Polytope of Type {45,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {45,18}*1620
if this polytope has a name.
Group : SmallGroup(1620,130)
Rank : 3
Schlafli Type : {45,18}
Number of vertices, edges, etc : 45, 405, 18
Order of s₀s₁s₂ : 90
Order of s₀s₁s₂s₁ : 18
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) :
   3-fold quotients : {45,6}*540
   5-fold quotients : {9,18}*324
   9-fold quotients : {45,2}*180, {15,6}*180
   15-fold quotients : {9,6}*108
   27-fold quotients : {15,2}*60
   45-fold quotients : {9,2}*36, {3,6}*36
   81-fold quotients : {5,2}*20
   135-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope