Polytope of Type {90,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {90,10}*1800b
Also Known As : {90,10|2}. if this polytope has another name.
Group : SmallGroup(1800,317)
Rank : 3
Schlafli Type : {90,10}
Number of vertices, edges, etc : 90, 450, 10
Order of s0s1s2 : 90
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {30,10}*600b
   5-fold quotients : {18,10}*360, {90,2}*360
   9-fold quotients : {10,10}*200a
   10-fold quotients : {45,2}*180
   15-fold quotients : {6,10}*120, {30,2}*120
   25-fold quotients : {18,2}*72
   30-fold quotients : {15,2}*60
   45-fold quotients : {2,10}*40, {10,2}*40
   50-fold quotients : {9,2}*36
   75-fold quotients : {6,2}*24
   90-fold quotients : {2,5}*20, {5,2}*20
   150-fold quotients : {3,2}*12
   225-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) : 
s0 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 16, 61)( 17, 63)( 18, 62)
( 19, 64)( 20, 66)( 21, 65)( 22, 67)( 23, 69)( 24, 68)( 25, 70)( 26, 72)
( 27, 71)( 28, 73)( 29, 75)( 30, 74)( 31, 46)( 32, 48)( 33, 47)( 34, 49)
( 35, 51)( 36, 50)( 37, 52)( 38, 54)( 39, 53)( 40, 55)( 41, 57)( 42, 56)
( 43, 58)( 44, 60)( 45, 59)( 76,152)( 77,151)( 78,153)( 79,155)( 80,154)
( 81,156)( 82,158)( 83,157)( 84,159)( 85,161)( 86,160)( 87,162)( 88,164)
( 89,163)( 90,165)( 91,212)( 92,211)( 93,213)( 94,215)( 95,214)( 96,216)
( 97,218)( 98,217)( 99,219)(100,221)(101,220)(102,222)(103,224)(104,223)
(105,225)(106,197)(107,196)(108,198)(109,200)(110,199)(111,201)(112,203)
(113,202)(114,204)(115,206)(116,205)(117,207)(118,209)(119,208)(120,210)
(121,182)(122,181)(123,183)(124,185)(125,184)(126,186)(127,188)(128,187)
(129,189)(130,191)(131,190)(132,192)(133,194)(134,193)(135,195)(136,167)
(137,166)(138,168)(139,170)(140,169)(141,171)(142,173)(143,172)(144,174)
(145,176)(146,175)(147,177)(148,179)(149,178)(150,180)(227,228)(230,231)
(233,234)(236,237)(239,240)(241,286)(242,288)(243,287)(244,289)(245,291)
(246,290)(247,292)(248,294)(249,293)(250,295)(251,297)(252,296)(253,298)
(254,300)(255,299)(256,271)(257,273)(258,272)(259,274)(260,276)(261,275)
(262,277)(263,279)(264,278)(265,280)(266,282)(267,281)(268,283)(269,285)
(270,284)(301,377)(302,376)(303,378)(304,380)(305,379)(306,381)(307,383)
(308,382)(309,384)(310,386)(311,385)(312,387)(313,389)(314,388)(315,390)
(316,437)(317,436)(318,438)(319,440)(320,439)(321,441)(322,443)(323,442)
(324,444)(325,446)(326,445)(327,447)(328,449)(329,448)(330,450)(331,422)
(332,421)(333,423)(334,425)(335,424)(336,426)(337,428)(338,427)(339,429)
(340,431)(341,430)(342,432)(343,434)(344,433)(345,435)(346,407)(347,406)
(348,408)(349,410)(350,409)(351,411)(352,413)(353,412)(354,414)(355,416)
(356,415)(357,417)(358,419)(359,418)(360,420)(361,392)(362,391)(363,393)
(364,395)(365,394)(366,396)(367,398)(368,397)(369,399)(370,401)(371,400)
(372,402)(373,404)(374,403)(375,405);;
s1 := (  1, 91)(  2, 93)(  3, 92)(  4,103)(  5,105)(  6,104)(  7,100)(  8,102)
(  9,101)( 10, 97)( 11, 99)( 12, 98)( 13, 94)( 14, 96)( 15, 95)( 16, 76)
( 17, 78)( 18, 77)( 19, 88)( 20, 90)( 21, 89)( 22, 85)( 23, 87)( 24, 86)
( 25, 82)( 26, 84)( 27, 83)( 28, 79)( 29, 81)( 30, 80)( 31,136)( 32,138)
( 33,137)( 34,148)( 35,150)( 36,149)( 37,145)( 38,147)( 39,146)( 40,142)
( 41,144)( 42,143)( 43,139)( 44,141)( 45,140)( 46,121)( 47,123)( 48,122)
( 49,133)( 50,135)( 51,134)( 52,130)( 53,132)( 54,131)( 55,127)( 56,129)
( 57,128)( 58,124)( 59,126)( 60,125)( 61,106)( 62,108)( 63,107)( 64,118)
( 65,120)( 66,119)( 67,115)( 68,117)( 69,116)( 70,112)( 71,114)( 72,113)
( 73,109)( 74,111)( 75,110)(151,167)(152,166)(153,168)(154,179)(155,178)
(156,180)(157,176)(158,175)(159,177)(160,173)(161,172)(162,174)(163,170)
(164,169)(165,171)(181,212)(182,211)(183,213)(184,224)(185,223)(186,225)
(187,221)(188,220)(189,222)(190,218)(191,217)(192,219)(193,215)(194,214)
(195,216)(196,197)(199,209)(200,208)(201,210)(202,206)(203,205)(204,207)
(226,316)(227,318)(228,317)(229,328)(230,330)(231,329)(232,325)(233,327)
(234,326)(235,322)(236,324)(237,323)(238,319)(239,321)(240,320)(241,301)
(242,303)(243,302)(244,313)(245,315)(246,314)(247,310)(248,312)(249,311)
(250,307)(251,309)(252,308)(253,304)(254,306)(255,305)(256,361)(257,363)
(258,362)(259,373)(260,375)(261,374)(262,370)(263,372)(264,371)(265,367)
(266,369)(267,368)(268,364)(269,366)(270,365)(271,346)(272,348)(273,347)
(274,358)(275,360)(276,359)(277,355)(278,357)(279,356)(280,352)(281,354)
(282,353)(283,349)(284,351)(285,350)(286,331)(287,333)(288,332)(289,343)
(290,345)(291,344)(292,340)(293,342)(294,341)(295,337)(296,339)(297,338)
(298,334)(299,336)(300,335)(376,392)(377,391)(378,393)(379,404)(380,403)
(381,405)(382,401)(383,400)(384,402)(385,398)(386,397)(387,399)(388,395)
(389,394)(390,396)(406,437)(407,436)(408,438)(409,449)(410,448)(411,450)
(412,446)(413,445)(414,447)(415,443)(416,442)(417,444)(418,440)(419,439)
(420,441)(421,422)(424,434)(425,433)(426,435)(427,431)(428,430)(429,432);;
s2 := (  1,229)(  2,230)(  3,231)(  4,226)(  5,227)(  6,228)(  7,238)(  8,239)
(  9,240)( 10,235)( 11,236)( 12,237)( 13,232)( 14,233)( 15,234)( 16,244)
( 17,245)( 18,246)( 19,241)( 20,242)( 21,243)( 22,253)( 23,254)( 24,255)
( 25,250)( 26,251)( 27,252)( 28,247)( 29,248)( 30,249)( 31,259)( 32,260)
( 33,261)( 34,256)( 35,257)( 36,258)( 37,268)( 38,269)( 39,270)( 40,265)
( 41,266)( 42,267)( 43,262)( 44,263)( 45,264)( 46,274)( 47,275)( 48,276)
