Polytope of Type {2,90,4}

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

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

Permutation Representation (Magma) :

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

to this polytope