Polytope of Type {20,36}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,36}*1440
Also Known As : {20,36|2}. if this polytope has another name.
Group : SmallGroup(1440,552)
Rank : 3
Schlafli Type : {20,36}
Number of vertices, edges, etc : 20, 360, 36
Order of s0s1s2 : 180
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
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 : {10,36}*720, {20,18}*720a
3-fold quotients : {20,12}*480
4-fold quotients : {10,18}*360
5-fold quotients : {4,36}*288a
6-fold quotients : {10,12}*240, {20,6}*240a
9-fold quotients : {20,4}*160
10-fold quotients : {2,36}*144, {4,18}*144a
12-fold quotients : {10,6}*120
15-fold quotients : {4,12}*96a
18-fold quotients : {20,2}*80, {10,4}*80
20-fold quotients : {2,18}*72
30-fold quotients : {2,12}*48, {4,6}*48a
36-fold quotients : {10,2}*40
40-fold quotients : {2,9}*36
45-fold quotients : {4,4}*32
60-fold quotients : {2,6}*24
72-fold quotients : {5,2}*20
90-fold quotients : {2,4}*16, {4,2}*16
120-fold quotients : {2,3}*12
180-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 4, 13)( 5, 14)( 6, 15)( 7, 10)( 8, 11)( 9, 12)( 19, 28)( 20, 29)
( 21, 30)( 22, 25)( 23, 26)( 24, 27)( 34, 43)( 35, 44)( 36, 45)( 37, 40)
( 38, 41)( 39, 42)( 49, 58)( 50, 59)( 51, 60)( 52, 55)( 53, 56)( 54, 57)
( 64, 73)( 65, 74)( 66, 75)( 67, 70)( 68, 71)( 69, 72)( 79, 88)( 80, 89)
( 81, 90)( 82, 85)( 83, 86)( 84, 87)( 94,103)( 95,104)( 96,105)( 97,100)
( 98,101)( 99,102)(109,118)(110,119)(111,120)(112,115)(113,116)(114,117)
(124,133)(125,134)(126,135)(127,130)(128,131)(129,132)(139,148)(140,149)
(141,150)(142,145)(143,146)(144,147)(154,163)(155,164)(156,165)(157,160)
(158,161)(159,162)(169,178)(170,179)(171,180)(172,175)(173,176)(174,177)
(181,226)(182,227)(183,228)(184,238)(185,239)(186,240)(187,235)(188,236)
(189,237)(190,232)(191,233)(192,234)(193,229)(194,230)(195,231)(196,241)
(197,242)(198,243)(199,253)(200,254)(201,255)(202,250)(203,251)(204,252)
(205,247)(206,248)(207,249)(208,244)(209,245)(210,246)(211,256)(212,257)
(213,258)(214,268)(215,269)(216,270)(217,265)(218,266)(219,267)(220,262)
(221,263)(222,264)(223,259)(224,260)(225,261)(271,316)(272,317)(273,318)
(274,328)(275,329)(276,330)(277,325)(278,326)(279,327)(280,322)(281,323)
(282,324)(283,319)(284,320)(285,321)(286,331)(287,332)(288,333)(289,343)
(290,344)(291,345)(292,340)(293,341)(294,342)(295,337)(296,338)(297,339)
(298,334)(299,335)(300,336)(301,346)(302,347)(303,348)(304,358)(305,359)
(306,360)(307,355)(308,356)(309,357)(310,352)(311,353)(312,354)(313,349)
(314,350)(315,351);;
s1 := ( 1,184)( 2,186)( 3,185)( 4,181)( 5,183)( 6,182)( 7,193)( 8,195)
( 9,194)( 10,190)( 11,192)( 12,191)( 13,187)( 14,189)( 15,188)( 16,216)
