Polytope of Type {10,90}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,90}*1800c
if this polytope has a name.
Group : SmallGroup(1800,317)
Rank : 3
Schlafli Type : {10,90}
Number of vertices, edges, etc : 10, 450, 90
Order of s0s1s2 : 90
Order of s0s1s2s1 : 10
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 : {10,45}*900
3-fold quotients : {10,30}*600c
5-fold quotients : {2,90}*360
6-fold quotients : {10,15}*300
9-fold quotients : {10,10}*200b
10-fold quotients : {2,45}*180
15-fold quotients : {2,30}*120
18-fold quotients : {10,5}*100
25-fold quotients : {2,18}*72
30-fold quotients : {2,15}*60
45-fold quotients : {2,10}*40
50-fold quotients : {2,9}*36
75-fold quotients : {2,6}*24
90-fold quotients : {2,5}*20
150-fold quotients : {2,3}*12
225-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 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);;
s1 := ( 1, 4)( 2, 6)( 3, 5)( 7, 13)( 8, 15)( 9, 14)( 11, 12)( 16, 64)
( 17, 66)( 18, 65)( 19, 61)( 20, 63)( 21, 62)( 22, 73)( 23, 75)( 24, 74)
( 25, 70)( 26, 72)( 27, 71)( 28, 67)( 29, 69)( 30, 68)( 31, 49)( 32, 51)
( 33, 50)( 34, 46)( 35, 48)( 36, 47)( 37, 58)( 38, 60)( 39, 59)( 40, 55)
( 41, 57)( 42, 56)( 43, 52)( 44, 54)( 45, 53)( 76,155)( 77,154)( 78,156)
( 79,152)( 80,151)( 81,153)( 82,164)( 83,163)( 84,165)( 85,161)( 86,160)
( 87,162)( 88,158)( 89,157)( 90,159)( 91,215)( 92,214)( 93,216)( 94,212)
( 95,211)( 96,213)( 97,224)( 98,223)( 99,225)(100,221)(101,220)(102,222)
(103,218)(104,217)(105,219)(106,200)(107,199)(108,201)(109,197)(110,196)
(111,198)(112,209)(113,208)(114,210)(115,206)(116,205)(117,207)(118,203)
(119,202)(120,204)(121,185)(122,184)(123,186)(124,182)(125,181)(126,183)
(127,194)(128,193)(129,195)(130,191)(131,190)(132,192)(133,188)(134,187)
(135,189)(136,170)(137,169)(138,171)(139,167)(140,166)(141,168)(142,179)
(143,178)(144,180)(145,176)(146,175)(147,177)(148,173)(149,172)(150,174)
(226,229)(227,231)(228,230)(232,238)(233,240)(234,239)(236,237)(241,289)
(242,291)(243,290)(244,286)(245,288)(246,287)(247,298)(248,300)(249,299)
(250,295)(251,297)(252,296)(253,292)(254,294)(255,293)(256,274)(257,276)
(258,275)(259,271)(260,273)(261,272)(262,283)(263,285)(264,284)(265,280)
(266,282)(267,281)(268,277)(269,279)(270,278)(301,380)(302,379)(303,381)
(304,377)(305,376)(306,378)(307,389)(308,388)(309,390)(310,386)(311,385)
(312,387)(313,383)(314,382)(315,384)(316,440)(317,439)(318,441)(319,437)
(320,436)(321,438)(322,449)(323,448)(324,450)(325,446)(326,445)(327,447)
(328,443)(329,442)(330,444)(331,425)(332,424)(333,426)(334,422)(335,421)
(336,423)(337,434)(338,433)(339,435)(340,431)(341,430)(342,432)(343,428)
(344,427)(345,429)(346,410)(347,409)(348,411)(349,407)(350,406)(351,408)
(352,419)(353,418)(354,420)(355,416)(356,415)(357,417)(358,413)(359,412)
(360,414)(361,395)(362,394)(363,396)(364,392)(365,391)(366,393)(367,404)
(368,403)(369,405)(370,401)(371,400)(372,402)(373,398)(374,397)(375,399);;
s2 := ( 1,316)( 2,318)( 3,317)( 4,328)( 5,330)( 6,329)( 7,325)( 8,327)
( 9,326)( 10,322)( 11,324)( 12,323)( 13,319)( 14,321)( 15,320)( 16,301)
( 17,303)( 18,302)( 19,313)( 20,315)( 21,314)( 22,310)( 23,312)( 24,311)
( 25,307)( 26,309)( 27,308)( 28,304)( 29,306)( 30,305)( 31,361)( 32,363)
( 33,362)( 34,373)( 35,375)( 36,374)( 37,370)( 38,372)( 39,371)( 40,367)
( 41,369)( 42,368)( 43,364)( 44,366)( 45,365)( 46,346)( 47,348)( 48,347)
( 49,358)( 50,360)( 51,359)( 52,355)( 53,357)( 54,356)( 55,352)( 56,354)
( 57,353)( 58,349)( 59,351)( 60,350)( 61,331)( 62,333)( 63,332)( 64,343)
( 65,345)( 66,344)( 67,340)( 68,342)( 69,341)( 70,337)( 71,339)( 72,338)
( 73,334)( 74,336)( 75,335)( 76,241)( 77,243)( 78,242)( 79,253)( 80,255)
( 81,254)( 82,250)( 83,252)( 84,251)( 85,247)( 86,249)( 87,248)( 88,244)
( 89,246)( 90,245)( 91,226)( 92,228)( 93,227)( 94,238)( 95,240)( 96,239)
( 97,235)( 98,237)( 99,236)(100,232)(101,234)(102,233)(103,229)(104,231)
(105,230)(106,286)(107,288)(108,287)(109,298)(110,300)(111,299)(112,295)
(113,297)(114,296)(115,292)(116,294)(117,293)(118,289)(119,291)(120,290)
(121,271)(122,273)(123,272)(124,283)(125,285)(126,284)(127,280)(128,282)
(129,281)(130,277)(131,279)(132,278)(133,274)(134,276)(135,275)(136,256)
(137,258)(138,257)(139,268)(140,270)(141,269)(142,265)(143,267)(144,266)
(145,262)(146,264)(147,263)(148,259)(149,261)(150,260)(151,392)(152,391)
(153,393)(154,404)(155,403)(156,405)(157,401)(158,400)(159,402)(160,398)
(161,397)(162,399)(163,395)(164,394)(165,396)(166,377)(167,376)(168,378)
(169,389)(170,388)(171,390)(172,386)(173,385)(174,387)(175,383)(176,382)
(177,384)(178,380)(179,379)(180,381)(181,437)(182,436)(183,438)(184,449)
(185,448)(186,450)(187,446)(188,445)(189,447)(190,443)(191,442)(192,444)
(193,440)(194,439)(195,441)(196,422)(197,421)(198,423)(199,434)(200,433)
(201,435)(202,431)(203,430)(204,432)(205,428)(206,427)(207,429)(208,425)
(209,424)(210,426)(211,407)(212,406)(213,408)(214,419)(215,418)(216,420)
(217,416)(218,415)(219,417)(220,413)(221,412)(222,414)(223,410)(224,409)
(225,411);;
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*s0*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,
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(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);
s1 := Sym(450)!( 1, 4)( 2, 6)( 3, 5)( 7, 13)( 8, 15)( 9, 14)( 11, 12)
( 16, 64)( 17, 66)( 18, 65)( 19, 61)( 20, 63)( 21, 62)( 22, 73)( 23, 75)
( 24, 74)( 25, 70)( 26, 72)( 27, 71)( 28, 67)( 29, 69)( 30, 68)( 31, 49)
( 32, 51)( 33, 50)( 34, 46)( 35, 48)( 36, 47)( 37, 58)( 38, 60)( 39, 59)
( 40, 55)( 41, 57)( 42, 56)( 43, 52)( 44, 54)( 45, 53)( 76,155)( 77,154)
( 78,156)( 79,152)( 80,151)( 81,153)( 82,164)( 83,163)( 84,165)( 85,161)
( 86,160)( 87,162)( 88,158)( 89,157)( 90,159)( 91,215)( 92,214)( 93,216)
( 94,212)( 95,211)( 96,213)( 97,224)( 98,223)( 99,225)(100,221)(101,220)
(102,222)(103,218)(104,217)(105,219)(106,200)(107,199)(108,201)(109,197)
(110,196)(111,198)(112,209)(113,208)(114,210)(115,206)(116,205)(117,207)
(118,203)(119,202)(120,204)(121,185)(122,184)(123,186)(124,182)(125,181)
(126,183)(127,194)(128,193)(129,195)(130,191)(131,190)(132,192)(133,188)
(134,187)(135,189)(136,170)(137,169)(138,171)(139,167)(140,166)(141,168)
(142,179)(143,178)(144,180)(145,176)(146,175)(147,177)(148,173)(149,172)
(150,174)(226,229)(227,231)(228,230)(232,238)(233,240)(234,239)(236,237)
(241,289)(242,291)(243,290)(244,286)(245,288)(246,287)(247,298)(248,300)
(249,299)(250,295)(251,297)(252,296)(253,292)(254,294)(255,293)(256,274)
(257,276)(258,275)(259,271)(260,273)(261,272)(262,283)(263,285)(264,284)
(265,280)(266,282)(267,281)(268,277)(269,279)(270,278)(301,380)(302,379)
(303,381)(304,377)(305,376)(306,378)(307,389)(308,388)(309,390)(310,386)
(311,385)(312,387)(313,383)(314,382)(315,384)(316,440)(317,439)(318,441)
(319,437)(320,436)(321,438)(322,449)(323,448)(324,450)(325,446)(326,445)
(327,447)(328,443)(329,442)(330,444)(331,425)(332,424)(333,426)(334,422)
(335,421)(336,423)(337,434)(338,433)(339,435)(340,431)(341,430)(342,432)
(343,428)(344,427)(345,429)(346,410)(347,409)(348,411)(349,407)(350,406)
(351,408)(352,419)(353,418)(354,420)(355,416)(356,415)(357,417)(358,413)
(359,412)(360,414)(361,395)(362,394)(363,396)(364,392)(365,391)(366,393)
(367,404)(368,403)(369,405)(370,401)(371,400)(372,402)(373,398)(374,397)
(375,399);
s2 := Sym(450)!( 1,316)( 2,318)( 3,317)( 4,328)( 5,330)( 6,329)( 7,325)
( 8,327)( 9,326)( 10,322)( 11,324)( 12,323)( 13,319)( 14,321)( 15,320)
( 16,301)( 17,303)( 18,302)( 19,313)( 20,315)( 21,314)( 22,310)( 23,312)
( 24,311)( 25,307)( 26,309)( 27,308)( 28,304)( 29,306)( 30,305)( 31,361)
( 32,363)( 33,362)( 34,373)( 35,375)( 36,374)( 37,370)( 38,372)( 39,371)
( 40,367)( 41,369)( 42,368)( 43,364)( 44,366)( 45,365)( 46,346)( 47,348)
( 48,347)( 49,358)( 50,360)( 51,359)( 52,355)( 53,357)( 54,356)( 55,352)
( 56,354)( 57,353)( 58,349)( 59,351)( 60,350)( 61,331)( 62,333)( 63,332)
( 64,343)( 65,345)( 66,344)( 67,340)( 68,342)( 69,341)( 70,337)( 71,339)
( 72,338)( 73,334)( 74,336)( 75,335)( 76,241)( 77,243)( 78,242)( 79,253)
( 80,255)( 81,254)( 82,250)( 83,252)( 84,251)( 85,247)( 86,249)( 87,248)
( 88,244)( 89,246)( 90,245)( 91,226)( 92,228)( 93,227)( 94,238)( 95,240)
( 96,239)( 97,235)( 98,237)( 99,236)(100,232)(101,234)(102,233)(103,229)
(104,231)(105,230)(106,286)(107,288)(108,287)(109,298)(110,300)(111,299)
(112,295)(113,297)(114,296)(115,292)(116,294)(117,293)(118,289)(119,291)
(120,290)(121,271)(122,273)(123,272)(124,283)(125,285)(126,284)(127,280)
(128,282)(129,281)(130,277)(131,279)(132,278)(133,274)(134,276)(135,275)
(136,256)(137,258)(138,257)(139,268)(140,270)(141,269)(142,265)(143,267)
(144,266)(145,262)(146,264)(147,263)(148,259)(149,261)(150,260)(151,392)
(152,391)(153,393)(154,404)(155,403)(156,405)(157,401)(158,400)(159,402)
(160,398)(161,397)(162,399)(163,395)(164,394)(165,396)(166,377)(167,376)
(168,378)(169,389)(170,388)(171,390)(172,386)(173,385)(174,387)(175,383)
(176,382)(177,384)(178,380)(179,379)(180,381)(181,437)(182,436)(183,438)
(184,449)(185,448)(186,450)(187,446)(188,445)(189,447)(190,443)(191,442)
(192,444)(193,440)(194,439)(195,441)(196,422)(197,421)(198,423)(199,434)
(200,433)(201,435)(202,431)(203,430)(204,432)(205,428)(206,427)(207,429)
(208,425)(209,424)(210,426)(211,407)(212,406)(213,408)(214,419)(215,418)
(216,420)(217,416)(218,415)(219,417)(220,413)(221,412)(222,414)(223,410)
(224,409)(225,411);
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*s0*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,
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