Polytope of Type {135,6}

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