Polytope of Type {6,90}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,90}*1620a
if this polytope has a name.
Group : SmallGroup(1620,131)
Rank : 3
Schlafli Type : {6,90}
Number of vertices, edges, etc : 9, 405, 135
Order of s0s1s2 : 45
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}*324a
15-fold quotients : {6,6}*108
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, 91)
( 47, 93)( 48, 92)( 49, 94)( 50, 96)( 51, 95)( 52, 97)( 53, 99)( 54, 98)
( 55,100)( 56,102)( 57,101)( 58,103)( 59,105)( 60,104)( 61,106)( 62,108)
( 63,107)( 64,109)( 65,111)( 66,110)( 67,112)( 68,114)( 69,113)( 70,115)
( 71,117)( 72,116)( 73,118)( 74,120)( 75,119)( 76,121)( 77,123)( 78,122)
( 79,124)( 80,126)( 81,125)( 82,127)( 83,129)( 84,128)( 85,130)( 86,132)
( 87,131)( 88,133)( 89,135)( 90,134)(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)(181,226)(182,228)(183,227)(184,229)(185,231)
(186,230)(187,232)(188,234)(189,233)(190,235)(191,237)(192,236)(193,238)
(194,240)(195,239)(196,241)(197,243)(198,242)(199,244)(200,246)(201,245)
(202,247)(203,249)(204,248)(205,250)(206,252)(207,251)(208,253)(209,255)
(210,254)(211,256)(212,258)(213,257)(214,259)(215,261)(216,260)(217,262)
(218,264)(219,263)(220,265)(221,267)(222,266)(223,268)(224,270)(225,269)
(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)(316,361)
(317,363)(318,362)(319,364)(320,366)(321,365)(322,367)(323,369)(324,368)
(325,370)(326,372)(327,371)(328,373)(329,375)(330,374)(331,376)(332,378)
(333,377)(334,379)(335,381)(336,380)(337,382)(338,384)(339,383)(340,385)
(341,387)(342,386)(343,388)(344,390)(345,389)(346,391)(347,393)(348,392)
(349,394)(350,396)(351,395)(352,397)(353,399)(354,398)(355,400)(356,402)
(357,401)(358,403)(359,405)(360,404);;
s1 := ( 1, 46)( 2, 47)( 3, 48)( 4, 52)( 5, 53)( 6, 54)( 7, 49)( 8, 50)
( 9, 51)( 10, 82)( 11, 83)( 12, 84)( 13, 88)( 14, 89)( 15, 90)( 16, 85)
( 17, 86)( 18, 87)( 19, 73)( 20, 74)( 21, 75)( 22, 79)( 23, 80)( 24, 81)
( 25, 76)( 26, 77)( 27, 78)( 28, 64)( 29, 65)( 30, 66)( 31, 70)( 32, 71)
( 33, 72)( 34, 67)( 35, 68)( 36, 69)( 37, 55)( 38, 56)( 39, 57)( 40, 61)
( 41, 62)( 42, 63)( 43, 58)( 44, 59)( 45, 60)( 94, 97)( 95, 98)( 96, 99)
(100,127)(101,128)(102,129)(103,133)(104,134)(105,135)(106,130)(107,131)
(108,132)(109,118)(110,119)(111,120)(112,124)(113,125)(114,126)(115,121)
(116,122)(117,123)(136,319)(137,320)(138,321)(139,316)(140,317)(141,318)
(142,322)(143,323)(144,324)(145,355)(146,356)(147,357)(148,352)(149,353)
(150,354)(151,358)(152,359)(153,360)(154,346)(155,347)(156,348)(157,343)
(158,344)(159,345)(160,349)(161,350)(162,351)(163,337)(164,338)(165,339)
(166,334)(167,335)(168,336)(169,340)(170,341)(171,342)(172,328)(173,329)
(174,330)(175,325)(176,326)(177,327)(178,331)(179,332)(180,333)(181,274)
(182,275)(183,276)(184,271)(185,272)(186,273)(187,277)(188,278)(189,279)
(190,310)(191,311)(192,312)(193,307)(194,308)(195,309)(196,313)(197,314)
(198,315)(199,301)(200,302)(201,303)(202,298)(203,299)(204,300)(205,304)
(206,305)(207,306)(208,292)(209,293)(210,294)(211,289)(212,290)(213,291)
(214,295)(215,296)(216,297)(217,283)(218,284)(219,285)(220,280)(221,281)
(222,282)(223,286)(224,287)(225,288)(226,364)(227,365)(228,366)(229,361)
(230,362)(231,363)(232,367)(233,368)(234,369)(235,400)(236,401)(237,402)
(238,397)(239,398)(240,399)(241,403)(242,404)(243,405)(244,391)(245,392)
(246,393)(247,388)(248,389)(249,390)(250,394)(251,395)(252,396)(253,382)
(254,383)(255,384)(256,379)(257,380)(258,381)(259,385)(260,386)(261,387)
(262,373)(263,374)(264,375)(265,370)(266,371)(267,372)(268,376)(269,377)
(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)( 46,191)( 47,190)( 48,192)
( 49,197)( 50,196)( 51,198)( 52,194)( 53,193)( 54,195)( 55,182)( 56,181)
( 57,183)( 58,188)( 59,187)( 60,189)( 61,185)( 62,184)( 63,186)( 64,218)
( 65,217)( 66,219)( 67,224)( 68,223)( 69,225)( 70,221)( 71,220)( 72,222)
( 73,209)( 74,208)( 75,210)( 76,215)( 77,214)( 78,216)( 79,212)( 80,211)
( 81,213)( 82,200)( 83,199)( 84,201)( 85,206)( 86,205)( 87,207)( 88,203)
( 89,202)( 90,204)( 91,237)( 92,236)( 93,235)( 94,243)( 95,242)( 96,241)
( 97,240)( 98,239)( 99,238)(100,228)(101,227)(102,226)(103,234)(104,233)
(105,232)(106,231)(107,230)(108,229)(109,264)(110,263)(111,262)(112,270)
(113,269)(114,268)(115,267)(116,266)(117,265)(118,255)(119,254)(120,253)
(121,261)(122,260)(123,259)(124,258)(125,257)(126,256)(127,246)(128,245)
(129,244)(130,252)(131,251)(132,250)(133,249)(134,248)(135,247)(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)(316,329)(317,328)(318,330)
(319,326)(320,325)(321,327)(322,332)(323,331)(324,333)(334,356)(335,355)
(336,357)(337,353)(338,352)(339,354)(340,359)(341,358)(342,360)(343,347)
(344,346)(345,348)(349,350)(361,375)(362,374)(363,373)(364,372)(365,371)
(366,370)(367,378)(368,377)(369,376)(379,402)(380,401)(381,400)(382,399)
(383,398)(384,397)(385,405)(386,404)(387,403)(388,393)(389,392)(390,391)
(394,396);;
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*s2*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*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, 91)( 47, 93)( 48, 92)( 49, 94)( 50, 96)( 51, 95)( 52, 97)( 53, 99)
( 54, 98)( 55,100)( 56,102)( 57,101)( 58,103)( 59,105)( 60,104)( 61,106)
( 62,108)( 63,107)( 64,109)( 65,111)( 66,110)( 67,112)( 68,114)( 69,113)
( 70,115)( 71,117)( 72,116)( 73,118)( 74,120)( 75,119)( 76,121)( 77,123)
( 78,122)( 79,124)( 80,126)( 81,125)( 82,127)( 83,129)( 84,128)( 85,130)
( 86,132)( 87,131)( 88,133)( 89,135)( 90,134)(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)(181,226)(182,228)(183,227)(184,229)
(185,231)(186,230)(187,232)(188,234)(189,233)(190,235)(191,237)(192,236)
(193,238)(194,240)(195,239)(196,241)(197,243)(198,242)(199,244)(200,246)
(201,245)(202,247)(203,249)(204,248)(205,250)(206,252)(207,251)(208,253)
(209,255)(210,254)(211,256)(212,258)(213,257)(214,259)(215,261)(216,260)
(217,262)(218,264)(219,263)(220,265)(221,267)(222,266)(223,268)(224,270)
(225,269)(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)
(316,361)(317,363)(318,362)(319,364)(320,366)(321,365)(322,367)(323,369)
(324,368)(325,370)(326,372)(327,371)(328,373)(329,375)(330,374)(331,376)
(332,378)(333,377)(334,379)(335,381)(336,380)(337,382)(338,384)(339,383)
(340,385)(341,387)(342,386)(343,388)(344,390)(345,389)(346,391)(347,393)
(348,392)(349,394)(350,396)(351,395)(352,397)(353,399)(354,398)(355,400)
(356,402)(357,401)(358,403)(359,405)(360,404);
s1 := Sym(405)!( 1, 46)( 2, 47)( 3, 48)( 4, 52)( 5, 53)( 6, 54)( 7, 49)
( 8, 50)( 9, 51)( 10, 82)( 11, 83)( 12, 84)( 13, 88)( 14, 89)( 15, 90)
( 16, 85)( 17, 86)( 18, 87)( 19, 73)( 20, 74)( 21, 75)( 22, 79)( 23, 80)
( 24, 81)( 25, 76)( 26, 77)( 27, 78)( 28, 64)( 29, 65)( 30, 66)( 31, 70)
( 32, 71)( 33, 72)( 34, 67)( 35, 68)( 36, 69)( 37, 55)( 38, 56)( 39, 57)
( 40, 61)( 41, 62)( 42, 63)( 43, 58)( 44, 59)( 45, 60)( 94, 97)( 95, 98)
( 96, 99)(100,127)(101,128)(102,129)(103,133)(104,134)(105,135)(106,130)
(107,131)(108,132)(109,118)(110,119)(111,120)(112,124)(113,125)(114,126)
(115,121)(116,122)(117,123)(136,319)(137,320)(138,321)(139,316)(140,317)
(141,318)(142,322)(143,323)(144,324)(145,355)(146,356)(147,357)(148,352)
(149,353)(150,354)(151,358)(152,359)(153,360)(154,346)(155,347)(156,348)
(157,343)(158,344)(159,345)(160,349)(161,350)(162,351)(163,337)(164,338)
(165,339)(166,334)(167,335)(168,336)(169,340)(170,341)(171,342)(172,328)
(173,329)(174,330)(175,325)(176,326)(177,327)(178,331)(179,332)(180,333)
(181,274)(182,275)(183,276)(184,271)(185,272)(186,273)(187,277)(188,278)
(189,279)(190,310)(191,311)(192,312)(193,307)(194,308)(195,309)(196,313)
(197,314)(198,315)(199,301)(200,302)(201,303)(202,298)(203,299)(204,300)
(205,304)(206,305)(207,306)(208,292)(209,293)(210,294)(211,289)(212,290)
(213,291)(214,295)(215,296)(216,297)(217,283)(218,284)(219,285)(220,280)
(221,281)(222,282)(223,286)(224,287)(225,288)(226,364)(227,365)(228,366)
(229,361)(230,362)(231,363)(232,367)(233,368)(234,369)(235,400)(236,401)
(237,402)(238,397)(239,398)(240,399)(241,403)(242,404)(243,405)(244,391)
(245,392)(246,393)(247,388)(248,389)(249,390)(250,394)(251,395)(252,396)
(253,382)(254,383)(255,384)(256,379)(257,380)(258,381)(259,385)(260,386)
(261,387)(262,373)(263,374)(264,375)(265,370)(266,371)(267,372)(268,376)
(269,377)(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)( 46,191)( 47,190)
( 48,192)( 49,197)( 50,196)( 51,198)( 52,194)( 53,193)( 54,195)( 55,182)
( 56,181)( 57,183)( 58,188)( 59,187)( 60,189)( 61,185)( 62,184)( 63,186)
( 64,218)( 65,217)( 66,219)( 67,224)( 68,223)( 69,225)( 70,221)( 71,220)
( 72,222)( 73,209)( 74,208)( 75,210)( 76,215)( 77,214)( 78,216)( 79,212)
( 80,211)( 81,213)( 82,200)( 83,199)( 84,201)( 85,206)( 86,205)( 87,207)
( 88,203)( 89,202)( 90,204)( 91,237)( 92,236)( 93,235)( 94,243)( 95,242)
( 96,241)( 97,240)( 98,239)( 99,238)(100,228)(101,227)(102,226)(103,234)
(104,233)(105,232)(106,231)(107,230)(108,229)(109,264)(110,263)(111,262)
(112,270)(113,269)(114,268)(115,267)(116,266)(117,265)(118,255)(119,254)
(120,253)(121,261)(122,260)(123,259)(124,258)(125,257)(126,256)(127,246)
(128,245)(129,244)(130,252)(131,251)(132,250)(133,249)(134,248)(135,247)
(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)(316,329)(317,328)
(318,330)(319,326)(320,325)(321,327)(322,332)(323,331)(324,333)(334,356)
(335,355)(336,357)(337,353)(338,352)(339,354)(340,359)(341,358)(342,360)
(343,347)(344,346)(345,348)(349,350)(361,375)(362,374)(363,373)(364,372)
(365,371)(366,370)(367,378)(368,377)(369,376)(379,402)(380,401)(381,400)
(382,399)(383,398)(384,397)(385,405)(386,404)(387,403)(388,393)(389,392)
(390,391)(394,396);
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*s2*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*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