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Polytope of Type {6,45}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,45}*1620d
if this polytope has a name.
Group : SmallGroup(1620,137)
Rank : 3
Schlafli Type : {6,45}
Number of vertices, edges, etc : 18, 405, 135
Order of s0s1s2 : 90
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {6,15}*540
   5-fold quotients : {6,9}*324d
   9-fold quotients : {6,15}*180
   15-fold quotients : {6,3}*108
   27-fold quotients : {2,15}*60
   45-fold quotients : {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 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 46, 91)
( 47, 92)( 48, 93)( 49, 97)( 50, 98)( 51, 99)( 52, 94)( 53, 95)( 54, 96)
( 55,100)( 56,101)( 57,102)( 58,106)( 59,107)( 60,108)( 61,103)( 62,104)
( 63,105)( 64,109)( 65,110)( 66,111)( 67,115)( 68,116)( 69,117)( 70,112)
( 71,113)( 72,114)( 73,118)( 74,119)( 75,120)( 76,124)( 77,125)( 78,126)
( 79,121)( 80,122)( 81,123)( 82,127)( 83,128)( 84,129)( 85,133)( 86,134)
( 87,135)( 88,130)( 89,131)( 90,132)(139,142)(140,143)(141,144)(148,151)
(149,152)(150,153)(157,160)(158,161)(159,162)(166,169)(167,170)(168,171)
(175,178)(176,179)(177,180)(181,226)(182,227)(183,228)(184,232)(185,233)
(186,234)(187,229)(188,230)(189,231)(190,235)(191,236)(192,237)(193,241)
(194,242)(195,243)(196,238)(197,239)(198,240)(199,244)(200,245)(201,246)
(202,250)(203,251)(204,252)(205,247)(206,248)(207,249)(208,253)(209,254)
(210,255)(211,259)(212,260)(213,261)(214,256)(215,257)(216,258)(217,262)
(218,263)(219,264)(220,268)(221,269)(222,270)(223,265)(224,266)(225,267)
(274,277)(275,278)(276,279)(283,286)(284,287)(285,288)(292,295)(293,296)
(294,297)(301,304)(302,305)(303,306)(310,313)(311,314)(312,315)(316,361)
(317,362)(318,363)(319,367)(320,368)(321,369)(322,364)(323,365)(324,366)
(325,370)(326,371)(327,372)(328,376)(329,377)(330,378)(331,373)(332,374)
(333,375)(334,379)(335,380)(336,381)(337,385)(338,386)(339,387)(340,382)
(341,383)(342,384)(343,388)(344,389)(345,390)(346,394)(347,395)(348,396)
(349,391)(350,392)(351,393)(352,397)(353,398)(354,399)(355,403)(356,404)
(357,405)(358,400)(359,401)(360,402);;
s1 := (  1, 46)(  2, 48)(  3, 47)(  4, 50)(  5, 49)(  6, 51)(  7, 54)(  8, 53)
(  9, 52)( 10, 82)( 11, 84)( 12, 83)( 13, 86)( 14, 85)( 15, 87)( 16, 90)
( 17, 89)( 18, 88)( 19, 73)( 20, 75)( 21, 74)( 22, 77)( 23, 76)( 24, 78)
( 25, 81)( 26, 80)( 27, 79)( 28, 64)( 29, 66)( 30, 65)( 31, 68)( 32, 67)
( 33, 69)( 34, 72)( 35, 71)( 36, 70)( 37, 55)( 38, 57)( 39, 56)( 40, 59)
( 41, 58)( 42, 60)( 43, 63)( 44, 62)( 45, 61)( 92, 93)( 94, 95)( 97, 99)
(100,127)(101,129)(102,128)(103,131)(104,130)(105,132)(106,135)(107,134)
(108,133)(109,118)(110,120)(111,119)(112,122)(113,121)(114,123)(115,126)
(116,125)(117,124)(136,317)(137,316)(138,318)(139,321)(140,320)(141,319)
