Polytope of Type {18,10}

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