Polytope of Type {6,135}

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