Polytope of Type {2,4,90}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,90}*1440c
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}*480c
5-fold quotients : {2,4,18}*288c
6-fold quotients : {2,4,15}*240
10-fold quotients : {2,4,9}*144
15-fold quotients : {2,4,6}*96b
30-fold quotients : {2,4,3}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3,185)( 4,186)( 5,183)( 6,184)( 7,189)( 8,190)( 9,187)( 10,188)
( 11,193)( 12,194)( 13,191)( 14,192)( 15,197)( 16,198)( 17,195)( 18,196)
( 19,201)( 20,202)( 21,199)( 22,200)( 23,205)( 24,206)( 25,203)( 26,204)
( 27,209)( 28,210)( 29,207)( 30,208)( 31,213)( 32,214)( 33,211)( 34,212)
( 35,217)( 36,218)( 37,215)( 38,216)( 39,221)( 40,222)( 41,219)( 42,220)
( 43,225)( 44,226)( 45,223)( 46,224)( 47,229)( 48,230)( 49,227)( 50,228)
( 51,233)( 52,234)( 53,231)( 54,232)( 55,237)( 56,238)( 57,235)( 58,236)
( 59,241)( 60,242)( 61,239)( 62,240)( 63,245)( 64,246)( 65,243)( 66,244)
( 67,249)( 68,250)( 69,247)( 70,248)( 71,253)( 72,254)( 73,251)( 74,252)
( 75,257)( 76,258)( 77,255)( 78,256)( 79,261)( 80,262)( 81,259)( 82,260)
( 83,265)( 84,266)( 85,263)( 86,264)( 87,269)( 88,270)( 89,267)( 90,268)
( 91,273)( 92,274)( 93,271)( 94,272)( 95,277)( 96,278)( 97,275)( 98,276)
( 99,281)(100,282)(101,279)(102,280)(103,285)(104,286)(105,283)(106,284)
(107,289)(108,290)(109,287)(110,288)(111,293)(112,294)(113,291)(114,292)
(115,297)(116,298)(117,295)(118,296)(119,301)(120,302)(121,299)(122,300)
(123,305)(124,306)(125,303)(126,304)(127,309)(128,310)(129,307)(130,308)
(131,313)(132,314)(133,311)(134,312)(135,317)(136,318)(137,315)(138,316)
(139,321)(140,322)(141,319)(142,320)(143,325)(144,326)(145,323)(146,324)
(147,329)(148,330)(149,327)(150,328)(151,333)(152,334)(153,331)(154,332)
(155,337)(156,338)(157,335)(158,336)(159,341)(160,342)(161,339)(162,340)
(163,345)(164,346)(165,343)(166,344)(167,349)(168,350)(169,347)(170,348)
(171,353)(172,354)(173,351)(174,352)(175,357)(176,358)(177,355)(178,356)
(179,361)(180,362)(181,359)(182,360);;
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*s3*s2*s1*s2*s3*s2*s3*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*s1*s3*s2*s1*s3*s2*s1*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)!( 3,185)( 4,186)( 5,183)( 6,184)( 7,189)( 8,190)( 9,187)
( 10,188)( 11,193)( 12,194)( 13,191)( 14,192)( 15,197)( 16,198)( 17,195)
( 18,196)( 19,201)( 20,202)( 21,199)( 22,200)( 23,205)( 24,206)( 25,203)
( 26,204)( 27,209)( 28,210)( 29,207)( 30,208)( 31,213)( 32,214)( 33,211)
( 34,212)( 35,217)( 36,218)( 37,215)( 38,216)( 39,221)( 40,222)( 41,219)
( 42,220)( 43,225)( 44,226)( 45,223)( 46,224)( 47,229)( 48,230)( 49,227)
( 50,228)( 51,233)( 52,234)( 53,231)( 54,232)( 55,237)( 56,238)( 57,235)
( 58,236)( 59,241)( 60,242)( 61,239)( 62,240)( 63,245)( 64,246)( 65,243)
( 66,244)( 67,249)( 68,250)( 69,247)( 70,248)( 71,253)( 72,254)( 73,251)
( 74,252)( 75,257)( 76,258)( 77,255)( 78,256)( 79,261)( 80,262)( 81,259)
( 82,260)( 83,265)( 84,266)( 85,263)( 86,264)( 87,269)( 88,270)( 89,267)
( 90,268)( 91,273)( 92,274)( 93,271)( 94,272)( 95,277)( 96,278)( 97,275)
( 98,276)( 99,281)(100,282)(101,279)(102,280)(103,285)(104,286)(105,283)
(106,284)(107,289)(108,290)(109,287)(110,288)(111,293)(112,294)(113,291)
(114,292)(115,297)(116,298)(117,295)(118,296)(119,301)(120,302)(121,299)
(122,300)(123,305)(124,306)(125,303)(126,304)(127,309)(128,310)(129,307)
(130,308)(131,313)(132,314)(133,311)(134,312)(135,317)(136,318)(137,315)
(138,316)(139,321)(140,322)(141,319)(142,320)(143,325)(144,326)(145,323)
(146,324)(147,329)(148,330)(149,327)(150,328)(151,333)(152,334)(153,331)
(154,332)(155,337)(156,338)(157,335)(158,336)(159,341)(160,342)(161,339)
(162,340)(163,345)(164,346)(165,343)(166,344)(167,349)(168,350)(169,347)
(170,348)(171,353)(172,354)(173,351)(174,352)(175,357)(176,358)(177,355)
(178,356)(179,361)(180,362)(181,359)(182,360);
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*s3*s2*s1*s2*s3*s2*s3*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*s1*s3*s2*s1*s3*s2*s1*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