Polytope of Type {8,90}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,90}*1440
Also Known As : {8,90|2}. if this polytope has another name.
Group : SmallGroup(1440,875)
Rank : 3
Schlafli Type : {8,90}
Number of vertices, edges, etc : 8, 360, 90
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 : {4,90}*720a
3-fold quotients : {8,30}*480
4-fold quotients : {2,90}*360
5-fold quotients : {8,18}*288
6-fold quotients : {4,30}*240a
8-fold quotients : {2,45}*180
9-fold quotients : {8,10}*160
10-fold quotients : {4,18}*144a
12-fold quotients : {2,30}*120
15-fold quotients : {8,6}*96
18-fold quotients : {4,10}*80
20-fold quotients : {2,18}*72
24-fold quotients : {2,15}*60
30-fold quotients : {4,6}*48a
36-fold quotients : {2,10}*40
40-fold quotients : {2,9}*36
45-fold quotients : {8,2}*32
60-fold quotients : {2,6}*24
72-fold quotients : {2,5}*20
90-fold quotients : {4,2}*16
120-fold quotients : {2,3}*12
180-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 91,136)( 92,137)( 93,138)( 94,139)( 95,140)( 96,141)( 97,142)( 98,143)
( 99,144)(100,145)(101,146)(102,147)(103,148)(104,149)(105,150)(106,151)
(107,152)(108,153)(109,154)(110,155)(111,156)(112,157)(113,158)(114,159)
(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)(122,167)
(123,168)(124,169)(125,170)(126,171)(127,172)(128,173)(129,174)(130,175)
(131,176)(132,177)(133,178)(134,179)(135,180)(181,271)(182,272)(183,273)
(184,274)(185,275)(186,276)(187,277)(188,278)(189,279)(190,280)(191,281)
(192,282)(193,283)(194,284)(195,285)(196,286)(197,287)(198,288)(199,289)
(200,290)(201,291)(202,292)(203,293)(204,294)(205,295)(206,296)(207,297)
(208,298)(209,299)(210,300)(211,301)(212,302)(213,303)(214,304)(215,305)
(216,306)(217,307)(218,308)(219,309)(220,310)(221,311)(222,312)(223,313)
(224,314)(225,315)(226,316)(227,317)(228,318)(229,319)(230,320)(231,321)
(232,322)(233,323)(234,324)(235,325)(236,326)(237,327)(238,328)(239,329)
(240,330)(241,331)(242,332)(243,333)(244,334)(245,335)(246,336)(247,337)
(248,338)(249,339)(250,340)(251,341)(252,342)(253,343)(254,344)(255,345)
(256,346)(257,347)(258,348)(259,349)(260,350)(261,351)(262,352)(263,353)
(264,354)(265,355)(266,356)(267,357)(268,358)(269,359)(270,360);;
s1 := ( 1,181)( 2,183)( 3,182)( 4,193)( 5,195)( 6,194)( 7,190)( 8,192)
( 9,191)( 10,187)( 11,189)( 12,188)( 13,184)( 14,186)( 15,185)( 16,213)
( 17,212)( 18,211)( 19,225)( 20,224)( 21,223)( 22,222)( 23,221)( 24,220)
( 25,219)( 26,218)( 27,217)( 28,216)( 29,215)( 30,214)( 31,198)( 32,197)
( 33,196)( 34,210)( 35,209)( 36,208)( 37,207)( 38,206)( 39,205)( 40,204)
( 41,203)( 42,202)( 43,201)( 44,200)( 45,199)( 46,226)( 47,228)( 48,227)
( 49,238)( 50,240)( 51,239)( 52,235)( 53,237)( 54,236)( 55,232)( 56,234)
( 57,233)( 58,229)( 59,231)( 60,230)( 61,258)( 62,257)( 63,256)( 64,270)
( 65,269)( 66,268)( 67,267)( 68,266)( 69,265)( 70,264)( 71,263)( 72,262)
( 73,261)( 74,260)( 75,259)( 76,243)( 77,242)( 78,241)( 79,255)( 80,254)
( 81,253)( 82,252)( 83,251)( 84,250)( 85,249)( 86,248)( 87,247)( 88,246)
( 89,245)( 90,244)( 91,316)( 92,318)( 93,317)( 94,328)( 95,330)( 96,329)
( 97,325)( 98,327)( 99,326)(100,322)(101,324)(102,323)(103,319)(104,321)
(105,320)(106,348)(107,347)(108,346)(109,360)(110,359)(111,358)(112,357)
(113,356)(114,355)(115,354)(116,353)(117,352)(118,351)(119,350)(120,349)
(121,333)(122,332)(123,331)(124,345)(125,344)(126,343)(127,342)(128,341)
(129,340)(130,339)(131,338)(132,337)(133,336)(134,335)(135,334)(136,271)
(137,273)(138,272)(139,283)(140,285)(141,284)(142,280)(143,282)(144,281)
(145,277)(146,279)(147,278)(148,274)(149,276)(150,275)(151,303)(152,302)
(153,301)(154,315)(155,314)(156,313)(157,312)(158,311)(159,310)(160,309)
(161,308)(162,307)(163,306)(164,305)(165,304)(166,288)(167,287)(168,286)
(169,300)(170,299)(171,298)(172,297)(173,296)(174,295)(175,294)(176,293)
(177,292)(178,291)(179,290)(180,289);;
s2 := ( 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)( 91,109)( 92,111)( 93,110)( 94,106)
( 95,108)( 96,107)( 97,118)( 98,120)( 99,119)(100,115)(101,117)(102,116)
(103,112)(104,114)(105,113)(121,126)(122,125)(123,124)(127,135)(128,134)
(129,133)(130,132)(136,154)(137,156)(138,155)(139,151)(140,153)(141,152)
(142,163)(143,165)(144,164)(145,160)(146,162)(147,161)(148,157)(149,159)
(150,158)(166,171)(167,170)(168,169)(172,180)(173,179)(174,178)(175,177)
(181,199)(182,201)(183,200)(184,196)(185,198)(186,197)(187,208)(188,210)
(189,209)(190,205)(191,207)(192,206)(193,202)(194,204)(195,203)(211,216)
(212,215)(213,214)(217,225)(218,224)(219,223)(220,222)(226,244)(227,246)
(228,245)(229,241)(230,243)(231,242)(232,253)(233,255)(234,254)(235,250)
(236,252)(237,251)(238,247)(239,249)(240,248)(256,261)(257,260)(258,259)
(262,270)(263,269)(264,268)(265,267)(271,289)(272,291)(273,290)(274,286)
(275,288)(276,287)(277,298)(278,300)(279,299)(280,295)(281,297)(282,296)
(283,292)(284,294)(285,293)(301,306)(302,305)(303,304)(307,315)(308,314)
(309,313)(310,312)(316,334)(317,336)(318,335)(319,331)(320,333)(321,332)
(322,343)(323,345)(324,344)(325,340)(326,342)(327,341)(328,337)(329,339)
(330,338)(346,351)(347,350)(348,349)(352,360)(353,359)(354,358)(355,357);;
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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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)!( 91,136)( 92,137)( 93,138)( 94,139)( 95,140)( 96,141)( 97,142)
( 98,143)( 99,144)(100,145)(101,146)(102,147)(103,148)(104,149)(105,150)
(106,151)(107,152)(108,153)(109,154)(110,155)(111,156)(112,157)(113,158)
(114,159)(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)
(122,167)(123,168)(124,169)(125,170)(126,171)(127,172)(128,173)(129,174)
(130,175)(131,176)(132,177)(133,178)(134,179)(135,180)(181,271)(182,272)
(183,273)(184,274)(185,275)(186,276)(187,277)(188,278)(189,279)(190,280)
(191,281)(192,282)(193,283)(194,284)(195,285)(196,286)(197,287)(198,288)
(199,289)(200,290)(201,291)(202,292)(203,293)(204,294)(205,295)(206,296)
(207,297)(208,298)(209,299)(210,300)(211,301)(212,302)(213,303)(214,304)
(215,305)(216,306)(217,307)(218,308)(219,309)(220,310)(221,311)(222,312)
(223,313)(224,314)(225,315)(226,316)(227,317)(228,318)(229,319)(230,320)
(231,321)(232,322)(233,323)(234,324)(235,325)(236,326)(237,327)(238,328)
(239,329)(240,330)(241,331)(242,332)(243,333)(244,334)(245,335)(246,336)
(247,337)(248,338)(249,339)(250,340)(251,341)(252,342)(253,343)(254,344)
(255,345)(256,346)(257,347)(258,348)(259,349)(260,350)(261,351)(262,352)
(263,353)(264,354)(265,355)(266,356)(267,357)(268,358)(269,359)(270,360);
s1 := Sym(360)!( 1,181)( 2,183)( 3,182)( 4,193)( 5,195)( 6,194)( 7,190)
( 8,192)( 9,191)( 10,187)( 11,189)( 12,188)( 13,184)( 14,186)( 15,185)
( 16,213)( 17,212)( 18,211)( 19,225)( 20,224)( 21,223)( 22,222)( 23,221)
( 24,220)( 25,219)( 26,218)( 27,217)( 28,216)( 29,215)( 30,214)( 31,198)
( 32,197)( 33,196)( 34,210)( 35,209)( 36,208)( 37,207)( 38,206)( 39,205)
( 40,204)( 41,203)( 42,202)( 43,201)( 44,200)( 45,199)( 46,226)( 47,228)
( 48,227)( 49,238)( 50,240)( 51,239)( 52,235)( 53,237)( 54,236)( 55,232)
( 56,234)( 57,233)( 58,229)( 59,231)( 60,230)( 61,258)( 62,257)( 63,256)
( 64,270)( 65,269)( 66,268)( 67,267)( 68,266)( 69,265)( 70,264)( 71,263)
( 72,262)( 73,261)( 74,260)( 75,259)( 76,243)( 77,242)( 78,241)( 79,255)
( 80,254)( 81,253)( 82,252)( 83,251)( 84,250)( 85,249)( 86,248)( 87,247)
( 88,246)( 89,245)( 90,244)( 91,316)( 92,318)( 93,317)( 94,328)( 95,330)
( 96,329)( 97,325)( 98,327)( 99,326)(100,322)(101,324)(102,323)(103,319)
(104,321)(105,320)(106,348)(107,347)(108,346)(109,360)(110,359)(111,358)
(112,357)(113,356)(114,355)(115,354)(116,353)(117,352)(118,351)(119,350)
(120,349)(121,333)(122,332)(123,331)(124,345)(125,344)(126,343)(127,342)
(128,341)(129,340)(130,339)(131,338)(132,337)(133,336)(134,335)(135,334)
(136,271)(137,273)(138,272)(139,283)(140,285)(141,284)(142,280)(143,282)
(144,281)(145,277)(146,279)(147,278)(148,274)(149,276)(150,275)(151,303)
(152,302)(153,301)(154,315)(155,314)(156,313)(157,312)(158,311)(159,310)
(160,309)(161,308)(162,307)(163,306)(164,305)(165,304)(166,288)(167,287)
(168,286)(169,300)(170,299)(171,298)(172,297)(173,296)(174,295)(175,294)
(176,293)(177,292)(178,291)(179,290)(180,289);
s2 := 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)( 91,109)( 92,111)( 93,110)
( 94,106)( 95,108)( 96,107)( 97,118)( 98,120)( 99,119)(100,115)(101,117)
(102,116)(103,112)(104,114)(105,113)(121,126)(122,125)(123,124)(127,135)
(128,134)(129,133)(130,132)(136,154)(137,156)(138,155)(139,151)(140,153)
(141,152)(142,163)(143,165)(144,164)(145,160)(146,162)(147,161)(148,157)
(149,159)(150,158)(166,171)(167,170)(168,169)(172,180)(173,179)(174,178)
(175,177)(181,199)(182,201)(183,200)(184,196)(185,198)(186,197)(187,208)
(188,210)(189,209)(190,205)(191,207)(192,206)(193,202)(194,204)(195,203)
(211,216)(212,215)(213,214)(217,225)(218,224)(219,223)(220,222)(226,244)
(227,246)(228,245)(229,241)(230,243)(231,242)(232,253)(233,255)(234,254)
(235,250)(236,252)(237,251)(238,247)(239,249)(240,248)(256,261)(257,260)
(258,259)(262,270)(263,269)(264,268)(265,267)(271,289)(272,291)(273,290)
(274,286)(275,288)(276,287)(277,298)(278,300)(279,299)(280,295)(281,297)
(282,296)(283,292)(284,294)(285,293)(301,306)(302,305)(303,304)(307,315)
(308,314)(309,313)(310,312)(316,334)(317,336)(318,335)(319,331)(320,333)
(321,332)(322,343)(323,345)(324,344)(325,340)(326,342)(327,341)(328,337)
(329,339)(330,338)(346,351)(347,350)(348,349)(352,360)(353,359)(354,358)
(355,357);
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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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