Polytope of Type {90,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {90,10}*1800c
if this polytope has a 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 : 10
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) :
   2-fold quotients : {45,10}*900
   3-fold quotients : {30,10}*600c
   5-fold quotients : {90,2}*360
   6-fold quotients : {15,10}*300
   9-fold quotients : {10,10}*200c
   10-fold quotients : {45,2}*180
   15-fold quotients : {30,2}*120
   18-fold quotients : {5,10}*100
   25-fold quotients : {18,2}*72
   30-fold quotients : {15,2}*60
   45-fold quotients : {10,2}*40
   50-fold quotients : {9,2}*36
   75-fold quotients : {6,2}*24
   90-fold quotients : {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)(  4, 13)(  5, 15)(  6, 14)(  7, 10)(  8, 12)(  9, 11)( 16, 61)
( 17, 63)( 18, 62)( 19, 73)( 20, 75)( 21, 74)( 22, 70)( 23, 72)( 24, 71)
( 25, 67)( 26, 69)( 27, 68)( 28, 64)( 29, 66)( 30, 65)( 31, 46)( 32, 48)
( 33, 47)( 34, 58)( 35, 60)( 36, 59)( 37, 55)( 38, 57)( 39, 56)( 40, 52)
( 41, 54)( 42, 53)( 43, 49)( 44, 51)( 45, 50)( 76,152)( 77,151)( 78,153)
( 79,164)( 80,163)( 81,165)( 82,161)( 83,160)( 84,162)( 85,158)( 86,157)
( 87,159)( 88,155)( 89,154)( 90,156)( 91,212)( 92,211)( 93,213)( 94,224)
( 95,223)( 96,225)( 97,221)( 98,220)( 99,222)(100,218)(101,217)(102,219)
(103,215)(104,214)(105,216)(106,197)(107,196)(108,198)(109,209)(110,208)
(111,210)(112,206)(113,205)(114,207)(115,203)(116,202)(117,204)(118,200)
(119,199)(120,201)(121,182)(122,181)(123,183)(124,194)(125,193)(126,195)
(127,191)(128,190)(129,192)(130,188)(131,187)(132,189)(133,185)(134,184)
(135,186)(136,167)(137,166)(138,168)(139,179)(140,178)(141,180)(142,176)
(143,175)(144,177)(145,173)(146,172)(147,174)(148,170)(149,169)(150,171)
(227,228)(229,238)(230,240)(231,239)(232,235)(233,237)(234,236)(241,286)
(242,288)(243,287)(244,298)(245,300)(246,299)(247,295)(248,297)(249,296)
(250,292)(251,294)(252,293)(253,289)(254,291)(255,290)(256,271)(257,273)
(258,272)(259,283)(260,285)(261,284)(262,280)(263,282)(264,281)(265,277)
(266,279)(267,278)(268,274)(269,276)(270,275)(301,377)(302,376)(303,378)
(304,389)(305,388)(306,390)(307,386)(308,385)(309,387)(310,383)(311,382)
(312,384)(313,380)(314,379)(315,381)(316,437)(317,436)(318,438)(319,449)
(320,448)(321,450)(322,446)(323,445)(324,447)(325,443)(326,442)(327,444)
(328,440)(329,439)(330,441)(331,422)(332,421)(333,423)(334,434)(335,433)
(336,435)(337,431)(338,430)(339,432)(340,428)(341,427)(342,429)(343,425)
(344,424)(345,426)(346,407)(347,406)(348,408)(349,419)(350,418)(351,420)
(352,416)(353,415)(354,417)(355,413)(356,412)(357,414)(358,410)(359,409)
(360,411)(361,392)(362,391)(363,393)(364,404)(365,403)(366,405)(367,401)
(368,400)(369,402)(370,398)(371,397)(372,399)(373,395)(374,394)(375,396);;
s1 := (  1,319)(  2,321)(  3,320)(  4,316)(  5,318)(  6,317)(  7,328)(  8,330)
(  9,329)( 10,325)( 11,327)( 12,326)( 13,322)( 14,324)( 15,323)( 16,304)
( 17,306)( 18,305)( 19,301)( 20,303)( 21,302)( 22,313)( 23,315)( 24,314)
( 25,310)( 26,312)( 27,311)( 28,307)( 29,309)( 30,308)( 31,364)( 32,366)
( 33,365)( 34,361)( 35,363)( 36,362)( 37,373)( 38,375)( 39,374)( 40,370)
( 41,372)( 42,371)( 43,367)( 44,369)( 45,368)( 46,349)( 47,351)( 48,350)
( 49,346)( 50,348)( 51,347)( 52,358)( 53,360)( 54,359)( 55,355)( 56,357)
( 57,356)( 58,352)( 59,354)( 60,353)( 61,334)( 62,336)( 63,335)( 64,331)
( 65,333)( 66,332)( 67,343)( 68,345)( 69,344)( 70,340)( 71,342)( 72,341)
( 73,337)( 74,339)( 75,338)( 76,244)( 77,246)( 78,245)( 79,241)( 80,243)
( 81,242)( 82,253)( 83,255)( 84,254)( 85,250)( 86,252)( 87,251)( 88,247)
( 89,249)( 90,248)( 91,229)( 92,231)( 93,230)( 94,226)( 95,228)( 96,227)
( 97,238)( 98,240)( 99,239)(100,235)(101,237)(102,236)(103,232)(104,234)
(105,233)(106,289)(107,291)(108,290)(109,286)(110,288)(111,287)(112,298)
(113,300)(114,299)(115,295)(116,297)(117,296)(118,292)(119,294)(120,293)
(121,274)(122,276)(123,275)(124,271)(125,273)(126,272)(127,283)(128,285)
(129,284)(130,280)(131,282)(132,281)(133,277)(134,279)(135,278)(136,259)
(137,261)(138,260)(139,256)(140,258)(141,257)(142,268)(143,270)(144,269)
(145,265)(146,267)(147,266)(148,262)(149,264)(150,263)(151,395)(152,394)
(153,396)(154,392)(155,391)(156,393)(157,404)(158,403)(159,405)(160,401)
(161,400)(162,402)(163,398)(164,397)(165,399)(166,380)(167,379)(168,381)
(169,377)(170,376)(171,378)(172,389)(173,388)(174,390)(175,386)(176,385)
(177,387)(178,383)(179,382)(180,384)(181,440)(182,439)(183,441)(184,437)
(185,436)(186,438)(187,449)(188,448)(189,450)(190,446)(191,445)(192,447)
(193,443)(194,442)(195,444)(196,425)(197,424)(198,426)(199,422)(200,421)
(201,423)(202,434)(203,433)(204,435)(205,431)(206,430)(207,432)(208,428)
(209,427)(210,429)(211,410)(212,409)(213,411)(214,407)(215,406)(216,408)
(217,419)(218,418)(219,420)(220,416)(221,415)(222,417)(223,413)(224,412)
(225,414);;
s2 := (  4, 13)(  5, 14)(  6, 15)(  7, 10)(  8, 11)(  9, 12)( 19, 28)( 20, 29)
( 21, 30)( 22, 25)( 23, 26)( 24, 27)( 34, 43)( 35, 44)( 36, 45)( 37, 40)
( 38, 41)( 39, 42)( 49, 58)( 50, 59)( 51, 60)( 52, 55)( 53, 56)( 54, 57)
( 64, 73)( 65, 74)( 66, 75)( 67, 70)( 68, 71)( 69, 72)( 79, 88)( 80, 89)
( 81, 90)( 82, 85)( 83, 86)( 84, 87)( 94,103)( 95,104)( 96,105)( 97,100)
( 98,101)( 99,102)(109,118)(110,119)(111,120)(112,115)(113,116)(114,117)
(124,133)(125,134)(126,135)(127,130)(128,131)(129,132)(139,148)(140,149)
(141,150)(142,145)(143,146)(144,147)(154,163)(155,164)(156,165)(157,160)
(158,161)(159,162)(169,178)(170,179)(171,180)(172,175)(173,176)(174,177)
(184,193)(185,194)(186,195)(187,190)(188,191)(189,192)(199,208)(200,209)
(201,210)(202,205)(203,206)(204,207)(214,223)(215,224)(216,225)(217,220)
(218,221)(219,222)(229,238)(230,239)(231,240)(232,235)(233,236)(234,237)
(244,253)(245,254)(246,255)(247,250)(248,251)(249,252)(259,268)(260,269)
(261,270)(262,265)(263,266)(264,267)(274,283)(275,284)(276,285)(277,280)
(278,281)(279,282)(289,298)(290,299)(291,300)(292,295)(293,296)(294,297)
(304,313)(305,314)(306,315)(307,310)(308,311)(309,312)(319,328)(320,329)
(321,330)(322,325)(323,326)(324,327)(334,343)(335,344)(336,345)(337,340)
(338,341)(339,342)(349,358)(350,359)(351,360)(352,355)(353,356)(354,357)
(364,373)(365,374)(366,375)(367,370)(368,371)(369,372)(379,388)(380,389)
(381,390)(382,385)(383,386)(384,387)(394,403)(395,404)(396,405)(397,400)
(398,401)(399,402)(409,418)(410,419)(411,420)(412,415)(413,416)(414,417)
(424,433)(425,434)(426,435)(427,430)(428,431)(429,432)(439,448)(440,449)
(441,450)(442,445)(443,446)(444,447);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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)(  4, 13)(  5, 15)(  6, 14)(  7, 10)(  8, 12)(  9, 11)
( 16, 61)( 17, 63)( 18, 62)( 19, 73)( 20, 75)( 21, 74)( 22, 70)( 23, 72)
( 24, 71)( 25, 67)( 26, 69)( 27, 68)( 28, 64)( 29, 66)( 30, 65)( 31, 46)
( 32, 48)( 33, 47)( 34, 58)( 35, 60)( 36, 59)( 37, 55)( 38, 57)( 39, 56)
( 40, 52)( 41, 54)( 42, 53)( 43, 49)( 44, 51)( 45, 50)( 76,152)( 77,151)
( 78,153)( 79,164)( 80,163)( 81,165)( 82,161)( 83,160)( 84,162)( 85,158)
( 86,157)( 87,159)( 88,155)( 89,154)( 90,156)( 91,212)( 92,211)( 93,213)
( 94,224)( 95,223)( 96,225)( 97,221)( 98,220)( 99,222)(100,218)(101,217)
(102,219)(103,215)(104,214)(105,216)(106,197)(107,196)(108,198)(109,209)
(110,208)(111,210)(112,206)(113,205)(114,207)(115,203)(116,202)(117,204)
(118,200)(119,199)(120,201)(121,182)(122,181)(123,183)(124,194)(125,193)
(126,195)(127,191)(128,190)(129,192)(130,188)(131,187)(132,189)(133,185)
(134,184)(135,186)(136,167)(137,166)(138,168)(139,179)(140,178)(141,180)
(142,176)(143,175)(144,177)(145,173)(146,172)(147,174)(148,170)(149,169)
(150,171)(227,228)(229,238)(230,240)(231,239)(232,235)(233,237)(234,236)
(241,286)(242,288)(243,287)(244,298)(245,300)(246,299)(247,295)(248,297)
(249,296)(250,292)(251,294)(252,293)(253,289)(254,291)(255,290)(256,271)
(257,273)(258,272)(259,283)(260,285)(261,284)(262,280)(263,282)(264,281)
(265,277)(266,279)(267,278)(268,274)(269,276)(270,275)(301,377)(302,376)
(303,378)(304,389)(305,388)(306,390)(307,386)(308,385)(309,387)(310,383)
(311,382)(312,384)(313,380)(314,379)(315,381)(316,437)(317,436)(318,438)
(319,449)(320,448)(321,450)(322,446)(323,445)(324,447)(325,443)(326,442)
(327,444)(328,440)(329,439)(330,441)(331,422)(332,421)(333,423)(334,434)
(335,433)(336,435)(337,431)(338,430)(339,432)(340,428)(341,427)(342,429)
(343,425)(344,424)(345,426)(346,407)(347,406)(348,408)(349,419)(350,418)
(351,420)(352,416)(353,415)(354,417)(355,413)(356,412)(357,414)(358,410)
(359,409)(360,411)(361,392)(362,391)(363,393)(364,404)(365,403)(366,405)
(367,401)(368,400)(369,402)(370,398)(371,397)(372,399)(373,395)(374,394)
(375,396);
s1 := Sym(450)!(  1,319)(  2,321)(  3,320)(  4,316)(  5,318)(  6,317)(  7,328)
(  8,330)(  9,329)( 10,325)( 11,327)( 12,326)( 13,322)( 14,324)( 15,323)
( 16,304)( 17,306)( 18,305)( 19,301)( 20,303)( 21,302)( 22,313)( 23,315)
( 24,314)( 25,310)( 26,312)( 27,311)( 28,307)( 29,309)( 30,308)( 31,364)
( 32,366)( 33,365)( 34,361)( 35,363)( 36,362)( 37,373)( 38,375)( 39,374)
( 40,370)( 41,372)( 42,371)( 43,367)( 44,369)( 45,368)( 46,349)( 47,351)
( 48,350)( 49,346)( 50,348)( 51,347)( 52,358)( 53,360)( 54,359)( 55,355)
( 56,357)( 57,356)( 58,352)( 59,354)( 60,353)( 61,334)( 62,336)( 63,335)
( 64,331)( 65,333)( 66,332)( 67,343)( 68,345)( 69,344)( 70,340)( 71,342)
( 72,341)( 73,337)( 74,339)( 75,338)( 76,244)( 77,246)( 78,245)( 79,241)
( 80,243)( 81,242)( 82,253)( 83,255)( 84,254)( 85,250)( 86,252)( 87,251)
( 88,247)( 89,249)( 90,248)( 91,229)( 92,231)( 93,230)( 94,226)( 95,228)
( 96,227)( 97,238)( 98,240)( 99,239)(100,235)(101,237)(102,236)(103,232)
(104,234)(105,233)(106,289)(107,291)(108,290)(109,286)(110,288)(111,287)
(112,298)(113,300)(114,299)(115,295)(116,297)(117,296)(118,292)(119,294)
(120,293)(121,274)(122,276)(123,275)(124,271)(125,273)(126,272)(127,283)
(128,285)(129,284)(130,280)(131,282)(132,281)(133,277)(134,279)(135,278)
(136,259)(137,261)(138,260)(139,256)(140,258)(141,257)(142,268)(143,270)
(144,269)(145,265)(146,267)(147,266)(148,262)(149,264)(150,263)(151,395)
(152,394)(153,396)(154,392)(155,391)(156,393)(157,404)(158,403)(159,405)
(160,401)(161,400)(162,402)(163,398)(164,397)(165,399)(166,380)(167,379)
(168,381)(169,377)(170,376)(171,378)(172,389)(173,388)(174,390)(175,386)
(176,385)(177,387)(178,383)(179,382)(180,384)(181,440)(182,439)(183,441)
(184,437)(185,436)(186,438)(187,449)(188,448)(189,450)(190,446)(191,445)
(192,447)(193,443)(194,442)(195,444)(196,425)(197,424)(198,426)(199,422)
(200,421)(201,423)(202,434)(203,433)(204,435)(205,431)(206,430)(207,432)
(208,428)(209,427)(210,429)(211,410)(212,409)(213,411)(214,407)(215,406)
(216,408)(217,419)(218,418)(219,420)(220,416)(221,415)(222,417)(223,413)
(224,412)(225,414);
s2 := Sym(450)!(  4, 13)(  5, 14)(  6, 15)(  7, 10)(  8, 11)(  9, 12)( 19, 28)
( 20, 29)( 21, 30)( 22, 25)( 23, 26)( 24, 27)( 34, 43)( 35, 44)( 36, 45)
( 37, 40)( 38, 41)( 39, 42)( 49, 58)( 50, 59)( 51, 60)( 52, 55)( 53, 56)
( 54, 57)( 64, 73)( 65, 74)( 66, 75)( 67, 70)( 68, 71)( 69, 72)( 79, 88)
( 80, 89)( 81, 90)( 82, 85)( 83, 86)( 84, 87)( 94,103)( 95,104)( 96,105)
( 97,100)( 98,101)( 99,102)(109,118)(110,119)(111,120)(112,115)(113,116)
(114,117)(124,133)(125,134)(126,135)(127,130)(128,131)(129,132)(139,148)
(140,149)(141,150)(142,145)(143,146)(144,147)(154,163)(155,164)(156,165)
(157,160)(158,161)(159,162)(169,178)(170,179)(171,180)(172,175)(173,176)
(174,177)(184,193)(185,194)(186,195)(187,190)(188,191)(189,192)(199,208)
(200,209)(201,210)(202,205)(203,206)(204,207)(214,223)(215,224)(216,225)
(217,220)(218,221)(219,222)(229,238)(230,239)(231,240)(232,235)(233,236)
(234,237)(244,253)(245,254)(246,255)(247,250)(248,251)(249,252)(259,268)
(260,269)(261,270)(262,265)(263,266)(264,267)(274,283)(275,284)(276,285)
(277,280)(278,281)(279,282)(289,298)(290,299)(291,300)(292,295)(293,296)
(294,297)(304,313)(305,314)(306,315)(307,310)(308,311)(309,312)(319,328)
(320,329)(321,330)(322,325)(323,326)(324,327)(334,343)(335,344)(336,345)
(337,340)(338,341)(339,342)(349,358)(350,359)(351,360)(352,355)(353,356)
(354,357)(364,373)(365,374)(366,375)(367,370)(368,371)(369,372)(379,388)
(380,389)(381,390)(382,385)(383,386)(384,387)(394,403)(395,404)(396,405)
(397,400)(398,401)(399,402)(409,418)(410,419)(411,420)(412,415)(413,416)
(414,417)(424,433)(425,434)(426,435)(427,430)(428,431)(429,432)(439,448)
(440,449)(441,450)(442,445)(443,446)(444,447);
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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