( 49,271)( 50,272)( 51,273)( 52,283)( 53,284)( 54,285)( 55,280)( 56,281)
( 57,282)( 58,277)( 59,278)( 60,279)( 61,289)( 62,290)( 63,291)( 64,286)
( 65,287)( 66,288)( 67,298)( 68,299)( 69,300)( 70,295)( 71,296)( 72,297)
( 73,292)( 74,293)( 75,294)( 76,304)( 77,305)( 78,306)( 79,301)( 80,302)
( 81,303)( 82,313)( 83,314)( 84,315)( 85,310)( 86,311)( 87,312)( 88,307)
( 89,308)( 90,309)( 91,319)( 92,320)( 93,321)( 94,316)( 95,317)( 96,318)
( 97,328)( 98,329)( 99,330)(100,325)(101,326)(102,327)(103,322)(104,323)
(105,324)(106,334)(107,335)(108,336)(109,331)(110,332)(111,333)(112,343)
(113,344)(114,345)(115,340)(116,341)(117,342)(118,337)(119,338)(120,339)
(121,349)(122,350)(123,351)(124,346)(125,347)(126,348)(127,358)(128,359)
(129,360)(130,355)(131,356)(132,357)(133,352)(134,353)(135,354)(136,364)
(137,365)(138,366)(139,361)(140,362)(141,363)(142,373)(143,374)(144,375)
(145,370)(146,371)(147,372)(148,367)(149,368)(150,369)(151,379)(152,380)
(153,381)(154,376)(155,377)(156,378)(157,388)(158,389)(159,390)(160,385)
(161,386)(162,387)(163,382)(164,383)(165,384)(166,394)(167,395)(168,396)
(169,391)(170,392)(171,393)(172,403)(173,404)(174,405)(175,400)(176,401)
(177,402)(178,397)(179,398)(180,399)(181,409)(182,410)(183,411)(184,406)
(185,407)(186,408)(187,418)(188,419)(189,420)(190,415)(191,416)(192,417)
(193,412)(194,413)(195,414)(196,424)(197,425)(198,426)(199,421)(200,422)
(201,423)(202,433)(203,434)(204,435)(205,430)(206,431)(207,432)(208,427)
(209,428)(210,429)(211,439)(212,440)(213,441)(214,436)(215,437)(216,438)
(217,448)(218,449)(219,450)(220,445)(221,446)(222,447)(223,442)(224,443)
(225,444);;
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*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(450)!(  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 16, 61)( 17, 63)
( 18, 62)( 19, 64)( 20, 66)( 21, 65)( 22, 67)( 23, 69)( 24, 68)( 25, 70)
( 26, 72)( 27, 71)( 28, 73)( 29, 75)( 30, 74)( 31, 46)( 32, 48)( 33, 47)
( 34, 49)( 35, 51)( 36, 50)( 37, 52)( 38, 54)( 39, 53)( 40, 55)( 41, 57)
( 42, 56)( 43, 58)( 44, 60)( 45, 59)( 76,152)( 77,151)( 78,153)( 79,155)
( 80,154)( 81,156)( 82,158)( 83,157)( 84,159)( 85,161)( 86,160)( 87,162)
( 88,164)( 89,163)( 90,165)( 91,212)( 92,211)( 93,213)( 94,215)( 95,214)
( 96,216)( 97,218)( 98,217)( 99,219)(100,221)(101,220)(102,222)(103,224)
(104,223)(105,225)(106,197)(107,196)(108,198)(109,200)(110,199)(111,201)
(112,203)(113,202)(114,204)(115,206)(116,205)(117,207)(118,209)(119,208)
(120,210)(121,182)(122,181)(123,183)(124,185)(125,184)(126,186)(127,188)
(128,187)(129,189)(130,191)(131,190)(132,192)(133,194)(134,193)(135,195)
(136,167)(137,166)(138,168)(139,170)(140,169)(141,171)(142,173)(143,172)
