Polytope of Type {90,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {90,8}*1440
Also Known As : {90,8|2}. if this polytope has another name.
Group : SmallGroup(1440,875)
Rank : 3
Schlafli Type : {90,8}
Number of vertices, edges, etc : 90, 360, 8
Order of s₀s₁s₂ : 360
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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 : {90,4}*720a
   3-fold quotients : {30,8}*480
   4-fold quotients : {90,2}*360
   5-fold quotients : {18,8}*288
   6-fold quotients : {30,4}*240a
   8-fold quotients : {45,2}*180
   9-fold quotients : {10,8}*160
   10-fold quotients : {18,4}*144a
   12-fold quotients : {30,2}*120
   15-fold quotients : {6,8}*96
   18-fold quotients : {10,4}*80
   20-fold quotients : {18,2}*72
   24-fold quotients : {15,2}*60
   30-fold quotients : {6,4}*48a
   36-fold quotients : {10,2}*40
   40-fold quotients : {9,2}*36
   45-fold quotients : {2,8}*32
   60-fold quotients : {6,2}*24
   72-fold quotients : {5,2}*20
   90-fold quotients : {2,4}*16
   120-fold quotients : {3,2}*12
   180-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope