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Polytope of Type {18,45}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,45}*1620
if this polytope has a name.
Group : SmallGroup(1620,130)
Rank : 3
Schlafli Type : {18,45}
Number of vertices, edges, etc : 18, 405, 45
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 : {6,45}*540
   5-fold quotients : {18,9}*324
   9-fold quotients : {2,45}*180, {6,15}*180
   15-fold quotients : {6,9}*108
   27-fold quotients : {2,15}*60
   45-fold quotients : {2,9}*36, {6,3}*36
   81-fold quotients : {2,5}*20
   135-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  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);;
s1 := (  1, 46)(  2, 48)(  3, 47)(  4, 52)(  5, 54)(  6, 53)(  7, 49)(  8, 51)
(  9, 50)( 10, 82)( 11, 84)( 12, 83)( 13, 88)( 14, 90)( 15, 89)( 16, 85)
( 17, 87)( 18, 86)( 19, 73)( 20, 75)( 21, 74)( 22, 79)( 23, 81)( 24, 80)
( 25, 76)( 26, 78)( 27, 77)( 28, 64)( 29, 66)( 30, 65)( 31, 70)( 32, 72)
( 33, 71)( 34, 67)( 35, 69)( 36, 68)( 37, 55)( 38, 57)( 39, 56)( 40, 61)
( 41, 63)( 42, 62)( 43, 58)( 44, 60)( 45, 59)( 91, 92)( 94, 98)( 95, 97)
( 96, 99)(100,128)(101,127)(102,129)(103,134)(104,133)(105,135)(106,131)
(107,130)(108,132)(109,119)(110,118)(111,120)(112,125)(113,124)(114,126)
(115,122)(116,121)(117,123)(136,319)(137,321)(138,320)(139,316)(140,318)
(141,317)(142,322)(143,324)(144,323)(145,355)(146,357)(147,356)(148,352)
(149,354)(150,353)(151,358)(152,360)(153,359)(154,346)(155,348)(156,347)
(157,343)(158,345)(159,344)(160,349)(161,351)(162,350)(163,337)(164,339)
(165,338)(166,334)(167,336)(168,335)(169,340)(170,342)(171,341)(172,328)
(173,330)(174,329)(175,325)(176,327)(177,326)(178,331)(179,333)(180,332)
(181,274)(182,276)(183,275)(184,271)(185,273)(186,272)(187,277)(188,279)
(189,278)(190,310)(191,312)(192,311)(193,307)(194,309)(195,308)(196,313)
(197,315)(198,314)(199,301)(200,303)(201,302)(202,298)(203,300)(204,299)
(205,304)(206,306)(207,305)(208,292)(209,294)(210,293)(211,289)(212,291)
(213,290)(214,295)(215,297)(216,296)(217,283)(218,285)(219,284)(220,280)
(221,282)(222,281)(223,286)(224,288)(225,287)(226,365)(227,364)(228,366)
(229,362)(230,361)(231,363)(232,368)(233,367)(234,369)(235,401)(236,400)
(237,402)(238,398)(239,397)(240,399)(241,404)(242,403)(243,405)(244,392)
(245,391)(246,393)(247,389)(248,388)(249,390)(250,395)(251,394)(252,396)
(253,383)(254,382)(255,384)(256,380)(257,379)(258,381)(259,386)(260,385)
(261,387)(262,374)(263,373)(264,375)(265,371)(266,370)(267,372)(268,377)
(269,376)(270,378);;
s2 := (  1,145)(  2,147)(  3,146)(  4,151)(  5,153)(  6,152)(  7,148)(  8,150)
(  9,149)( 10,136)( 11,138)( 12,137)( 13,142)( 14,144)( 15,143)( 16,139)
( 17,141)( 18,140)( 19,172)( 20,174)( 21,173)( 22,178)( 23,180)( 24,179)
( 25,175)( 26,177)( 27,176)( 28,163)( 29,165)( 30,164)( 31,169)( 32,171)
( 33,170)( 34,166)( 35,168)( 36,167)( 37,154)( 38,156)( 39,155)( 40,160)
( 41,162)( 42,161)( 43,157)( 44,159)( 45,158)( 46,236)( 47,235)( 48,237)
( 49,242)( 50,241)( 51,243)( 52,239)( 53,238)( 54,240)( 55,227)( 56,226)
( 57,228)( 58,233)( 59,232)( 60,234)( 61,230)( 62,229)( 63,231)( 64,263)
( 65,262)( 66,264)( 67,269)( 68,268)( 69,270)( 70,266)( 71,265)( 72,267)
( 73,254)( 74,253)( 75,255)( 76,260)( 77,259)( 78,261)( 79,257)( 80,256)
( 81,258)( 82,245)( 83,244)( 84,246)( 85,251)( 86,250)( 87,252)( 88,248)
( 89,247)( 90,249)( 91,191)( 92,190)( 93,192)( 94,197)( 95,196)( 96,198)
( 97,194)( 98,193)( 99,195)(100,182)(101,181)(102,183)(103,188)(104,187)
(105,189)(106,185)(107,184)(108,186)(109,218)(110,217)(111,219)(112,224)
(113,223)(114,225)(115,221)(116,220)(117,222)(118,209)(119,208)(120,210)
(121,215)(122,214)(123,216)(124,212)(125,211)(126,213)(127,200)(128,199)
(129,201)(130,206)(131,205)(132,207)(133,203)(134,202)(135,204)(271,283)
(272,285)(273,284)(274,280)(275,282)(276,281)(277,286)(278,288)(279,287)
(289,310)(290,312)(291,311)(292,307)(293,309)(294,308)(295,313)(296,315)
(297,314)(298,301)(299,303)(300,302)(305,306)(316,374)(317,373)(318,375)
(319,371)(320,370)(321,372)(322,377)(323,376)(324,378)(325,365)(326,364)
(327,366)(328,362)(329,361)(330,363)(331,368)(332,367)(333,369)(334,401)
(335,400)(336,402)(337,398)(338,397)(339,399)(340,404)(341,403)(342,405)
(343,392)(344,391)(345,393)(346,389)(347,388)(348,390)(349,395)(350,394)
(351,396)(352,383)(353,382)(354,384)(355,380)(356,379)(357,381)(358,386)
(359,385)(360,387);;
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*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := 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);
s1 := Sym(405)!(  1, 46)(  2, 48)(  3, 47)(  4, 52)(  5, 54)(  6, 53)(  7, 49)
(  8, 51)(  9, 50)( 10, 82)( 11, 84)( 12, 83)( 13, 88)( 14, 90)( 15, 89)
( 16, 85)( 17, 87)( 18, 86)( 19, 73)( 20, 75)( 21, 74)( 22, 79)( 23, 81)
( 24, 80)( 25, 76)( 26, 78)( 27, 77)( 28, 64)( 29, 66)( 30, 65)( 31, 70)
( 32, 72)( 33, 71)( 34, 67)( 35, 69)( 36, 68)( 37, 55)( 38, 57)( 39, 56)
( 40, 61)( 41, 63)( 42, 62)( 43, 58)( 44, 60)( 45, 59)( 91, 92)( 94, 98)
( 95, 97)( 96, 99)(100,128)(101,127)(102,129)(103,134)(104,133)(105,135)
(106,131)(107,130)(108,132)(109,119)(110,118)(111,120)(112,125)(113,124)
(114,126)(115,122)(116,121)(117,123)(136,319)(137,321)(138,320)(139,316)
(140,318)(141,317)(142,322)(143,324)(144,323)(145,355)(146,357)(147,356)
(148,352)(149,354)(150,353)(151,358)(152,360)(153,359)(154,346)(155,348)
(156,347)(157,343)(158,345)(159,344)(160,349)(161,351)(162,350)(163,337)
(164,339)(165,338)(166,334)(167,336)(168,335)(169,340)(170,342)(171,341)
(172,328)(173,330)(174,329)(175,325)(176,327)(177,326)(178,331)(179,333)
(180,332)(181,274)(182,276)(183,275)(184,271)(185,273)(186,272)(187,277)
(188,279)(189,278)(190,310)(191,312)(192,311)(193,307)(194,309)(195,308)
(196,313)(197,315)(198,314)(199,301)(200,303)(201,302)(202,298)(203,300)
(204,299)(205,304)(206,306)(207,305)(208,292)(209,294)(210,293)(211,289)
(212,291)(213,290)(214,295)(215,297)(216,296)(217,283)(218,285)(219,284)
(220,280)(221,282)(222,281)(223,286)(224,288)(225,287)(226,365)(227,364)
(228,366)(229,362)(230,361)(231,363)(232,368)(233,367)(234,369)(235,401)
(236,400)(237,402)(238,398)(239,397)(240,399)(241,404)(242,403)(243,405)
(244,392)(245,391)(246,393)(247,389)(248,388)(249,390)(250,395)(251,394)
(252,396)(253,383)(254,382)(255,384)(256,380)(257,379)(258,381)(259,386)
(260,385)(261,387)(262,374)(263,373)(264,375)(265,371)(266,370)(267,372)
(268,377)(269,376)(270,378);
s2 := Sym(405)!(  1,145)(  2,147)(  3,146)(  4,151)(  5,153)(  6,152)(  7,148)
(  8,150)(  9,149)( 10,136)( 11,138)( 12,137)( 13,142)( 14,144)( 15,143)
( 16,139)( 17,141)( 18,140)( 19,172)( 20,174)( 21,173)( 22,178)( 23,180)
( 24,179)( 25,175)( 26,177)( 27,176)( 28,163)( 29,165)( 30,164)( 31,169)
( 32,171)( 33,170)( 34,166)( 35,168)( 36,167)( 37,154)( 38,156)( 39,155)
( 40,160)( 41,162)( 42,161)( 43,157)( 44,159)( 45,158)( 46,236)( 47,235)
( 48,237)( 49,242)( 50,241)( 51,243)( 52,239)( 53,238)( 54,240)( 55,227)
( 56,226)( 57,228)( 58,233)( 59,232)( 60,234)( 61,230)( 62,229)( 63,231)
( 64,263)( 65,262)( 66,264)( 67,269)( 68,268)( 69,270)( 70,266)( 71,265)
( 72,267)( 73,254)( 74,253)( 75,255)( 76,260)( 77,259)( 78,261)( 79,257)
( 80,256)( 81,258)( 82,245)( 83,244)( 84,246)( 85,251)( 86,250)( 87,252)
( 88,248)( 89,247)( 90,249)( 91,191)( 92,190)( 93,192)( 94,197)( 95,196)
( 96,198)( 97,194)( 98,193)( 99,195)(100,182)(101,181)(102,183)(103,188)
(104,187)(105,189)(106,185)(107,184)(108,186)(109,218)(110,217)(111,219)
(112,224)(113,223)(114,225)(115,221)(116,220)(117,222)(118,209)(119,208)
(120,210)(121,215)(122,214)(123,216)(124,212)(125,211)(126,213)(127,200)
(128,199)(129,201)(130,206)(131,205)(132,207)(133,203)(134,202)(135,204)
(271,283)(272,285)(273,284)(274,280)(275,282)(276,281)(277,286)(278,288)
(279,287)(289,310)(290,312)(291,311)(292,307)(293,309)(294,308)(295,313)
(296,315)(297,314)(298,301)(299,303)(300,302)(305,306)(316,374)(317,373)
(318,375)(319,371)(320,370)(321,372)(322,377)(323,376)(324,378)(325,365)
(326,364)(327,366)(328,362)(329,361)(330,363)(331,368)(332,367)(333,369)
(334,401)(335,400)(336,402)(337,398)(338,397)(339,399)(340,404)(341,403)
(342,405)(343,392)(344,391)(345,393)(346,389)(347,388)(348,390)(349,395)
(350,394)(351,396)(352,383)(353,382)(354,384)(355,380)(356,379)(357,381)
(358,386)(359,385)(360,387);
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*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*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 >; 
 
References : None.
to this polytope