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