137)(114,138)(115,142)(116,143)(117,144)(118,148)(119,149)
(120,150)(121,145)(122,146)(123,147)(124,151)(125,152)(126,153)(127,157)
(128,158)(129,159)(130,154)(131,155)(132,156)(133,160)(134,161)(135,162)
(163,193)(164,194)(165,195)(166,190)(167,191)(168,192)(169,196)(170,197)
(171,198)(172,202)(173,203)(174,204)(175,199)(176,200)(177,201)(178,205)
(179,206)(180,207)(181,211)(182,212)(183,213)(184,208)(185,209)(186,210)
(187,214)(188,215)(189,216);
poly := sub<Sym(216)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope