Polytope of Type {4,90}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,90}*720c
if this polytope has a name.
Group : SmallGroup(720,397)
Rank : 3
Schlafli Type : {4,90}
Number of vertices, edges, etc : 4, 180, 90
Order of s0s1s2 : 45
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,90,2} of size 1440
Vertex Figure Of :
{2,4,90} of size 1440
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,45}*360
3-fold quotients : {4,30}*240c
5-fold quotients : {4,18}*144c
6-fold quotients : {4,15}*120
10-fold quotients : {4,9}*72
15-fold quotients : {4,6}*48b
30-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,90}*1440
Permutation Representation (GAP) :
s0 := ( 1,183)( 2,184)( 3,181)( 4,182)( 5,187)( 6,188)( 7,185)( 8,186)
( 9,191)( 10,192)( 11,189)( 12,190)( 13,195)( 14,196)( 15,193)( 16,194)
( 17,199)( 18,200)( 19,197)( 20,198)( 21,203)( 22,204)( 23,201)( 24,202)
( 25,207)( 26,208)( 27,205)( 28,206)( 29,211)( 30,212)( 31,209)( 32,210)
( 33,215)( 34,216)( 35,213)( 36,214)( 37,219)( 38,220)( 39,217)( 40,218)
( 41,223)( 42,224)( 43,221)( 44,222)( 45,227)( 46,228)( 47,225)( 48,226)
( 49,231)( 50,232)( 51,229)( 52,230)( 53,235)( 54,236)( 55,233)( 56,234)
( 57,239)( 58,240)( 59,237)( 60,238)( 61,243)( 62,244)( 63,241)( 64,242)
( 65,247)( 66,248)( 67,245)( 68,246)( 69,251)( 70,252)( 71,249)( 72,250)
( 73,255)( 74,256)( 75,253)( 76,254)( 77,259)( 78,260)( 79,257)( 80,258)
( 81,263)( 82,264)( 83,261)( 84,262)( 85,267)( 86,268)( 87,265)( 88,266)
( 89,271)( 90,272)( 91,269)( 92,270)( 93,275)( 94,276)( 95,273)( 96,274)
( 97,279)( 98,280)( 99,277)(100,278)(101,283)(102,284)(103,281)(104,282)
(105,287)(106,288)(107,285)(108,286)(109,291)(110,292)(111,289)(112,290)
(113,295)(114,296)(115,293)(116,294)(117,299)(118,300)(119,297)(120,298)
(121,303)(122,304)(123,301)(124,302)(125,307)(126,308)(127,305)(128,306)
(129,311)(130,312)(131,309)(132,310)(133,315)(134,316)(135,313)(136,314)
(137,319)(138,320)(139,317)(140,318)(141,323)(142,324)(143,321)(144,322)
(145,327)(146,328)(147,325)(148,326)(149,331)(150,332)(151,329)(152,330)
(153,335)(154,336)(155,333)(156,334)(157,339)(158,340)(159,337)(160,338)
(161,343)(162,344)(163,341)(164,342)(165,347)(166,348)(167,345)(168,346)
(169,351)(170,352)(171,349)(172,350)(173,355)(174,356)(175,353)(176,354)
(177,359)(178,360)(179,357)(180,358);;
s1 := ( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 13, 49)( 14, 51)( 15, 50)
( 16, 52)( 17, 57)( 18, 59)( 19, 58)( 20, 60)( 21, 53)( 22, 55)( 23, 54)
( 24, 56)( 25, 37)( 26, 39)( 27, 38)( 28, 40)( 29, 45)( 30, 47)( 31, 46)
( 32, 48)( 33, 41)( 34, 43)( 35, 42)( 36, 44)( 61,125)( 62,127)( 63,126)
( 64,128)( 65,121)( 66,123)( 67,122)( 68,124)( 69,129)( 70,131)( 71,130)
( 72,132)( 73,173)( 74,175)( 75,174)( 76,176)( 77,169)( 78,171)( 79,170)
( 80,172)( 81,177)( 82,179)( 83,178)( 84,180)( 85,161)( 86,163)( 87,162)
( 88,164)( 89,157)( 90,159)( 91,158)( 92,160)( 93,165)( 94,167)( 95,166)
( 96,168)( 97,149)( 98,151)( 99,150)(100,152)(101,145)(102,147)(103,146)
(104,148)(105,153)(106,155)(107,154)(108,156)(109,137)(110,139)(111,138)
(112,140)(113,133)(114,135)(115,134)(116,136)(117,141)(118,143)(119,142)
(120,144)(182,183)(185,189)(186,191)(187,190)(188,192)(193,229)(194,231)
(195,230)(196,232)(197,237)(198,239)(199,238)(200,240)(201,233)(202,235)
(203,234)(204,236)(205,217)(206,219)(207,218)(208,220)(209,225)(210,227)
(211,226)(212,228)(213,221)(214,223)(215,222)(216,224)(241,305)(242,307)
(243,306)(244,308)(245,301)(246,303)(247,302)(248,304)(249,309)(250,311)
(251,310)(252,312)(253,353)(254,355)(255,354)(256,356)(257,349)(258,351)
(259,350)(260,352)(261,357)(262,359)(263,358)(264,360)(265,341)(266,343)
(267,342)(268,344)(269,337)(270,339)(271,338)(272,340)(273,345)(274,347)
(275,346)(276,348)(277,329)(278,331)(279,330)(280,332)(281,325)(282,327)
(283,326)(284,328)(285,333)(286,335)(287,334)(288,336)(289,317)(290,319)
(291,318)(292,320)(293,313)(294,315)(295,314)(296,316)(297,321)(298,323)
(299,322)(300,324);;
s2 := ( 1,313)( 2,316)( 3,315)( 4,314)( 5,321)( 6,324)( 7,323)( 8,322)
( 9,317)( 10,320)( 11,319)( 12,318)( 13,301)( 14,304)( 15,303)( 16,302)
( 17,309)( 18,312)( 19,311)( 20,310)( 21,305)( 22,308)( 23,307)( 24,306)
( 25,349)( 26,352)( 27,351)( 28,350)( 29,357)( 30,360)( 31,359)( 32,358)
( 33,353)( 34,356)( 35,355)( 36,354)( 37,337)( 38,340)( 39,339)( 40,338)
( 41,345)( 42,348)( 43,347)( 44,346)( 45,341)( 46,344)( 47,343)( 48,342)
( 49,325)( 50,328)( 51,327)( 52,326)( 53,333)( 54,336)( 55,335)( 56,334)
( 57,329)( 58,332)( 59,331)( 60,330)( 61,253)( 62,256)( 63,255)( 64,254)
( 65,261)( 66,264)( 67,263)( 68,262)( 69,257)( 70,260)( 71,259)( 72,258)
( 73,241)( 74,244)( 75,243)( 76,242)( 77,249)( 78,252)( 79,251)( 80,250)
( 81,245)( 82,248)( 83,247)( 84,246)( 85,289)( 86,292)( 87,291)( 88,290)
( 89,297)( 90,300)( 91,299)( 92,298)( 93,293)( 94,296)( 95,295)( 96,294)
( 97,277)( 98,280)( 99,279)(100,278)(101,285)(102,288)(103,287)(104,286)
(105,281)(106,284)(107,283)(108,282)(109,265)(110,268)(111,267)(112,266)
(113,273)(114,276)(115,275)(116,274)(117,269)(118,272)(119,271)(120,270)
(121,193)(122,196)(123,195)(124,194)(125,201)(126,204)(127,203)(128,202)
(129,197)(130,200)(131,199)(132,198)(133,181)(134,184)(135,183)(136,182)
(137,189)(138,192)(139,191)(140,190)(141,185)(142,188)(143,187)(144,186)
(145,229)(146,232)(147,231)(148,230)(149,237)(150,240)(151,239)(152,238)
(153,233)(154,236)(155,235)(156,234)(157,217)(158,220)(159,219)(160,218)
(161,225)(162,228)(163,227)(164,226)(165,221)(166,224)(167,223)(168,222)
(169,205)(170,208)(171,207)(172,206)(173,213)(174,216)(175,215)(176,214)
(177,209)(178,212)(179,211)(180,210);;
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*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*s0*s2*s1*s0*s2*s1*s0*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)!( 1,183)( 2,184)( 3,181)( 4,182)( 5,187)( 6,188)( 7,185)
( 8,186)( 9,191)( 10,192)( 11,189)( 12,190)( 13,195)( 14,196)( 15,193)
( 16,194)( 17,199)( 18,200)( 19,197)( 20,198)( 21,203)( 22,204)( 23,201)
( 24,202)( 25,207)( 26,208)( 27,205)( 28,206)( 29,211)( 30,212)( 31,209)
( 32,210)( 33,215)( 34,216)( 35,213)( 36,214)( 37,219)( 38,220)( 39,217)
( 40,218)( 41,223)( 42,224)( 43,221)( 44,222)( 45,227)( 46,228)( 47,225)
( 48,226)( 49,231)( 50,232)( 51,229)( 52,230)( 53,235)( 54,236)( 55,233)
( 56,234)( 57,239)( 58,240)( 59,237)( 60,238)( 61,243)( 62,244)( 63,241)
( 64,242)( 65,247)( 66,248)( 67,245)( 68,246)( 69,251)( 70,252)( 71,249)
( 72,250)( 73,255)( 74,256)( 75,253)( 76,254)( 77,259)( 78,260)( 79,257)
( 80,258)( 81,263)( 82,264)( 83,261)( 84,262)( 85,267)( 86,268)( 87,265)
( 88,266)( 89,271)( 90,272)( 91,269)( 92,270)( 93,275)( 94,276)( 95,273)
( 96,274)( 97,279)( 98,280)( 99,277)(100,278)(101,283)(102,284)(103,281)
(104,282)(105,287)(106,288)(107,285)(108,286)(109,291)(110,292)(111,289)
(112,290)(113,295)(114,296)(115,293)(116,294)(117,299)(118,300)(119,297)
(120,298)(121,303)(122,304)(123,301)(124,302)(125,307)(126,308)(127,305)
(128,306)(129,311)(130,312)(131,309)(132,310)(133,315)(134,316)(135,313)
(136,314)(137,319)(138,320)(139,317)(140,318)(141,323)(142,324)(143,321)
(144,322)(145,327)(146,328)(147,325)(148,326)(149,331)(150,332)(151,329)
(152,330)(153,335)(154,336)(155,333)(156,334)(157,339)(158,340)(159,337)
(160,338)(161,343)(162,344)(163,341)(164,342)(165,347)(166,348)(167,345)
(168,346)(169,351)(170,352)(171,349)(172,350)(173,355)(174,356)(175,353)
(176,354)(177,359)(178,360)(179,357)(180,358);
s1 := Sym(360)!( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 13, 49)( 14, 51)
( 15, 50)( 16, 52)( 17, 57)( 18, 59)( 19, 58)( 20, 60)( 21, 53)( 22, 55)
( 23, 54)( 24, 56)( 25, 37)( 26, 39)( 27, 38)( 28, 40)( 29, 45)( 30, 47)
( 31, 46)( 32, 48)( 33, 41)( 34, 43)( 35, 42)( 36, 44)( 61,125)( 62,127)
( 63,126)( 64,128)( 65,121)( 66,123)( 67,122)( 68,124)( 69,129)( 70,131)
( 71,130)( 72,132)( 73,173)( 74,175)( 75,174)( 76,176)( 77,169)( 78,171)
( 79,170)( 80,172)( 81,177)( 82,179)( 83,178)( 84,180)( 85,161)( 86,163)
( 87,162)( 88,164)( 89,157)( 90,159)( 91,158)( 92,160)( 93,165)( 94,167)
( 95,166)( 96,168)( 97,149)( 98,151)( 99,150)(100,152)(101,145)(102,147)
(103,146)(104,148)(105,153)(106,155)(107,154)(108,156)(109,137)(110,139)
(111,138)(112,140)(113,133)(114,135)(115,134)(116,136)(117,141)(118,143)
(119,142)(120,144)(182,183)(185,189)(186,191)(187,190)(188,192)(193,229)
(194,231)(195,230)(196,232)(197,237)(198,239)(199,238)(200,240)(201,233)
(202,235)(203,234)(204,236)(205,217)(206,219)(207,218)(208,220)(209,225)
(210,227)(211,226)(212,228)(213,221)(214,223)(215,222)(216,224)(241,305)
(242,307)(243,306)(244,308)(245,301)(246,303)(247,302)(248,304)(249,309)
(250,311)(251,310)(252,312)(253,353)(254,355)(255,354)(256,356)(257,349)
(258,351)(259,350)(260,352)(261,357)(262,359)(263,358)(264,360)(265,341)
(266,343)(267,342)(268,344)(269,337)(270,339)(271,338)(272,340)(273,345)
(274,347)(275,346)(276,348)(277,329)(278,331)(279,330)(280,332)(281,325)
(282,327)(283,326)(284,328)(285,333)(286,335)(287,334)(288,336)(289,317)
(290,319)(291,318)(292,320)(293,313)(294,315)(295,314)(296,316)(297,321)
(298,323)(299,322)(300,324);
s2 := Sym(360)!( 1,313)( 2,316)( 3,315)( 4,314)( 5,321)( 6,324)( 7,323)
( 8,322)( 9,317)( 10,320)( 11,319)( 12,318)( 13,301)( 14,304)( 15,303)
( 16,302)( 17,309)( 18,312)( 19,311)( 20,310)( 21,305)( 22,308)( 23,307)
( 24,306)( 25,349)( 26,352)( 27,351)( 28,350)( 29,357)( 30,360)( 31,359)
( 32,358)( 33,353)( 34,356)( 35,355)( 36,354)( 37,337)( 38,340)( 39,339)
( 40,338)( 41,345)( 42,348)( 43,347)( 44,346)( 45,341)( 46,344)( 47,343)
( 48,342)( 49,325)( 50,328)( 51,327)( 52,326)( 53,333)( 54,336)( 55,335)
( 56,334)( 57,329)( 58,332)( 59,331)( 60,330)( 61,253)( 62,256)( 63,255)
( 64,254)( 65,261)( 66,264)( 67,263)( 68,262)( 69,257)( 70,260)( 71,259)
( 72,258)( 73,241)( 74,244)( 75,243)( 76,242)( 77,249)( 78,252)( 79,251)
( 80,250)( 81,245)( 82,248)( 83,247)( 84,246)( 85,289)( 86,292)( 87,291)
( 88,290)( 89,297)( 90,300)( 91,299)( 92,298)( 93,293)( 94,296)( 95,295)
( 96,294)( 97,277)( 98,280)( 99,279)(100,278)(101,285)(102,288)(103,287)
(104,286)(105,281)(106,284)(107,283)(108,282)(109,265)(110,268)(111,267)
(112,266)(113,273)(114,276)(115,275)(116,274)(117,269)(118,272)(119,271)
(120,270)(121,193)(122,196)(123,195)(124,194)(125,201)(126,204)(127,203)
(128,202)(129,197)(130,200)(131,199)(132,198)(133,181)(134,184)(135,183)
(136,182)(137,189)(138,192)(139,191)(140,190)(141,185)(142,188)(143,187)
(144,186)(145,229)(146,232)(147,231)(148,230)(149,237)(150,240)(151,239)
(152,238)(153,233)(154,236)(155,235)(156,234)(157,217)(158,220)(159,219)
(160,218)(161,225)(162,228)(163,227)(164,226)(165,221)(166,224)(167,223)
(168,222)(169,205)(170,208)(171,207)(172,206)(173,213)(174,216)(175,215)
(176,214)(177,209)(178,212)(179,211)(180,210);
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*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*s0*s2*s1*s0*s2*s1*s0*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