(142,322)(143,324)(144,323)(145,353)(146,352)(147,354)(148,357)(149,356)
(150,355)(151,358)(152,360)(153,359)(154,344)(155,343)(156,345)(157,348)
(158,347)(159,346)(160,349)(161,351)(162,350)(163,335)(164,334)(165,336)
(166,339)(167,338)(168,337)(169,340)(170,342)(171,341)(172,326)(173,325)
(174,327)(175,330)(176,329)(177,328)(178,331)(179,333)(180,332)(181,272)
(182,271)(183,273)(184,276)(185,275)(186,274)(187,277)(188,279)(189,278)
(190,308)(191,307)(192,309)(193,312)(194,311)(195,310)(196,313)(197,315)
(198,314)(199,299)(200,298)(201,300)(202,303)(203,302)(204,301)(205,304)
(206,306)(207,305)(208,290)(209,289)(210,291)(211,294)(212,293)(213,292)
(214,295)(215,297)(216,296)(217,281)(218,280)(219,282)(220,285)(221,284)
(222,283)(223,286)(224,288)(225,287)(226,362)(227,361)(228,363)(229,366)
(230,365)(231,364)(232,367)(233,369)(234,368)(235,398)(236,397)(237,399)
(238,402)(239,401)(240,400)(241,403)(242,405)(243,404)(244,389)(245,388)
(246,390)(247,393)(248,392)(249,391)(250,394)(251,396)(252,395)(253,380)
(254,379)(255,381)(256,384)(257,383)(258,382)(259,385)(260,387)(261,386)
(262,371)(263,370)(264,372)(265,375)(266,374)(267,373)(268,376)(269,378)
(270,377);;
s2 := (  1,145)(  2,147)(  3,146)(  4,148)(  5,150)(  6,149)(  7,151)(  8,153)
(  9,152)( 10,136)( 11,138)( 12,137)( 13,139)( 14,141)( 15,140)( 16,142)
( 17,144)( 18,143)( 19,172)( 20,174)( 21,173)( 22,175)( 23,177)( 24,176)
( 25,178)( 26,180)( 27,179)( 28,163)( 29,165)( 30,164)( 31,166)( 32,168)
( 33,167)( 34,169)( 35,171)( 36,170)( 37,154)( 38,156)( 39,155)( 40,157)
( 41,159)( 42,158)( 43,160)( 44,162)( 45,161)( 46,243)( 47,242)( 48,241)
( 49,237)( 50,236)( 51,235)( 52,240)( 53,239)( 54,238)( 55,234)( 56,233)
( 57,232)( 58,228)( 59,227)( 60,226)( 61,231)( 62,230)( 63,229)( 64,270)
( 65,269)( 66,268)( 67,264)( 68,263)( 69,262)( 70,267)( 71,266)( 72,265)
( 73,261)( 74,260)( 75,259)( 76,255)( 77,254)( 78,253)( 79,258)( 80,257)
( 81,256)( 82,252)( 83,251)( 84,250)( 85,246)( 86,245)( 87,244)( 88,249)
( 89,248)( 90,247)( 91,195)( 92,194)( 93,193)( 94,198)( 95,197)( 96,196)
( 97,192)( 98,191)( 99,190)(100,186)(101,185)(102,184)(103,189)(104,188)
(105,187)(106,183)(107,182)(108,181)(109,222)(110,221)(111,220)(112,225)
(113,224)(114,223)(115,219)(116,218)(117,217)(118,213)(119,212)(120,211)
(121,216)(122,215)(123,214)(124,210)(125,209)(126,208)(127,204)(128,203)
(129,202)(130,207)(131,206)(132,205)(133,201)(134,200)(135,199)(271,281)
(272,280)(273,282)(274,284)(275,283)(276,285)(277,287)(278,286)(279,288)
(289,308)(290,307)(291,309)(292,311)(293,310)(294,312)(295,314)(296,313)
(297,315)(298,299)(301,302)(304,305)(316,376)(317,378)(318,377)(319,370)
(320,372)(321,371)(322,373)(323,375)(324,374)(325,367)(326,369)(327,368)
(328,361)(329,363)(330,362)(331,364)(332,366)(333,365)(334,403)(335,405)
(336,404)(337,397)(338,399)(339,398)(340,400)(341,402)(342,401)(343,394)
(344,396)(345,395)(346,388)(347,390)(348,389)(349,391)(350,393)(351,392)
(352,385)(353,387)(354,386)(355,379)(356,381)(357,380)(358,382)(359,384)
(360,383);;
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(405)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)
( 46, 91)( 47, 92)( 48, 93)( 49, 97)( 50, 98)( 51, 99)( 52, 94)( 53, 95)
( 54, 96)( 55,100)( 56,101)( 57,102)( 58,106)( 59,107)( 60,108)( 61,103)
( 62,104)( 63,105)( 64,109)( 65,110)( 66,111)( 67,115)( 68,116)( 69,117)
( 70,112)( 71,113)( 72,114)( 73,118)( 74,119)( 75,120)( 76,124)( 77,125)
( 78,126)( 79,121)( 80,122)( 81,123)( 82,127)( 83,128)( 84,129)( 85,133)
( 86,134)( 87,135)( 88,130)( 89,131)( 90,132)(139,142)(140,143)(141,144)
(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(166,169)(167,170)
(168,171)(175,178)(176,179)(177,180)(181,226)(182,227)(183,228)(184,232)
(185,233)(186,234)(187,229)(188,230)(189,231)(190,235)(191,236)(192,237)
(193,241)(194,242)(195,243)(196,238)(197,239)(198,240)(199,244)(200,245)
(201,246)(202,250)(203,251)(204,252)(205,247)(206,248)(207,249)(208,253)
(209,254)(210,255)(211,259)(212,260)(213,261)(214,256)(215,257)(216,258)
(217,262)(218,263)(219,264)(220,268)(221,269)(222,270)(223,265)(224,266)
(225,267)(274,277)(275,278)(276,279)(283,286)(284,287)(285,288)(292,295)
(293,296)(294,297)(301,304)(302,305)(303,306)(310,313)(311,314)(312,315)
(316,361)(317,362)(318,363)(319,367)(320,368)(321,369)(322,364)(323,365)
(324,366)(325,370)(326,371)(327,372)(328,376)(329,377)(330,378)(331,373)
(332,374)(333,375)(334,379)(335,380)(336,381)(337,385)(338,386)(339,387)
(340,382)(341,383)(342,384)(343,388)(344,389)(345,390)(346,394)(347,395)
(348,396)(349,391)(350,392)(351,393)(352,397)(353,398)(354,399)(355,403)
(356,404)(357,405)(358,400)(359,401)(360,402);
s1 := Sym(405)!(  1, 46)(  2, 48)(  3, 47)(  4, 50)(  5, 49)(  6, 51)(  7, 54)
(  8, 53)(  9, 52)( 10, 82)( 11, 84)( 12, 83)( 13, 86)( 14, 85)( 15, 87)
( 16, 90)( 17, 89)( 18, 88)( 19, 73)( 20, 75)( 21, 74)( 22, 77)( 23, 76)
( 24, 78)( 25, 81)( 26, 80)( 27, 79)( 28, 64)( 29, 66)( 30, 65)( 31, 68)
( 32, 67)( 33, 69)( 34, 72)( 35, 71)( 36, 70)( 37, 55)( 38, 57)( 39, 56)
( 40, 59)( 41, 58)( 42, 60)( 43, 63)( 44, 62)( 45, 61)( 92, 93)( 94, 95)
( 97, 99)(100,127)(101,129)(102,128)(103,131)(104,130)(105,132)(106,135)
(107,134)(108,133)(109,118)(110,120)(111,119)(112,122)(113,121)(114,123)
(115,126)(116,125)(117,124)(136,317)(137,316)(138,318)(139,321)(140,320)
(141,319)(142,322)(143,324)(144,323)(145,353)(146,352)(147,354)(148,357)
(149,356)(150,355)(151,358)(152,360)(153,359)(154,344)(155,343)(156,345)
(157,348)(158,347)(159,346)(160,349)(161,351)(162,350)(163,335)(164,334)
(165,336)(166,339)(167,338)(168,337)(169,340)(170,342)(171,341)(172,326)
(173,325)(174,327)(175,330)(176,329)(177,328)(178,331)(179,333)(180,332)
(181,272)(182,271)(183,273)(184,276)(185,275)(186,274)(187,277)(188,279)
(189,278)(190,308)(191,307)(192,309)(193,312)(194,311)(195,310)(196,313)
(197,315)(198,314)(199,299)(200,298)(201,300)(202,303)(203,302)(204,301)
(205,304)(206,306)(207,305)(208,290)(209,289)(210,291)(211,294)(212,293)
(213,292)(214,295)(215,297)(216,296)(217,281)(218,280)(219,282)(220,285)
(221,284)(222,283)(223,286)(224,288)(225,287)(226,362)(227,361)(228,363)
(229,366)(230,365)(231,364)(232,367)(233,369)(234,368)(235,398)(236,397)
(237,399)(238,402)(239,401)(240,400)(241,403)(242,405)(243,404)(244,389)
(245,388)(246,390)(247,393)(248,392)(249,391)(250,394)(251,396)(252,395)
(253,380)(254,379)(255,381)(256,384)(257,383)(258,382)(259,385)(260,387)
(261,386)(262,371)(263,370)(264,372)(265,375)(266,374)(267,373)(268,376)
(269,378)(270,377);
s2 := Sym(405)!(  1,145)(  2,147)(  3,146)(  4,148)(  5,150)(  6,149)(  7,151)
(  8,153)(  9,152)( 10,136)( 11,138)( 12,137)( 13,139)( 14,141)( 15,140)
( 16,142)( 17,144)( 18,143)( 19,172)( 20,174)( 21,173)( 22,175)( 23,177)
( 24,176)( 25,178)( 26,180)( 27,179)( 28,163)( 29,165)( 30,164)( 31,166)
( 32,168)( 33,167)( 34,169)( 35,171)( 36,170)( 37,154)( 38,156)( 39,155)
( 40,157)( 41,159)( 42,158)( 43,160)( 44,162)( 45,161)( 46,243)( 47,242)
( 48,241)( 49,237)( 50,236)( 51,235)( 52,240)( 53,239)( 54,238)( 55,234)
( 56,233)( 57,232)( 58,228)( 59,227)( 60,226)( 61,231)( 62,230)( 63,229)
( 64,270)( 65,269)( 66,268)( 67,264)( 68,263)( 69,262)( 70,267)( 71,266)
( 72,265)( 73,261)( 74,260)( 75,259)( 76,255)( 77,254)( 78,253)( 79,258)
( 80,257)( 81,256)( 82,252)( 83,251)( 84,250)( 85,246)( 86,245)( 87,244)
( 88,249)( 89,248)( 90,247)( 91,195)( 92,194)( 93,193)( 94,198)( 95,197)
( 96,196)( 97,192)( 98,191)( 99,190)(100,186)(101,185)(102,184)(103,189)
(104,188)(105,187)(106,183)(107,182)(108,181)(109,222)(110,221)(111,220)
(112,225)(113,224)(114,223)(115,219)(116,218)(117,217)(118,213)(119,212)
(120,211)(121,216)(122,215)(123,214)(124,210)(125,209)(126,208)(127,204)
(128,203)(129,202)(130,207)(131,206)(132,205)(133,201)(134,200)(135,199)
(271,281)(272,280)(273,282)(274,284)(275,283)(276,285)(277,287)(278,286)
(279,288)(289,308)(290,307)(291,309)(292,311)(293,310)(294,312)(295,314)
(296,313)(297,315)(298,299)(301,302)(304,305)(316,376)(317,378)(318,377)
(319,370)(320,372)(321,371)(322,373)(323,375)(324,374)(325,367)(326,369)
(327,368)(328,361)(329,363)(330,362)(331,364)(332,366)(333,365)(334,403)
(335,405)(336,404)(337,397)(338,399)(339,398)(340,400)(341,402)(342,401)
(343,394)(344,396)(345,395)(346,388)(347,390)(348,389)(349,391)(350,393)
(351,392)(352,385)(353,387)(354,386)(355,379)(356,381)(357,380)(358,382)
(359,384)(360,383);
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1 >; 
 
References : None.
to this polytope