(144,174)(145,176)(146,175)(147,177)(148,179)(149,178)(150,180)(227,228)
(230,231)(233,234)(236,237)(239,240)(241,286)(242,288)(243,287)(244,289)
(245,291)(246,290)(247,292)(248,294)(249,293)(250,295)(251,297)(252,296)
(253,298)(254,300)(255,299)(256,271)(257,273)(258,272)(259,274)(260,276)
(261,275)(262,277)(263,279)(264,278)(265,280)(266,282)(267,281)(268,283)
(269,285)(270,284)(301,377)(302,376)(303,378)(304,380)(305,379)(306,381)
(307,383)(308,382)(309,384)(310,386)(311,385)(312,387)(313,389)(314,388)
(315,390)(316,437)(317,436)(318,438)(319,440)(320,439)(321,441)(322,443)
(323,442)(324,444)(325,446)(326,445)(327,447)(328,449)(329,448)(330,450)
(331,422)(332,421)(333,423)(334,425)(335,424)(336,426)(337,428)(338,427)
(339,429)(340,431)(341,430)(342,432)(343,434)(344,433)(345,435)(346,407)
(347,406)(348,408)(349,410)(350,409)(351,411)(352,413)(353,412)(354,414)
(355,416)(356,415)(357,417)(358,419)(359,418)(360,420)(361,392)(362,391)
(363,393)(364,395)(365,394)(366,396)(367,398)(368,397)(369,399)(370,401)
(371,400)(372,402)(373,404)(374,403)(375,405);
s1 := Sym(450)!(  1, 91)(  2, 93)(  3, 92)(  4,103)(  5,105)(  6,104)(  7,100)
(  8,102)(  9,101)( 10, 97)( 11, 99)( 12, 98)( 13, 94)( 14, 96)( 15, 95)
( 16, 76)( 17, 78)( 18, 77)( 19, 88)( 20, 90)( 21, 89)( 22, 85)( 23, 87)
( 24, 86)( 25, 82)( 26, 84)( 27, 83)( 28, 79)( 29, 81)( 30, 80)( 31,136)
( 32,138)( 33,137)( 34,148)( 35,150)( 36,149)( 37,145)( 38,147)( 39,146)
( 40,142)( 41,144)( 42,143)( 43,139)( 44,141)( 45,140)( 46,121)( 47,123)
( 48,122)( 49,133)( 50,135)( 51,134)( 52,130)( 53,132)( 54,131)( 55,127)
( 56,129)( 57,128)( 58,124)( 59,126)( 60,125)( 61,106)( 62,108)( 63,107)
( 64,118)( 65,120)( 66,119)( 67,115)( 68,117)( 69,116)( 70,112)( 71,114)
( 72,113)( 73,109)( 74,111)( 75,110)(151,167)(152,166)(153,168)(154,179)
(155,178)(156,180)(157,176)(158,175)(159,177)(160,173)(161,172)(162,174)
(163,170)(164,169)(165,171)(181,212)(182,211)(183,213)(184,224)(185,223)
(186,225)(187,221)(188,220)(189,222)(190,218)(191,217)(192,219)(193,215)
(194,214)(195,216)(196,197)(199,209)(200,208)(201,210)(202,206)(203,205)
(204,207)(226,316)(227,318)(228,317)(229,328)(230,330)(231,329)(232,325)
(233,327)(234,326)(235,322)(236,324)(237,323)(238,319)(239,321)(240,320)
(241,301)(242,303)(243,302)(244,313)(245,315)(246,314)(247,310)(248,312)
(249,311)(250,307)(251,309)(252,308)(253,304)(254,306)(255,305)(256,361)
(257,363)(258,362)(259,373)(260,375)(261,374)(262,370)(263,372)(264,371)
(265,367)(266,369)(267,368)(268,364)(269,366)(270,365)(271,346)(272,348)
(273,347)(274,358)(275,360)(276,359)(277,355)(278,357)(279,356)(280,352)
(281,354)(282,353)(283,349)(284,351)(285,350)(286,331)(287,333)(288,332)
(289,343)(290,345)(291,344)(292,340)(293,342)(294,341)(295,337)(296,339)
(297,338)(298,334)(299,336)(300,335)(376,392)(377,391)(378,393)(379,404)
(380,403)(381,405)(382,401)(383,400)(384,402)(385,398)(386,397)(387,399)
(388,395)(389,394)(390,396)(406,437)(407,436)(408,438)(409,449)(410,448)
(411,450)(412,446)(413,445)(414,447)(415,443)(416,442)(417,444)(418,440)
(419,439)(420,441)(421,422)(424,434)(425,433)(426,435)(427,431)(428,430)
(429,432);
s2 := Sym(450)!(  1,229)(  2,230)(  3,231)(  4,226)(  5,227)(  6,228)(  7,238)
(  8,239)(  9,240)( 10,235)( 11,236)( 12,237)( 13,232)( 14,233)( 15,234)
( 16,244)( 17,245)( 18,246)( 19,241)( 20,242)( 21,243)( 22,253)( 23,254)
( 24,255)( 25,250)( 26,251)( 27,252)( 28,247)( 29,248)( 30,249)( 31,259)
( 32,260)( 33,261)( 34,256)( 35,257)( 36,258)( 37,268)( 38,269)( 39,270)
( 40,265)( 41,266)( 42,267)( 43,262)( 44,263)( 45,264)( 46,274)( 47,275)
( 48,276)( 49,271)( 50,272)( 51,273)( 52,283)( 53,284)( 54,285)( 55,280)
( 56,281)( 57,282)( 58,277)( 59,278)( 60,279)( 61,289)( 62,290)( 63,291)
( 64,286)( 65,287)( 66,288)( 67,298)( 68,299)( 69,300)( 70,295)( 71,296)
( 72,297)( 73,292)( 74,293)( 75,294)( 76,304)( 77,305)( 78,306)( 79,301)
( 80,302)( 81,303)( 82,313)( 83,314)( 84,315)( 85,310)( 86,311)( 87,312)
( 88,307)( 89,308)( 90,309)( 91,319)( 92,320)( 93,321)( 94,316)( 95,317)
( 96,318)( 97,328)( 98,329)( 99,330)(100,325)(101,326)(102,327)(103,322)
(104,323)(105,324)(106,334)(107,335)(108,336)(109,331)(110,332)(111,333)
(112,343)(113,344)(114,345)(115,340)(116,341)(117,342)(118,337)(119,338)
(120,339)(121,349)(122,350)(123,351)(124,346)(125,347)(126,348)(127,358)
(128,359)(129,360)(130,355)(131,356)(132,357)(133,352)(134,353)(135,354)
(136,364)(137,365)(138,366)(139,361)(140,362)(141,363)(142,373)(143,374)
(144,375)(145,370)(146,371)(147,372)(148,367)(149,368)(150,369)(151,379)
(152,380)(153,381)(154,376)(155,377)(156,378)(157,388)(158,389)(159,390)
(160,385)(161,386)(162,387)(163,382)(164,383)(165,384)(166,394)(167,395)
(168,396)(169,391)(170,392)(171,393)(172,403)(173,404)(174,405)(175,400)
(176,401)(177,402)(178,397)(179,398)(180,399)(181,409)(182,410)(183,411)
(184,406)(185,407)(186,408)(187,418)(188,419)(189,420)(190,415)(191,416)
(192,417)(193,412)(194,413)(195,414)(196,424)(197,425)(198,426)(199,421)
(200,422)(201,423)(202,433)(203,434)(204,435)(205,430)(206,431)(207,432)
(208,427)(209,428)(210,429)(211,439)(212,440)(213,441)(214,436)(215,437)
(216,438)(217,448)(218,449)(219,450)(220,445)(221,446)(222,447)(223,442)
(224,443)(225,444);
poly := sub<Sym(450)|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*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