Polytope of Type {2,4,90}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,90}*1440b
if this polytope has a name.
Group : SmallGroup(1440,4575)
Rank : 4
Schlafli Type : {2,4,90}
Number of vertices, edges, etc : 2, 4, 180, 90
Order of s₀s₁s₂s₃ : 90
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-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 : {2,4,45}*720
   3-fold quotients : {2,4,30}*480b
   5-fold quotients : {2,4,18}*288b
   6-fold quotients : {2,4,15}*240
   10-fold quotients : {2,4,9}*144
   15-fold quotients : {2,4,6}*96c
   30-fold quotients : {2,4,3}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  3,  5)(  4,  6)(  7,  9)(  8, 10)( 11, 13)( 12, 14)( 15, 17)( 16, 18)
( 19, 21)( 20, 22)( 23, 25)( 24, 26)( 27, 29)( 28, 30)( 31, 33)( 32, 34)
( 35, 37)( 36, 38)( 39, 41)( 40, 42)( 43, 45)( 44, 46)( 47, 49)( 48, 50)
( 51, 53)( 52, 54)( 55, 57)( 56, 58)( 59, 61)( 60, 62)( 63, 65)( 64, 66)
( 67, 69)( 68, 70)( 71, 73)( 72, 74)( 75, 77)( 76, 78)( 79, 81)( 80, 82)
( 83, 85)( 84, 86)( 87, 89)( 88, 90)( 91, 93)( 92, 94)( 95, 97)( 96, 98)
( 99,101)(100,102)(103,105)(104,106)(107,109)(108,110)(111,113)(112,114)
(115,117)(116,118)(119,121)(120,122)(123,125)(124,126)(127,129)(128,130)
(131,133)(132,134)(135,137)(136,138)(139,141)(140,142)(143,145)(144,146)
(147,149)(148,150)(151,153)(152,154)(155,157)(156,158)(159,161)(160,162)
(163,165)(164,166)(167,169)(168,170)(171,173)(172,174)(175,177)(176,178)
(179,181)(180,182)(183,185)(184,186)(187,189)(188,190)(191,193)(192,194)
(195,197)(196,198)(199,201)(200,202)(203,205)(204,206)(207,209)(208,210)
(211,213)(212,214)(215,217)(216,218)(219,221)(220,222)(223,225)(224,226)
(227,229)(228,230)(231,233)(232,234)(235,237)(236,238)(239,241)(240,242)
(243,245)(244,246)(247,249)(248,250)(251,253)(252,254)(255,257)(256,258)
(259,261)(260,262)(263,265)(264,266)(267,269)(268,270)(271,273)(272,274)
(275,277)(276,278)(279,281)(280,282)(283,285)(284,286)(287,289)(288,290)
(291,293)(292,294)(295,297)(296,298)(299,301)(300,302)(303,305)(304,306)
(307,309)(308,310)(311,313)(312,314)(315,317)(316,318)(319,321)(320,322)
(323,325)(324,326)(327,329)(328,330)(331,333)(332,334)(335,337)(336,338)
(339,341)(340,342)(343,345)(344,346)(347,349)(348,350)(351,353)(352,354)
(355,357)(356,358)(359,361)(360,362);;
s2 := (  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 15, 51)( 16, 53)( 17, 52)
( 18, 54)( 19, 59)( 20, 61)( 21, 60)( 22, 62)( 23, 55)( 24, 57)( 25, 56)
( 26, 58)( 27, 39)( 28, 41)( 29, 40)( 30, 42)( 31, 47)( 32, 49)( 33, 48)
( 34, 50)( 35, 43)( 36, 45)( 37, 44)( 38, 46)( 63,127)( 64,129)( 65,128)
( 66,130)( 67,123)( 68,125)( 69,124)( 70,126)( 71,131)( 72,133)( 73,132)
( 74,134)( 75,175)( 76,177)( 77,176)( 78,178)( 79,171)( 80,173)( 81,172)
( 82,174)( 83,179)( 84,181)( 85,180)( 86,182)( 87,163)( 88,165)( 89,164)
( 90,166)( 91,159)( 92,161)( 93,160)( 94,162)( 95,167)( 96,169)( 97,168)
( 98,170)( 99,151)(100,153)(101,152)(102,154)(103,147)(104,149)(105,148)
(106,150)(107,155)(108,157)(109,156)(110,158)(111,139)(112,141)(113,140)
(114,142)(115,135)(116,137)(117,136)(118,138)(119,143)(120,145)(121,144)
(122,146)(184,185)(187,191)(188,193)(189,192)(190,194)(195,231)(196,233)
(197,232)(198,234)(199,239)(200,241)(201,240)(202,242)(203,235)(204,237)
(205,236)(206,238)(207,219)(208,221)(209,220)(210,222)(211,227)(212,229)
(213,228)(214,230)(215,223)(216,225)(217,224)(218,226)(243,307)(244,309)
(245,308)(246,310)(247,303)(248,305)(249,304)(250,306)(251,311)(252,313)
(253,312)(254,314)(255,355)(256,357)(257,356)(258,358)(259,351)(260,353)
(261,352)(262,354)(263,359)(264,361)(265,360)(266,362)(267,343)(268,345)
(269,344)(270,346)(271,339)(272,341)(273,340)(274,342)(275,347)(276,349)
(277,348)(278,350)(279,331)(280,333)(281,332)(282,334)(283,327)(284,329)
(285,328)(286,330)(287,335)(288,337)(289,336)(290,338)(291,319)(292,321)
(293,320)(294,322)(295,315)(296,317)(297,316)(298,318)(299,323)(300,325)
(301,324)(302,326);;
s3 := (  3,315)(  4,318)(  5,317)(  6,316)(  7,323)(  8,326)(  9,325)( 10,324)
( 11,319)( 12,322)( 13,321)( 14,320)( 15,303)( 16,306)( 17,305)( 18,304)
( 19,311)( 20,314)( 21,313)( 22,312)( 23,307)( 24,310)( 25,309)( 26,308)
( 27,351)( 28,354)( 29,353)( 30,352)( 31,359)( 32,362)( 33,361)( 34,360)
( 35,355)( 36,358)( 37,357)( 38,356)( 39,339)( 40,342)( 41,341)( 42,340)
( 43,347)( 44,350)( 45,349)( 46,348)( 47,343)( 48,346)( 49,345)( 50,344)
( 51,327)( 52,330)( 53,329)( 54,328)( 55,335)( 56,338)( 57,337)( 58,336)
( 59,331)( 60,334)( 61,333)( 62,332)( 63,255)( 64,258)( 65,257)( 66,256)
( 67,263)( 68,266)( 69,265)( 70,264)( 71,259)( 72,262)( 73,261)( 74,260)
( 75,243)( 76,246)( 77,245)( 78,244)( 79,251)( 80,254)( 81,253)( 82,252)
( 83,247)( 84,250)( 85,249)( 86,248)( 87,291)( 88,294)( 89,293)( 90,292)
( 91,299)( 92,302)( 93,301)( 94,300)( 95,295)( 96,298)( 97,297)( 98,296)
( 99,279)(100,282)(101,281)(102,280)(103,287)(104,290)(105,289)(106,288)
(107,283)(108,286)(109,285)(110,284)(111,267)(112,270)(113,269)(114,268)
(115,275)(116,278)(117,277)(118,276)(119,271)(120,274)(121,273)(122,272)
(123,195)(124,198)(125,197)(126,196)(127,203)(128,206)(129,205)(130,204)
(131,199)(132,202)(133,201)(134,200)(135,183)(136,186)(137,185)(138,184)
(139,191)(140,194)(141,193)(142,192)(143,187)(144,190)(145,189)(146,188)
(147,231)(148,234)(149,233)(150,232)(151,239)(152,242)(153,241)(154,240)
(155,235)(156,238)(157,237)(158,236)(159,219)(160,222)(161,221)(162,220)
(163,227)(164,230)(165,229)(166,228)(167,223)(168,226)(169,225)(170,224)
(171,207)(172,210)(173,209)(174,208)(175,215)(176,218)(177,217)(178,216)
(179,211)(180,214)(181,213)(182,212);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s1*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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(362)!(1,2);
s1 := Sym(362)!(  3,  5)(  4,  6)(  7,  9)(  8, 10)( 11, 13)( 12, 14)( 15, 17)
( 16, 18)( 19, 21)( 20, 22)( 23, 25)( 24, 26)( 27, 29)( 28, 30)( 31, 33)
( 32, 34)( 35, 37)( 36, 38)( 39, 41)( 40, 42)( 43, 45)( 44, 46)( 47, 49)
( 48, 50)( 51, 53)( 52, 54)( 55, 57)( 56, 58)( 59, 61)( 60, 62)( 63, 65)
( 64, 66)( 67, 69)( 68, 70)( 71, 73)( 72, 74)( 75, 77)( 76, 78)( 79, 81)
( 80, 82)( 83, 85)( 84, 86)( 87, 89)( 88, 90)( 91, 93)( 92, 94)( 95, 97)
( 96, 98)( 99,101)(100,102)(103,105)(104,106)(107,109)(108,110)(111,113)
(112,114)(115,117)(116,118)(119,121)(120,122)(123,125)(124,126)(127,129)
(128,130)(131,133)(132,134)(135,137)(136,138)(139,141)(140,142)(143,145)
(144,146)(147,149)(148,150)(151,153)(152,154)(155,157)(156,158)(159,161)
(160,162)(163,165)(164,166)(167,169)(168,170)(171,173)(172,174)(175,177)
(176,178)(179,181)(180,182)(183,185)(184,186)(187,189)(188,190)(191,193)
(192,194)(195,197)(196,198)(199,201)(200,202)(203,205)(204,206)(207,209)
(208,210)(211,213)(212,214)(215,217)(216,218)(219,221)(220,222)(223,225)
(224,226)(227,229)(228,230)(231,233)(232,234)(235,237)(236,238)(239,241)
(240,242)(243,245)(244,246)(247,249)(248,250)(251,253)(252,254)(255,257)
(256,258)(259,261)(260,262)(263,265)(264,266)(267,269)(268,270)(271,273)
(272,274)(275,277)(276,278)(279,281)(280,282)(283,285)(284,286)(287,289)
(288,290)(291,293)(292,294)(295,297)(296,298)(299,301)(300,302)(303,305)
(304,306)(307,309)(308,310)(311,313)(312,314)(315,317)(316,318)(319,321)
(320,322)(323,325)(324,326)(327,329)(328,330)(331,333)(332,334)(335,337)
(336,338)(339,341)(340,342)(343,345)(344,346)(347,349)(348,350)(351,353)
(352,354)(355,357)(356,358)(359,361)(360,362);
s2 := Sym(362)!(  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 15, 51)( 16, 53)
( 17, 52)( 18, 54)( 19, 59)( 20, 61)( 21, 60)( 22, 62)( 23, 55)( 24, 57)
( 25, 56)( 26, 58)( 27, 39)( 28, 41)( 29, 40)( 30, 42)( 31, 47)( 32, 49)
( 33, 48)( 34, 50)( 35, 43)( 36, 45)( 37, 44)( 38, 46)( 63,127)( 64,129)
( 65,128)( 66,130)( 67,123)( 68,125)( 69,124)( 70,126)( 71,131)( 72,133)
( 73,132)( 74,134)( 75,175)( 76,177)( 77,176)( 78,178)( 79,171)( 80,173)
( 81,172)( 82,174)( 83,179)( 84,181)( 85,180)( 86,182)( 87,163)( 88,165)
( 89,164)( 90,166)( 91,159)( 92,161)( 93,160)( 94,162)( 95,167)( 96,169)
( 97,168)( 98,170)( 99,151)(100,153)(101,152)(102,154)(103,147)(104,149)
(105,148)(106,150)(107,155)(108,157)(109,156)(110,158)(111,139)(112,141)
(113,140)(114,142)(115,135)(116,137)(117,136)(118,138)(119,143)(120,145)
(121,144)(122,146)(184,185)(187,191)(188,193)(189,192)(190,194)(195,231)
(196,233)(197,232)(198,234)(199,239)(200,241)(201,240)(202,242)(203,235)
(204,237)(205,236)(206,238)(207,219)(208,221)(209,220)(210,222)(211,227)
(212,229)(213,228)(214,230)(215,223)(216,225)(217,224)(218,226)(243,307)
(244,309)(245,308)(246,310)(247,303)(248,305)(249,304)(250,306)(251,311)
(252,313)(253,312)(254,314)(255,355)(256,357)(257,356)(258,358)(259,351)
(260,353)(261,352)(262,354)(263,359)(264,361)(265,360)(266,362)(267,343)
(268,345)(269,344)(270,346)(271,339)(272,341)(273,340)(274,342)(275,347)
(276,349)(277,348)(278,350)(279,331)(280,333)(281,332)(282,334)(283,327)
(284,329)(285,328)(286,330)(287,335)(288,337)(289,336)(290,338)(291,319)
(292,321)(293,320)(294,322)(295,315)(296,317)(297,316)(298,318)(299,323)
(300,325)(301,324)(302,326);
s3 := Sym(362)!(  3,315)(  4,318)(  5,317)(  6,316)(  7,323)(  8,326)(  9,325)
( 10,324)( 11,319)( 12,322)( 13,321)( 14,320)( 15,303)( 16,306)( 17,305)
( 18,304)( 19,311)( 20,314)( 21,313)( 22,312)( 23,307)( 24,310)( 25,309)
( 26,308)( 27,351)( 28,354)( 29,353)( 30,352)( 31,359)( 32,362)( 33,361)
( 34,360)( 35,355)( 36,358)( 37,357)( 38,356)( 39,339)( 40,342)( 41,341)
( 42,340)( 43,347)( 44,350)( 45,349)( 46,348)( 47,343)( 48,346)( 49,345)
( 50,344)( 51,327)( 52,330)( 53,329)( 54,328)( 55,335)( 56,338)( 57,337)
( 58,336)( 59,331)( 60,334)( 61,333)( 62,332)( 63,255)( 64,258)( 65,257)
( 66,256)( 67,263)( 68,266)( 69,265)( 70,264)( 71,259)( 72,262)( 73,261)
( 74,260)( 75,243)( 76,246)( 77,245)( 78,244)( 79,251)( 80,254)( 81,253)
( 82,252)( 83,247)( 84,250)( 85,249)( 86,248)( 87,291)( 88,294)( 89,293)
( 90,292)( 91,299)( 92,302)( 93,301)( 94,300)( 95,295)( 96,298)( 97,297)
( 98,296)( 99,279)(100,282)(101,281)(102,280)(103,287)(104,290)(105,289)
(106,288)(107,283)(108,286)(109,285)(110,284)(111,267)(112,270)(113,269)
(114,268)(115,275)(116,278)(117,277)(118,276)(119,271)(120,274)(121,273)
(122,272)(123,195)(124,198)(125,197)(126,196)(127,203)(128,206)(129,205)
(130,204)(131,199)(132,202)(133,201)(134,200)(135,183)(136,186)(137,185)
(138,184)(139,191)(140,194)(141,193)(142,192)(143,187)(144,190)(145,189)
(146,188)(147,231)(148,234)(149,233)(150,232)(151,239)(152,242)(153,241)
(154,240)(155,235)(156,238)(157,237)(158,236)(159,219)(160,222)(161,221)
(162,220)(163,227)(164,230)(165,229)(166,228)(167,223)(168,226)(169,225)
(170,224)(171,207)(172,210)(173,209)(174,208)(175,215)(176,218)(177,217)
(178,216)(179,211)(180,214)(181,213)(182,212);
poly := sub<Sym(362)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s1*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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope