Polytope of Type {165,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {165,6}*1980
if this polytope has a name.
Group : SmallGroup(1980,103)
Rank : 3
Schlafli Type : {165,6}
Number of vertices, edges, etc : 165, 495, 6
Order of s₀s₁s₂ : 330
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 : {165,2}*660
   5-fold quotients : {33,6}*396
   9-fold quotients : {55,2}*220
   11-fold quotients : {15,6}*180
   15-fold quotients : {33,2}*132
   33-fold quotients : {15,2}*60
   45-fold quotients : {11,2}*44
   55-fold quotients : {3,6}*36
   99-fold quotients : {5,2}*20
   165-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2, 11)(  3, 10)(  4,  9)(  5,  8)(  6,  7)( 12, 45)( 13, 55)( 14, 54)
( 15, 53)( 16, 52)( 17, 51)( 18, 50)( 19, 49)( 20, 48)( 21, 47)( 22, 46)
( 23, 34)( 24, 44)( 25, 43)( 26, 42)( 27, 41)( 28, 40)( 29, 39)( 30, 38)
( 31, 37)( 32, 36)( 33, 35)( 56,111)( 57,121)( 58,120)( 59,119)( 60,118)
( 61,117)( 62,116)( 63,115)( 64,114)( 65,113)( 66,112)( 67,155)( 68,165)
( 69,164)( 70,163)( 71,162)( 72,161)( 73,160)( 74,159)( 75,158)( 76,157)
( 77,156)( 78,144)( 79,154)( 80,153)( 81,152)( 82,151)( 83,150)( 84,149)
( 85,148)( 86,147)( 87,146)( 88,145)( 89,133)( 90,143)( 91,142)( 92,141)
( 93,140)( 94,139)( 95,138)( 96,137)( 97,136)( 98,135)( 99,134)(100,122)
(101,132)(102,131)(103,130)(104,129)(105,128)(106,127)(107,126)(108,125)
(109,124)(110,123)(166,331)(167,341)(168,340)(169,339)(170,338)(171,337)
(172,336)(173,335)(174,334)(175,333)(176,332)(177,375)(178,385)(179,384)
(180,383)(181,382)(182,381)(183,380)(184,379)(185,378)(186,377)(187,376)
(188,364)(189,374)(190,373)(191,372)(192,371)(193,370)(194,369)(195,368)
(196,367)(197,366)(198,365)(199,353)(200,363)(201,362)(202,361)(203,360)
(204,359)(205,358)(206,357)(207,356)(208,355)(209,354)(210,342)(211,352)
(212,351)(213,350)(214,349)(215,348)(216,347)(217,346)(218,345)(219,344)
(220,343)(221,441)(222,451)(223,450)(224,449)(225,448)(226,447)(227,446)
(228,445)(229,444)(230,443)(231,442)(232,485)(233,495)(234,494)(235,493)
(236,492)(237,491)(238,490)(239,489)(240,488)(241,487)(242,486)(243,474)
(244,484)(245,483)(246,482)(247,481)(248,480)(249,479)(250,478)(251,477)
(252,476)(253,475)(254,463)(255,473)(256,472)(257,471)(258,470)(259,469)
(260,468)(261,467)(262,466)(263,465)(264,464)(265,452)(266,462)(267,461)
(268,460)(269,459)(270,458)(271,457)(272,456)(273,455)(274,454)(275,453)
(276,386)(277,396)(278,395)(279,394)(280,393)(281,392)(282,391)(283,390)
(284,389)(285,388)(286,387)(287,430)(288,440)(289,439)(290,438)(291,437)
(292,436)(293,435)(294,434)(295,433)(296,432)(297,431)(298,419)(299,429)
(300,428)(301,427)(302,426)(303,425)(304,424)(305,423)(306,422)(307,421)
(308,420)(309,408)(310,418)(311,417)(312,416)(313,415)(314,414)(315,413)
(316,412)(317,411)(318,410)(319,409)(320,397)(321,407)(322,406)(323,405)
(324,404)(325,403)(326,402)(327,401)(328,400)(329,399)(330,398);;
s1 := (  1,233)(  2,232)(  3,242)(  4,241)(  5,240)(  6,239)(  7,238)(  8,237)
(  9,236)( 10,235)( 11,234)( 12,222)( 13,221)( 14,231)( 15,230)( 16,229)
( 17,228)( 18,227)( 19,226)( 20,225)( 21,224)( 22,223)( 23,266)( 24,265)
( 25,275)( 26,274)( 27,273)( 28,272)( 29,271)( 30,270)( 31,269)( 32,268)
( 33,267)( 34,255)( 35,254)( 36,264)( 37,263)( 38,262)( 39,261)( 40,260)
( 41,259)( 42,258)( 43,257)( 44,256)( 45,244)( 46,243)( 47,253)( 48,252)
( 49,251)( 50,250)( 51,249)( 52,248)( 53,247)( 54,246)( 55,245)( 56,178)
( 57,177)( 58,187)( 59,186)( 60,185)( 61,184)( 62,183)( 63,182)( 64,181)
( 65,180)( 66,179)( 67,167)( 68,166)( 69,176)( 70,175)( 71,174)( 72,173)
( 73,172)( 74,171)( 75,170)( 76,169)( 77,168)( 78,211)( 79,210)( 80,220)
( 81,219)( 82,218)( 83,217)( 84,216)( 85,215)( 86,214)( 87,213)( 88,212)
( 89,200)( 90,199)( 91,209)( 92,208)( 93,207)( 94,206)( 95,205)( 96,204)
( 97,203)( 98,202)( 99,201)(100,189)(101,188)(102,198)(103,197)(104,196)
(105,195)(106,194)(107,193)(108,192)(109,191)(110,190)(111,288)(112,287)
(113,297)(114,296)(115,295)(116,294)(117,293)(118,292)(119,291)(120,290)
(121,289)(122,277)(123,276)(124,286)(125,285)(126,284)(127,283)(128,282)
(129,281)(130,280)(131,279)(132,278)(133,321)(134,320)(135,330)(136,329)
(137,328)(138,327)(139,326)(140,325)(141,324)(142,323)(143,322)(144,310)
(145,309)(146,319)(147,318)(148,317)(149,316)(150,315)(151,314)(152,313)
(153,312)(154,311)(155,299)(156,298)(157,308)(158,307)(159,306)(160,305)
(161,304)(162,303)(163,302)(164,301)(165,300)(331,398)(332,397)(333,407)
(334,406)(335,405)(336,404)(337,403)(338,402)(339,401)(340,400)(341,399)
(342,387)(343,386)(344,396)(345,395)(346,394)(347,393)(348,392)(349,391)
(350,390)(351,389)(352,388)(353,431)(354,430)(355,440)(356,439)(357,438)
(358,437)(359,436)(360,435)(361,434)(362,433)(363,432)(364,420)(365,419)
(366,429)(367,428)(368,427)(369,426)(370,425)(371,424)(372,423)(373,422)
(374,421)(375,409)(376,408)(377,418)(378,417)(379,416)(380,415)(381,414)
(382,413)(383,412)(384,411)(385,410)(441,453)(442,452)(443,462)(444,461)
(445,460)(446,459)(447,458)(448,457)(449,456)(450,455)(451,454)(463,486)
(464,485)(465,495)(466,494)(467,493)(468,492)(469,491)(470,490)(471,489)
(472,488)(473,487)(474,475)(476,484)(477,483)(478,482)(479,481);;
s2 := ( 56,111)( 57,112)( 58,113)( 59,114)( 60,115)( 61,116)( 62,117)( 63,118)
( 64,119)( 65,120)( 66,121)( 67,122)( 68,123)( 69,124)( 70,125)( 71,126)
( 72,127)( 73,128)( 74,129)( 75,130)( 76,131)( 77,132)( 78,133)( 79,134)
( 80,135)( 81,136)( 82,137)( 83,138)( 84,139)( 85,140)( 86,141)( 87,142)
( 88,143)( 89,144)( 90,145)( 91,146)( 92,147)( 93,148)( 94,149)( 95,150)
( 96,151)( 97,152)( 98,153)( 99,154)(100,155)(101,156)(102,157)(103,158)
(104,159)(105,160)(106,161)(107,162)(108,163)(109,164)(110,165)(221,276)
(222,277)(223,278)(224,279)(225,280)(226,281)(227,282)(228,283)(229,284)
(230,285)(231,286)(232,287)(233,288)(234,289)(235,290)(236,291)(237,292)
(238,293)(239,294)(240,295)(241,296)(242,297)(243,298)(244,299)(245,300)
(246,301)(247,302)(248,303)(249,304)(250,305)(251,306)(252,307)(253,308)
(254,309)(255,310)(256,311)(257,312)(258,313)(259,314)(260,315)(261,316)
(262,317)(263,318)(264,319)(265,320)(266,321)(267,322)(268,323)(269,324)
(270,325)(271,326)(272,327)(273,328)(274,329)(275,330)(386,441)(387,442)
(388,443)(389,444)(390,445)(391,446)(392,447)(393,448)(394,449)(395,450)
(396,451)(397,452)(398,453)(399,454)(400,455)(401,456)(402,457)(403,458)
(404,459)(405,460)(406,461)(407,462)(408,463)(409,464)(410,465)(411,466)
(412,467)(413,468)(414,469)(415,470)(416,471)(417,472)(418,473)(419,474)
(420,475)(421,476)(422,477)(423,478)(424,479)(425,480)(426,481)(427,482)
(428,483)(429,484)(430,485)(431,486)(432,487)(433,488)(434,489)(435,490)
(436,491)(437,492)(438,493)(439,494)(440,495);;
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*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(495)!(  2, 11)(  3, 10)(  4,  9)(  5,  8)(  6,  7)( 12, 45)( 13, 55)
( 14, 54)( 15, 53)( 16, 52)( 17, 51)( 18, 50)( 19, 49)( 20, 48)( 21, 47)
( 22, 46)( 23, 34)( 24, 44)( 25, 43)( 26, 42)( 27, 41)( 28, 40)( 29, 39)
( 30, 38)( 31, 37)( 32, 36)( 33, 35)( 56,111)( 57,121)( 58,120)( 59,119)
( 60,118)( 61,117)( 62,116)( 63,115)( 64,114)( 65,113)( 66,112)( 67,155)
( 68,165)( 69,164)( 70,163)( 71,162)( 72,161)( 73,160)( 74,159)( 75,158)
( 76,157)( 77,156)( 78,144)( 79,154)( 80,153)( 81,152)( 82,151)( 83,150)
( 84,149)( 85,148)( 86,147)( 87,146)( 88,145)( 89,133)( 90,143)( 91,142)
( 92,141)( 93,140)( 94,139)( 95,138)( 96,137)( 97,136)( 98,135)( 99,134)
(100,122)(101,132)(102,131)(103,130)(104,129)(105,128)(106,127)(107,126)
(108,125)(109,124)(110,123)(166,331)(167,341)(168,340)(169,339)(170,338)
(171,337)(172,336)(173,335)(174,334)(175,333)(176,332)(177,375)(178,385)
(179,384)(180,383)(181,382)(182,381)(183,380)(184,379)(185,378)(186,377)
(187,376)(188,364)(189,374)(190,373)(191,372)(192,371)(193,370)(194,369)
(195,368)(196,367)(197,366)(198,365)(199,353)(200,363)(201,362)(202,361)
(203,360)(204,359)(205,358)(206,357)(207,356)(208,355)(209,354)(210,342)
(211,352)(212,351)(213,350)(214,349)(215,348)(216,347)(217,346)(218,345)
(219,344)(220,343)(221,441)(222,451)(223,450)(224,449)(225,448)(226,447)
(227,446)(228,445)(229,444)(230,443)(231,442)(232,485)(233,495)(234,494)
(235,493)(236,492)(237,491)(238,490)(239,489)(240,488)(241,487)(242,486)
(243,474)(244,484)(245,483)(246,482)(247,481)(248,480)(249,479)(250,478)
(251,477)(252,476)(253,475)(254,463)(255,473)(256,472)(257,471)(258,470)
(259,469)(260,468)(261,467)(262,466)(263,465)(264,464)(265,452)(266,462)
(267,461)(268,460)(269,459)(270,458)(271,457)(272,456)(273,455)(274,454)
(275,453)(276,386)(277,396)(278,395)(279,394)(280,393)(281,392)(282,391)
(283,390)(284,389)(285,388)(286,387)(287,430)(288,440)(289,439)(290,438)
(291,437)(292,436)(293,435)(294,434)(295,433)(296,432)(297,431)(298,419)
(299,429)(300,428)(301,427)(302,426)(303,425)(304,424)(305,423)(306,422)
(307,421)(308,420)(309,408)(310,418)(311,417)(312,416)(313,415)(314,414)
(315,413)(316,412)(317,411)(318,410)(319,409)(320,397)(321,407)(322,406)
(323,405)(324,404)(325,403)(326,402)(327,401)(328,400)(329,399)(330,398);
s1 := Sym(495)!(  1,233)(  2,232)(  3,242)(  4,241)(  5,240)(  6,239)(  7,238)
(  8,237)(  9,236)( 10,235)( 11,234)( 12,222)( 13,221)( 14,231)( 15,230)
( 16,229)( 17,228)( 18,227)( 19,226)( 20,225)( 21,224)( 22,223)( 23,266)
( 24,265)( 25,275)( 26,274)( 27,273)( 28,272)( 29,271)( 30,270)( 31,269)
( 32,268)( 33,267)( 34,255)( 35,254)( 36,264)( 37,263)( 38,262)( 39,261)
( 40,260)( 41,259)( 42,258)( 43,257)( 44,256)( 45,244)( 46,243)( 47,253)
( 48,252)( 49,251)( 50,250)( 51,249)( 52,248)( 53,247)( 54,246)( 55,245)
( 56,178)( 57,177)( 58,187)( 59,186)( 60,185)( 61,184)( 62,183)( 63,182)
( 64,181)( 65,180)( 66,179)( 67,167)( 68,166)( 69,176)( 70,175)( 71,174)
( 72,173)( 73,172)( 74,171)( 75,170)( 76,169)( 77,168)( 78,211)( 79,210)
( 80,220)( 81,219)( 82,218)( 83,217)( 84,216)( 85,215)( 86,214)( 87,213)
( 88,212)( 89,200)( 90,199)( 91,209)( 92,208)( 93,207)( 94,206)( 95,205)
( 96,204)( 97,203)( 98,202)( 99,201)(100,189)(101,188)(102,198)(103,197)
(104,196)(105,195)(106,194)(107,193)(108,192)(109,191)(110,190)(111,288)
(112,287)(113,297)(114,296)(115,295)(116,294)(117,293)(118,292)(119,291)
(120,290)(121,289)(122,277)(123,276)(124,286)(125,285)(126,284)(127,283)
(128,282)(129,281)(130,280)(131,279)(132,278)(133,321)(134,320)(135,330)
(136,329)(137,328)(138,327)(139,326)(140,325)(141,324)(142,323)(143,322)
(144,310)(145,309)(146,319)(147,318)(148,317)(149,316)(150,315)(151,314)
(152,313)(153,312)(154,311)(155,299)(156,298)(157,308)(158,307)(159,306)
(160,305)(161,304)(162,303)(163,302)(164,301)(165,300)(331,398)(332,397)
(333,407)(334,406)(335,405)(336,404)(337,403)(338,402)(339,401)(340,400)
(341,399)(342,387)(343,386)(344,396)(345,395)(346,394)(347,393)(348,392)
(349,391)(350,390)(351,389)(352,388)(353,431)(354,430)(355,440)(356,439)
(357,438)(358,437)(359,436)(360,435)(361,434)(362,433)(363,432)(364,420)
(365,419)(366,429)(367,428)(368,427)(369,426)(370,425)(371,424)(372,423)
(373,422)(374,421)(375,409)(376,408)(377,418)(378,417)(379,416)(380,415)
(381,414)(382,413)(383,412)(384,411)(385,410)(441,453)(442,452)(443,462)
(444,461)(445,460)(446,459)(447,458)(448,457)(449,456)(450,455)(451,454)
(463,486)(464,485)(465,495)(466,494)(467,493)(468,492)(469,491)(470,490)
(471,489)(472,488)(473,487)(474,475)(476,484)(477,483)(478,482)(479,481);
s2 := Sym(495)!( 56,111)( 57,112)( 58,113)( 59,114)( 60,115)( 61,116)( 62,117)
( 63,118)( 64,119)( 65,120)( 66,121)( 67,122)( 68,123)( 69,124)( 70,125)
( 71,126)( 72,127)( 73,128)( 74,129)( 75,130)( 76,131)( 77,132)( 78,133)
( 79,134)( 80,135)( 81,136)( 82,137)( 83,138)( 84,139)( 85,140)( 86,141)
( 87,142)( 88,143)( 89,144)( 90,145)( 91,146)( 92,147)( 93,148)( 94,149)
( 95,150)( 96,151)( 97,152)( 98,153)( 99,154)(100,155)(101,156)(102,157)
(103,158)(104,159)(105,160)(106,161)(107,162)(108,163)(109,164)(110,165)
(221,276)(222,277)(223,278)(224,279)(225,280)(226,281)(227,282)(228,283)
(229,284)(230,285)(231,286)(232,287)(233,288)(234,289)(235,290)(236,291)
(237,292)(238,293)(239,294)(240,295)(241,296)(242,297)(243,298)(244,299)
(245,300)(246,301)(247,302)(248,303)(249,304)(250,305)(251,306)(252,307)
(253,308)(254,309)(255,310)(256,311)(257,312)(258,313)(259,314)(260,315)
(261,316)(262,317)(263,318)(264,319)(265,320)(266,321)(267,322)(268,323)
(269,324)(270,325)(271,326)(272,327)(273,328)(274,329)(275,330)(386,441)
(387,442)(388,443)(389,444)(390,445)(391,446)(392,447)(393,448)(394,449)
(395,450)(396,451)(397,452)(398,453)(399,454)(400,455)(401,456)(402,457)
(403,458)(404,459)(405,460)(406,461)(407,462)(408,463)(409,464)(410,465)
(411,466)(412,467)(413,468)(414,469)(415,470)(416,471)(417,472)(418,473)
(419,474)(420,475)(421,476)(422,477)(423,478)(424,479)(425,480)(426,481)
(427,482)(428,483)(429,484)(430,485)(431,486)(432,487)(433,488)(434,489)
(435,490)(436,491)(437,492)(438,493)(439,494)(440,495);
poly := sub<Sym(495)|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*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