( 17,215)( 18,214)( 19,213)( 20,212)( 21,211)( 22,225)( 23,224)( 24,223)
( 25,222)( 26,221)( 27,220)( 28,219)( 29,218)( 30,217)( 31,201)( 32,200)
( 33,199)( 34,198)( 35,197)( 36,196)( 37,210)( 38,209)( 39,208)( 40,207)
( 41,206)( 42,205)( 43,204)( 44,203)( 45,202)( 46,229)( 47,231)( 48,230)
( 49,226)( 50,228)( 51,227)( 52,238)( 53,240)( 54,239)( 55,235)( 56,237)
( 57,236)( 58,232)( 59,234)( 60,233)( 61,261)( 62,260)( 63,259)( 64,258)
( 65,257)( 66,256)( 67,270)( 68,269)( 69,268)( 70,267)( 71,266)( 72,265)
( 73,264)( 74,263)( 75,262)( 76,246)( 77,245)( 78,244)( 79,243)( 80,242)
( 81,241)( 82,255)( 83,254)( 84,253)( 85,252)( 86,251)( 87,250)( 88,249)
( 89,248)( 90,247)( 91,274)( 92,276)( 93,275)( 94,271)( 95,273)( 96,272)
( 97,283)( 98,285)( 99,284)(100,280)(101,282)(102,281)(103,277)(104,279)
(105,278)(106,306)(107,305)(108,304)(109,303)(110,302)(111,301)(112,315)
(113,314)(114,313)(115,312)(116,311)(117,310)(118,309)(119,308)(120,307)
(121,291)(122,290)(123,289)(124,288)(125,287)(126,286)(127,300)(128,299)
(129,298)(130,297)(131,296)(132,295)(133,294)(134,293)(135,292)(136,319)
(137,321)(138,320)(139,316)(140,318)(141,317)(142,328)(143,330)(144,329)
(145,325)(146,327)(147,326)(148,322)(149,324)(150,323)(151,351)(152,350)
(153,349)(154,348)(155,347)(156,346)(157,360)(158,359)(159,358)(160,357)
(161,356)(162,355)(163,354)(164,353)(165,352)(166,336)(167,335)(168,334)
(169,333)(170,332)(171,331)(172,345)(173,344)(174,343)(175,342)(176,341)
(177,340)(178,339)(179,338)(180,337);;
s2 := ( 1, 16)( 2, 18)( 3, 17)( 4, 19)( 5, 21)( 6, 20)( 7, 22)( 8, 24)
( 9, 23)( 10, 25)( 11, 27)( 12, 26)( 13, 28)( 14, 30)( 15, 29)( 31, 33)
( 34, 36)( 37, 39)( 40, 42)( 43, 45)( 46, 61)( 47, 63)( 48, 62)( 49, 64)
( 50, 66)( 51, 65)( 52, 67)( 53, 69)( 54, 68)( 55, 70)( 56, 72)( 57, 71)
( 58, 73)( 59, 75)( 60, 74)( 76, 78)( 79, 81)( 82, 84)( 85, 87)( 88, 90)
( 91,106)( 92,108)( 93,107)( 94,109)( 95,111)( 96,110)( 97,112)( 98,114)
( 99,113)(100,115)(101,117)(102,116)(103,118)(104,120)(105,119)(121,123)
(124,126)(127,129)(130,132)(133,135)(136,151)(137,153)(138,152)(139,154)
(140,156)(141,155)(142,157)(143,159)(144,158)(145,160)(146,162)(147,161)
(148,163)(149,165)(150,164)(166,168)(169,171)(172,174)(175,177)(178,180)
(181,286)(182,288)(183,287)(184,289)(185,291)(186,290)(187,292)(188,294)
(189,293)(190,295)(191,297)(192,296)(193,298)(194,300)(195,299)(196,271)
(197,273)(198,272)(199,274)(200,276)(201,275)(202,277)(203,279)(204,278)
(205,280)(206,282)(207,281)(208,283)(209,285)(210,284)(211,303)(212,302)
(213,301)(214,306)(215,305)(216,304)(217,309)(218,308)(219,307)(220,312)
(221,311)(222,310)(223,315)(224,314)(225,313)(226,331)(227,333)(228,332)
(229,334)(230,336)(231,335)(232,337)(233,339)(234,338)(235,340)(236,342)
(237,341)(238,343)(239,345)(240,344)(241,316)(242,318)(243,317)(244,319)
(245,321)(246,320)(247,322)(248,324)(249,323)(250,325)(251,327)(252,326)
(253,328)(254,330)(255,329)(256,348)(257,347)(258,346)(259,351)(260,350)
(261,349)(262,354)(263,353)(264,352)(265,357)(266,356)(267,355)(268,360)
(269,359)(270,358);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*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*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(360)!( 4, 13)( 5, 14)( 6, 15)( 7, 10)( 8, 11)( 9, 12)( 19, 28)
( 20, 29)( 21, 30)( 22, 25)( 23, 26)( 24, 27)( 34, 43)( 35, 44)( 36, 45)
( 37, 40)( 38, 41)( 39, 42)( 49, 58)( 50, 59)( 51, 60)( 52, 55)( 53, 56)
( 54, 57)( 64, 73)( 65, 74)( 66, 75)( 67, 70)( 68, 71)( 69, 72)( 79, 88)
( 80, 89)( 81, 90)( 82, 85)( 83, 86)( 84, 87)( 94,103)( 95,104)( 96,105)
( 97,100)( 98,101)( 99,102)(109,118)(110,119)(111,120)(112,115)(113,116)
(114,117)(124,133)(125,134)(126,135)(127,130)(128,131)(129,132)(139,148)
(140,149)(141,150)(142,145)(143,146)(144,147)(154,163)(155,164)(156,165)
(157,160)(158,161)(159,162)(169,178)(170,179)(171,180)(172,175)(173,176)
(174,177)(181,226)(182,227)(183,228)(184,238)(185,239)(186,240)(187,235)
(188,236)(189,237)(190,232)(191,233)(192,234)(193,229)(194,230)(195,231)
(196,241)(197,242)(198,243)(199,253)(200,254)(201,255)(202,250)(203,251)
(204,252)(205,247)(206,248)(207,249)(208,244)(209,245)(210,246)(211,256)
(212,257)(213,258)(214,268)(215,269)(216,270)(217,265)(218,266)(219,267)
(220,262)(221,263)(222,264)(223,259)(224,260)(225,261)(271,316)(272,317)
(273,318)(274,328)(275,329)(276,330)(277,325)(278,326)(279,327)(280,322)
(281,323)(282,324)(283,319)(284,320)(285,321)(286,331)(287,332)(288,333)
(289,343)(290,344)(291,345)(292,340)(293,341)(294,342)(295,337)(296,338)
(297,339)(298,334)(299,335)(300,336)(301,346)(302,347)(303,348)(304,358)
(305,359)(306,360)(307,355)(308,356)(309,357)(310,352)(311,353)(312,354)
(313,349)(314,350)(315,351);
s1 := Sym(360)!( 1,184)( 2,186)( 3,185)( 4,181)( 5,183)( 6,182)( 7,193)
( 8,195)( 9,194)( 10,190)( 11,192)( 12,191)( 13,187)( 14,189)( 15,188)
( 16,216)( 17,215)( 18,214)( 19,213)( 20,212)( 21,211)( 22,225)( 23,224)
( 24,223)( 25,222)( 26,221)( 27,220)( 28,219)( 29,218)( 30,217)( 31,201)
( 32,200)( 33,199)( 34,198)( 35,197)( 36,196)( 37,210)( 38,209)( 39,208)
( 40,207)( 41,206)( 42,205)( 43,204)( 44,203)( 45,202)( 46,229)( 47,231)
( 48,230)( 49,226)( 50,228)( 51,227)( 52,238)( 53,240)( 54,239)( 55,235)
( 56,237)( 57,236)( 58,232)( 59,234)( 60,233)( 61,261)( 62,260)( 63,259)
( 64,258)( 65,257)( 66,256)( 67,270)( 68,269)( 69,268)( 70,267)( 71,266)
( 72,265)( 73,264)( 74,263)( 75,262)( 76,246)( 77,245)( 78,244)( 79,243)
( 80,242)( 81,241)( 82,255)( 83,254)( 84,253)( 85,252)( 86,251)( 87,250)
( 88,249)( 89,248)( 90,247)( 91,274)( 92,276)( 93,275)( 94,271)( 95,273)
( 96,272)( 97,283)( 98,285)( 99,284)(100,280)(101,282)(102,281)(103,277)
(104,279)(105,278)(106,306)(107,305)(108,304)(109,303)(110,302)(111,301)
(112,315)(113,314)(114,313)(115,312)(116,311)(117,310)(118,309)(119,308)
(120,307)(121,291)(122,290)(123,289)(124,288)(125,287)(126,286)(127,300)
(128,299)(129,298)(130,297)(131,296)(132,295)(133,294)(134,293)(135,292)
(136,319)(137,321)(138,320)(139,316)(140,318)(141,317)(142,328)(143,330)
(144,329)(145,325)(146,327)(147,326)(148,322)(149,324)(150,323)(151,351)
(152,350)(153,349)(154,348)(155,347)(156,346)(157,360)(158,359)(159,358)
(160,357)(161,356)(162,355)(163,354)(164,353)(165,352)(166,336)(167,335)
(168,334)(169,333)(170,332)(171,331)(172,345)(173,344)(174,343)(175,342)
(176,341)(177,340)(178,339)(179,338)(180,337);
s2 := Sym(360)!( 1, 16)( 2, 18)( 3, 17)( 4, 19)( 5, 21)( 6, 20)( 7, 22)
( 8, 24)( 9, 23)( 10, 25)( 11, 27)( 12, 26)( 13, 28)( 14, 30)( 15, 29)
( 31, 33)( 34, 36)( 37, 39)( 40, 42)( 43, 45)( 46, 61)( 47, 63)( 48, 62)
( 49, 64)( 50, 66)( 51, 65)( 52, 67)( 53, 69)( 54, 68)( 55, 70)( 56, 72)
( 57, 71)( 58, 73)( 59, 75)( 60, 74)( 76, 78)( 79, 81)( 82, 84)( 85, 87)
( 88, 90)( 91,106)( 92,108)( 93,107)( 94,109)( 95,111)( 96,110)( 97,112)
( 98,114)( 99,113)(100,115)(101,117)(102,116)(103,118)(104,120)(105,119)
(121,123)(124,126)(127,129)(130,132)(133,135)(136,151)(137,153)(138,152)
(139,154)(140,156)(141,155)(142,157)(143,159)(144,158)(145,160)(146,162)
(147,161)(148,163)(149,165)(150,164)(166,168)(169,171)(172,174)(175,177)
(178,180)(181,286)(182,288)(183,287)(184,289)(185,291)(186,290)(187,292)
(188,294)(189,293)(190,295)(191,297)(192,296)(193,298)(194,300)(195,299)
(196,271)(197,273)(198,272)(199,274)(200,276)(201,275)(202,277)(203,279)
(204,278)(205,280)(206,282)(207,281)(208,283)(209,285)(210,284)(211,303)
(212,302)(213,301)(214,306)(215,305)(216,304)(217,309)(218,308)(219,307)
(220,312)(221,311)(222,310)(223,315)(224,314)(225,313)(226,331)(227,333)
(228,332)(229,334)(230,336)(231,335)(232,337)(233,339)(234,338)(235,340)
(236,342)(237,341)(238,343)(239,345)(240,344)(241,316)(242,318)(243,317)
(244,319)(245,321)(246,320)(247,322)(248,324)(249,323)(250,325)(251,327)
(252,326)(253,328)(254,330)(255,329)(256,348)(257,347)(258,346)(259,351)
(260,350)(261,349)(262,354)(263,353)(264,352)(265,357)(266,356)(267,355)
(268,360)(269,359)(270,358);
poly := sub<Sym(360)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*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*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope