Polytope of Type {6,90}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,90}*1620c
if this polytope has a name.
Group : SmallGroup(1620,139)
Rank : 3
Schlafli Type : {6,90}
Number of vertices, edges, etc : 9, 405, 135
Order of s₀s₁s₂ : 15
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-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 : {6,30}*540
   5-fold quotients : {6,18}*324c
   15-fold quotients : {6,6}*108
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, 51)(  5, 50)(  6, 49)(  7, 53)(  8, 52)
(  9, 54)( 10, 82)( 11, 84)( 12, 83)( 13, 87)( 14, 86)( 15, 85)( 16, 89)
( 17, 88)( 18, 90)( 19, 73)( 20, 75)( 21, 74)( 22, 78)( 23, 77)( 24, 76)
( 25, 80)( 26, 79)( 27, 81)( 28, 64)( 29, 66)( 30, 65)( 31, 69)( 32, 68)
( 33, 67)( 34, 71)( 35, 70)( 36, 72)( 37, 55)( 38, 57)( 39, 56)( 40, 60)
( 41, 59)( 42, 58)( 43, 62)( 44, 61)( 45, 63)( 92, 93)( 94, 96)( 97, 98)
(100,127)(101,129)(102,128)(103,132)(104,131)(105,130)(106,134)(107,133)
(108,135)(109,118)(110,120)(111,119)(112,123)(113,122)(114,121)(115,125)
(116,124)(117,126)(136,317)(137,316)(138,318)(139,319)(140,321)(141,320)
(142,324)(143,323)(144,322)(145,353)(146,352)(147,354)(148,355)(149,357)
(150,356)(151,360)(152,359)(153,358)(154,344)(155,343)(156,345)(157,346)
(158,348)(159,347)(160,351)(161,350)(162,349)(163,335)(164,334)(165,336)
(166,337)(167,339)(168,338)(169,342)(170,341)(171,340)(172,326)(173,325)
(174,327)(175,328)(176,330)(177,329)(178,333)(179,332)(180,331)(181,272)
(182,271)(183,273)(184,274)(185,276)(186,275)(187,279)(188,278)(189,277)
(190,308)(191,307)(192,309)(193,310)(194,312)(195,311)(196,315)(197,314)
(198,313)(199,299)(200,298)(201,300)(202,301)(203,303)(204,302)(205,306)
(206,305)(207,304)(208,290)(209,289)(210,291)(211,292)(212,294)(213,293)
(214,297)(215,296)(216,295)(217,281)(218,280)(219,282)(220,283)(221,285)
(222,284)(223,288)(224,287)(225,286)(226,362)(227,361)(228,363)(229,364)
(230,366)(231,365)(232,369)(233,368)(234,367)(235,398)(236,397)(237,399)
(238,400)(239,402)(240,401)(241,405)(242,404)(243,403)(244,389)(245,388)
(246,390)(247,391)(248,393)(249,392)(250,396)(251,395)(252,394)(253,380)
(254,379)(255,381)(256,382)(257,384)(258,383)(259,387)(260,386)(261,385)
(262,371)(263,370)(264,372)(265,373)(266,375)(267,374)(268,378)(269,377)
(270,376);;
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)( 46,194)( 47,193)( 48,195)
( 49,191)( 50,190)( 51,192)( 52,197)( 53,196)( 54,198)( 55,185)( 56,184)
( 57,186)( 58,182)( 59,181)( 60,183)( 61,188)( 62,187)( 63,189)( 64,221)
( 65,220)( 66,222)( 67,218)( 68,217)( 69,219)( 70,224)( 71,223)( 72,225)
( 73,212)( 74,211)( 75,213)( 76,209)( 77,208)( 78,210)( 79,215)( 80,214)
( 81,216)( 82,203)( 83,202)( 84,204)( 85,200)( 86,199)( 87,201)( 88,206)
( 89,205)( 90,207)( 91,242)( 92,241)( 93,243)( 94,239)( 95,238)( 96,240)
( 97,236)( 98,235)( 99,237)(100,233)(101,232)(102,234)(103,230)(104,229)
(105,231)(106,227)(107,226)(108,228)(109,269)(110,268)(111,270)(112,266)
(113,265)(114,267)(115,263)(116,262)(117,264)(118,260)(119,259)(120,261)
(121,257)(122,256)(123,258)(124,254)(125,253)(126,255)(127,251)(128,250)
(129,252)(130,248)(131,247)(132,249)(133,245)(134,244)(135,246)(271,281)
(272,280)(273,282)(274,287)(275,286)(276,288)(277,284)(278,283)(279,285)
(289,308)(290,307)(291,309)(292,314)(293,313)(294,315)(295,311)(296,310)
(297,312)(298,299)(301,305)(302,304)(303,306)(316,330)(317,329)(318,328)
(319,327)(320,326)(321,325)(322,333)(323,332)(324,331)(334,357)(335,356)
(336,355)(337,354)(338,353)(339,352)(340,360)(341,359)(342,358)(343,348)
(344,347)(345,346)(349,351)(361,378)(362,377)(363,376)(364,375)(365,374)
(366,373)(367,372)(368,371)(369,370)(379,405)(380,404)(381,403)(382,402)
(383,401)(384,400)(385,399)(386,398)(387,397)(388,396)(389,395)(390,394)
(391,393);;
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, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
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, 51)(  5, 50)(  6, 49)(  7, 53)
(  8, 52)(  9, 54)( 10, 82)( 11, 84)( 12, 83)( 13, 87)( 14, 86)( 15, 85)
( 16, 89)( 17, 88)( 18, 90)( 19, 73)( 20, 75)( 21, 74)( 22, 78)( 23, 77)
( 24, 76)( 25, 80)( 26, 79)( 27, 81)( 28, 64)( 29, 66)( 30, 65)( 31, 69)
( 32, 68)( 33, 67)( 34, 71)( 35, 70)( 36, 72)( 37, 55)( 38, 57)( 39, 56)
( 40, 60)( 41, 59)( 42, 58)( 43, 62)( 44, 61)( 45, 63)( 92, 93)( 94, 96)
( 97, 98)(100,127)(101,129)(102,128)(103,132)(104,131)(105,130)(106,134)
(107,133)(108,135)(109,118)(110,120)(111,119)(112,123)(113,122)(114,121)
(115,125)(116,124)(117,126)(136,317)(137,316)(138,318)(139,319)(140,321)
(141,320)(142,324)(143,323)(144,322)(145,353)(146,352)(147,354)(148,355)
(149,357)(150,356)(151,360)(152,359)(153,358)(154,344)(155,343)(156,345)
(157,346)(158,348)(159,347)(160,351)(161,350)(162,349)(163,335)(164,334)
(165,336)(166,337)(167,339)(168,338)(169,342)(170,341)(171,340)(172,326)
(173,325)(174,327)(175,328)(176,330)(177,329)(178,333)(179,332)(180,331)
(181,272)(182,271)(183,273)(184,274)(185,276)(186,275)(187,279)(188,278)
(189,277)(190,308)(191,307)(192,309)(193,310)(194,312)(195,311)(196,315)
(197,314)(198,313)(199,299)(200,298)(201,300)(202,301)(203,303)(204,302)
(205,306)(206,305)(207,304)(208,290)(209,289)(210,291)(211,292)(212,294)
(213,293)(214,297)(215,296)(216,295)(217,281)(218,280)(219,282)(220,283)
(221,285)(222,284)(223,288)(224,287)(225,286)(226,362)(227,361)(228,363)
(229,364)(230,366)(231,365)(232,369)(233,368)(234,367)(235,398)(236,397)
(237,399)(238,400)(239,402)(240,401)(241,405)(242,404)(243,403)(244,389)
(245,388)(246,390)(247,391)(248,393)(249,392)(250,396)(251,395)(252,394)
(253,380)(254,379)(255,381)(256,382)(257,384)(258,383)(259,387)(260,386)
(261,385)(262,371)(263,370)(264,372)(265,373)(266,375)(267,374)(268,378)
(269,377)(270,376);
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)( 46,194)( 47,193)
( 48,195)( 49,191)( 50,190)( 51,192)( 52,197)( 53,196)( 54,198)( 55,185)
( 56,184)( 57,186)( 58,182)( 59,181)( 60,183)( 61,188)( 62,187)( 63,189)
( 64,221)( 65,220)( 66,222)( 67,218)( 68,217)( 69,219)( 70,224)( 71,223)
( 72,225)( 73,212)( 74,211)( 75,213)( 76,209)( 77,208)( 78,210)( 79,215)
( 80,214)( 81,216)( 82,203)( 83,202)( 84,204)( 85,200)( 86,199)( 87,201)
( 88,206)( 89,205)( 90,207)( 91,242)( 92,241)( 93,243)( 94,239)( 95,238)
( 96,240)( 97,236)( 98,235)( 99,237)(100,233)(101,232)(102,234)(103,230)
(104,229)(105,231)(106,227)(107,226)(108,228)(109,269)(110,268)(111,270)
(112,266)(113,265)(114,267)(115,263)(116,262)(117,264)(118,260)(119,259)
(120,261)(121,257)(122,256)(123,258)(124,254)(125,253)(126,255)(127,251)
(128,250)(129,252)(130,248)(131,247)(132,249)(133,245)(134,244)(135,246)
(271,281)(272,280)(273,282)(274,287)(275,286)(276,288)(277,284)(278,283)
(279,285)(289,308)(290,307)(291,309)(292,314)(293,313)(294,315)(295,311)
(296,310)(297,312)(298,299)(301,305)(302,304)(303,306)(316,330)(317,329)
(318,328)(319,327)(320,326)(321,325)(322,333)(323,332)(324,331)(334,357)
(335,356)(336,355)(337,354)(338,353)(339,352)(340,360)(341,359)(342,358)
(343,348)(344,347)(345,346)(349,351)(361,378)(362,377)(363,376)(364,375)
(365,374)(366,373)(367,372)(368,371)(369,370)(379,405)(380,404)(381,403)
(382,402)(383,401)(384,400)(385,399)(386,398)(387,397)(388,396)(389,395)
(390,394)(391,393);
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, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope