Polytope of Type {4,164}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,164}*1312
Also Known As : {4,164|2}. if this polytope has another name.
Group : SmallGroup(1312,116)
Rank : 3
Schlafli Type : {4,164}
Number of vertices, edges, etc : 4, 328, 164
Order of s₀s₁s₂ : 164
Order of s₀s₁s₂s₁ : 2
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) :
   2-fold quotients : {2,164}*656, {4,82}*656
   4-fold quotients : {2,82}*328
   8-fold quotients : {2,41}*164
   41-fold quotients : {4,4}*32
   82-fold quotients : {2,4}*16, {4,2}*16
   164-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,329)(  2,330)(  3,331)(  4,332)(  5,333)(  6,334)(  7,335)(  8,336)
(  9,337)( 10,338)( 11,339)( 12,340)( 13,341)( 14,342)( 15,343)( 16,344)
( 17,345)( 18,346)( 19,347)( 20,348)( 21,349)( 22,350)( 23,351)( 24,352)
( 25,353)( 26,354)( 27,355)( 28,356)( 29,357)( 30,358)( 31,359)( 32,360)
( 33,361)( 34,362)( 35,363)( 36,364)( 37,365)( 38,366)( 39,367)( 40,368)
( 41,369)( 42,370)( 43,371)( 44,372)( 45,373)( 46,374)( 47,375)( 48,376)
( 49,377)( 50,378)( 51,379)( 52,380)( 53,381)( 54,382)( 55,383)( 56,384)
( 57,385)( 58,386)( 59,387)( 60,388)( 61,389)( 62,390)( 63,391)( 64,392)
( 65,393)( 66,394)( 67,395)( 68,396)( 69,397)( 70,398)( 71,399)( 72,400)
( 73,401)( 74,402)( 75,403)( 76,404)( 77,405)( 78,406)( 79,407)( 80,408)
( 81,409)( 82,410)( 83,411)( 84,412)( 85,413)( 86,414)( 87,415)( 88,416)
( 89,417)( 90,418)( 91,419)( 92,420)( 93,421)( 94,422)( 95,423)( 96,424)
( 97,425)( 98,426)( 99,427)(100,428)(101,429)(102,430)(103,431)(104,432)
(105,433)(106,434)(107,435)(108,436)(109,437)(110,438)(111,439)(112,440)
(113,441)(114,442)(115,443)(116,444)(117,445)(118,446)(119,447)(120,448)
(121,449)(122,450)(123,451)(124,452)(125,453)(126,454)(127,455)(128,456)
(129,457)(130,458)(131,459)(132,460)(133,461)(134,462)(135,463)(136,464)
(137,465)(138,466)(139,467)(140,468)(141,469)(142,470)(143,471)(144,472)
(145,473)(146,474)(147,475)(148,476)(149,477)(150,478)(151,479)(152,480)
(153,481)(154,482)(155,483)(156,484)(157,485)(158,486)(159,487)(160,488)
(161,489)(162,490)(163,491)(164,492)(165,493)(166,494)(167,495)(168,496)
(169,497)(170,498)(171,499)(172,500)(173,501)(174,502)(175,503)(176,504)
(177,505)(178,506)(179,507)(180,508)(181,509)(182,510)(183,511)(184,512)
(185,513)(186,514)(187,515)(188,516)(189,517)(190,518)(191,519)(192,520)
(193,521)(194,522)(195,523)(196,524)(197,525)(198,526)(199,527)(200,528)
(201,529)(202,530)(203,531)(204,532)(205,533)(206,534)(207,535)(208,536)
(209,537)(210,538)(211,539)(212,540)(213,541)(214,542)(215,543)(216,544)
(217,545)(218,546)(219,547)(220,548)(221,549)(222,550)(223,551)(224,552)
(225,553)(226,554)(227,555)(228,556)(229,557)(230,558)(231,559)(232,560)
(233,561)(234,562)(235,563)(236,564)(237,565)(238,566)(239,567)(240,568)
(241,569)(242,570)(243,571)(244,572)(245,573)(246,574)(247,575)(248,576)
(249,577)(250,578)(251,579)(252,580)(253,581)(254,582)(255,583)(256,584)
(257,585)(258,586)(259,587)(260,588)(261,589)(262,590)(263,591)(264,592)
(265,593)(266,594)(267,595)(268,596)(269,597)(270,598)(271,599)(272,600)
(273,601)(274,602)(275,603)(276,604)(277,605)(278,606)(279,607)(280,608)
(281,609)(282,610)(283,611)(284,612)(285,613)(286,614)(287,615)(288,616)
(289,617)(290,618)(291,619)(292,620)(293,621)(294,622)(295,623)(296,624)
(297,625)(298,626)(299,627)(300,628)(301,629)(302,630)(303,631)(304,632)
(305,633)(306,634)(307,635)(308,636)(309,637)(310,638)(311,639)(312,640)
(313,641)(314,642)(315,643)(316,644)(317,645)(318,646)(319,647)(320,648)
(321,649)(322,650)(323,651)(324,652)(325,653)(326,654)(327,655)(328,656);;
s1 := (  2, 41)(  3, 40)(  4, 39)(  5, 38)(  6, 37)(  7, 36)(  8, 35)(  9, 34)
( 10, 33)( 11, 32)( 12, 31)( 13, 30)( 14, 29)( 15, 28)( 16, 27)( 17, 26)
( 18, 25)( 19, 24)( 20, 23)( 21, 22)( 43, 82)( 44, 81)( 45, 80)( 46, 79)
( 47, 78)( 48, 77)( 49, 76)( 50, 75)( 51, 74)( 52, 73)( 53, 72)( 54, 71)
( 55, 70)( 56, 69)( 57, 68)( 58, 67)( 59, 66)( 60, 65)( 61, 64)( 62, 63)
( 84,123)( 85,122)( 86,121)( 87,120)( 88,119)( 89,118)( 90,117)( 91,116)
( 92,115)( 93,114)( 94,113)( 95,112)( 96,111)( 97,110)( 98,109)( 99,108)
(100,107)(101,106)(102,105)(103,104)(125,164)(126,163)(127,162)(128,161)
(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)(136,153)
(137,152)(138,151)(139,150)(140,149)(141,148)(142,147)(143,146)(144,145)
(165,206)(166,246)(167,245)(168,244)(169,243)(170,242)(171,241)(172,240)
(173,239)(174,238)(175,237)(176,236)(177,235)(178,234)(179,233)(180,232)
(181,231)(182,230)(183,229)(184,228)(185,227)(186,226)(187,225)(188,224)
(189,223)(190,222)(191,221)(192,220)(193,219)(194,218)(195,217)(196,216)
(197,215)(198,214)(199,213)(200,212)(201,211)(202,210)(203,209)(204,208)
(205,207)(247,288)(248,328)(249,327)(250,326)(251,325)(252,324)(253,323)
(254,322)(255,321)(256,320)(257,319)(258,318)(259,317)(260,316)(261,315)
(262,314)(263,313)(264,312)(265,311)(266,310)(267,309)(268,308)(269,307)
(270,306)(271,305)(272,304)(273,303)(274,302)(275,301)(276,300)(277,299)
(278,298)(279,297)(280,296)(281,295)(282,294)(283,293)(284,292)(285,291)
(286,290)(287,289)(329,411)(330,451)(331,450)(332,449)(333,448)(334,447)
(335,446)(336,445)(337,444)(338,443)(339,442)(340,441)(341,440)(342,439)
(343,438)(344,437)(345,436)(346,435)(347,434)(348,433)(349,432)(350,431)
(351,430)(352,429)(353,428)(354,427)(355,426)(356,425)(357,424)(358,423)
(359,422)(360,421)(361,420)(362,419)(363,418)(364,417)(365,416)(366,415)
(367,414)(368,413)(369,412)(370,452)(371,492)(372,491)(373,490)(374,489)
(375,488)(376,487)(377,486)(378,485)(379,484)(380,483)(381,482)(382,481)
(383,480)(384,479)(385,478)(386,477)(387,476)(388,475)(389,474)(390,473)
(391,472)(392,471)(393,470)(394,469)(395,468)(396,467)(397,466)(398,465)
(399,464)(400,463)(401,462)(402,461)(403,460)(404,459)(405,458)(406,457)
(407,456)(408,455)(409,454)(410,453)(493,616)(494,656)(495,655)(496,654)
(497,653)(498,652)(499,651)(500,650)(501,649)(502,648)(503,647)(504,646)
(505,645)(506,644)(507,643)(508,642)(509,641)(510,640)(511,639)(512,638)
(513,637)(514,636)(515,635)(516,634)(517,633)(518,632)(519,631)(520,630)
(521,629)(522,628)(523,627)(524,626)(525,625)(526,624)(527,623)(528,622)
(529,621)(530,620)(531,619)(532,618)(533,617)(534,575)(535,615)(536,614)
(537,613)(538,612)(539,611)(540,610)(541,609)(542,608)(543,607)(544,606)
(545,605)(546,604)(547,603)(548,602)(549,601)(550,600)(551,599)(552,598)
(553,597)(554,596)(555,595)(556,594)(557,593)(558,592)(559,591)(560,590)
(561,589)(562,588)(563,587)(564,586)(565,585)(566,584)(567,583)(568,582)
(569,581)(570,580)(571,579)(572,578)(573,577)(574,576);;
s2 := (  1,166)(  2,165)(  3,205)(  4,204)(  5,203)(  6,202)(  7,201)(  8,200)
(  9,199)( 10,198)( 11,197)( 12,196)( 13,195)( 14,194)( 15,193)( 16,192)
( 17,191)( 18,190)( 19,189)( 20,188)( 21,187)( 22,186)( 23,185)( 24,184)
( 25,183)( 26,182)( 27,181)( 28,180)( 29,179)( 30,178)( 31,177)( 32,176)
( 33,175)( 34,174)( 35,173)( 36,172)( 37,171)( 38,170)( 39,169)( 40,168)
( 41,167)( 42,207)( 43,206)( 44,246)( 45,245)( 46,244)( 47,243)( 48,242)
( 49,241)( 50,240)( 51,239)( 52,238)( 53,237)( 54,236)( 55,235)( 56,234)
( 57,233)( 58,232)( 59,231)( 60,230)( 61,229)( 62,228)( 63,227)( 64,226)
( 65,225)( 66,224)( 67,223)( 68,222)( 69,221)( 70,220)( 71,219)( 72,218)
( 73,217)( 74,216)( 75,215)( 76,214)( 77,213)( 78,212)( 79,211)( 80,210)
( 81,209)( 82,208)( 83,248)( 84,247)( 85,287)( 86,286)( 87,285)( 88,284)
( 89,283)( 90,282)( 91,281)( 92,280)( 93,279)( 94,278)( 95,277)( 96,276)
( 97,275)( 98,274)( 99,273)(100,272)(101,271)(102,270)(103,269)(104,268)
(105,267)(106,266)(107,265)(108,264)(109,263)(110,262)(111,261)(112,260)
(113,259)(114,258)(115,257)(116,256)(117,255)(118,254)(119,253)(120,252)
(121,251)(122,250)(123,249)(124,289)(125,288)(126,328)(127,327)(128,326)
(129,325)(130,324)(131,323)(132,322)(133,321)(134,320)(135,319)(136,318)
(137,317)(138,316)(139,315)(140,314)(141,313)(142,312)(143,311)(144,310)
(145,309)(146,308)(147,307)(148,306)(149,305)(150,304)(151,303)(152,302)
(153,301)(154,300)(155,299)(156,298)(157,297)(158,296)(159,295)(160,294)
(161,293)(162,292)(163,291)(164,290)(329,494)(330,493)(331,533)(332,532)
(333,531)(334,530)(335,529)(336,528)(337,527)(338,526)(339,525)(340,524)
(341,523)(342,522)(343,521)(344,520)(345,519)(346,518)(347,517)(348,516)
(349,515)(350,514)(351,513)(352,512)(353,511)(354,510)(355,509)(356,508)
(357,507)(358,506)(359,505)(360,504)(361,503)(362,502)(363,501)(364,500)
(365,499)(366,498)(367,497)(368,496)(369,495)(370,535)(371,534)(372,574)
(373,573)(374,572)(375,571)(376,570)(377,569)(378,568)(379,567)(380,566)
(381,565)(382,564)(383,563)(384,562)(385,561)(386,560)(387,559)(388,558)
(389,557)(390,556)(391,555)(392,554)(393,553)(394,552)(395,551)(396,550)
(397,549)(398,548)(399,547)(400,546)(401,545)(402,544)(403,543)(404,542)
(405,541)(406,540)(407,539)(408,538)(409,537)(410,536)(411,576)(412,575)
(413,615)(414,614)(415,613)(416,612)(417,611)(418,610)(419,609)(420,608)
(421,607)(422,606)(423,605)(424,604)(425,603)(426,602)(427,601)(428,600)
(429,599)(430,598)(431,597)(432,596)(433,595)(434,594)(435,593)(436,592)
(437,591)(438,590)(439,589)(440,588)(441,587)(442,586)(443,585)(444,584)
(445,583)(446,582)(447,581)(448,580)(449,579)(450,578)(451,577)(452,617)
(453,616)(454,656)(455,655)(456,654)(457,653)(458,652)(459,651)(460,650)
(461,649)(462,648)(463,647)(464,646)(465,645)(466,644)(467,643)(468,642)
(469,641)(470,640)(471,639)(472,638)(473,637)(474,636)(475,635)(476,634)
(477,633)(478,632)(479,631)(480,630)(481,629)(482,628)(483,627)(484,626)
(485,625)(486,624)(487,623)(488,622)(489,621)(490,620)(491,619)(492,618);;
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*s2*s1*s0*s1*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(656)!(  1,329)(  2,330)(  3,331)(  4,332)(  5,333)(  6,334)(  7,335)
(  8,336)(  9,337)( 10,338)( 11,339)( 12,340)( 13,341)( 14,342)( 15,343)
( 16,344)( 17,345)( 18,346)( 19,347)( 20,348)( 21,349)( 22,350)( 23,351)
( 24,352)( 25,353)( 26,354)( 27,355)( 28,356)( 29,357)( 30,358)( 31,359)
( 32,360)( 33,361)( 34,362)( 35,363)( 36,364)( 37,365)( 38,366)( 39,367)
( 40,368)( 41,369)( 42,370)( 43,371)( 44,372)( 45,373)( 46,374)( 47,375)
( 48,376)( 49,377)( 50,378)( 51,379)( 52,380)( 53,381)( 54,382)( 55,383)
( 56,384)( 57,385)( 58,386)( 59,387)( 60,388)( 61,389)( 62,390)( 63,391)
( 64,392)( 65,393)( 66,394)( 67,395)( 68,396)( 69,397)( 70,398)( 71,399)
( 72,400)( 73,401)( 74,402)( 75,403)( 76,404)( 77,405)( 78,406)( 79,407)
( 80,408)( 81,409)( 82,410)( 83,411)( 84,412)( 85,413)( 86,414)( 87,415)
( 88,416)( 89,417)( 90,418)( 91,419)( 92,420)( 93,421)( 94,422)( 95,423)
( 96,424)( 97,425)( 98,426)( 99,427)(100,428)(101,429)(102,430)(103,431)
(104,432)(105,433)(106,434)(107,435)(108,436)(109,437)(110,438)(111,439)
(112,440)(113,441)(114,442)(115,443)(116,444)(117,445)(118,446)(119,447)
(120,448)(121,449)(122,450)(123,451)(124,452)(125,453)(126,454)(127,455)
(128,456)(129,457)(130,458)(131,459)(132,460)(133,461)(134,462)(135,463)
(136,464)(137,465)(138,466)(139,467)(140,468)(141,469)(142,470)(143,471)
(144,472)(145,473)(146,474)(147,475)(148,476)(149,477)(150,478)(151,479)
(152,480)(153,481)(154,482)(155,483)(156,484)(157,485)(158,486)(159,487)
(160,488)(161,489)(162,490)(163,491)(164,492)(165,493)(166,494)(167,495)
(168,496)(169,497)(170,498)(171,499)(172,500)(173,501)(174,502)(175,503)
(176,504)(177,505)(178,506)(179,507)(180,508)(181,509)(182,510)(183,511)
(184,512)(185,513)(186,514)(187,515)(188,516)(189,517)(190,518)(191,519)
(192,520)(193,521)(194,522)(195,523)(196,524)(197,525)(198,526)(199,527)
(200,528)(201,529)(202,530)(203,531)(204,532)(205,533)(206,534)(207,535)
(208,536)(209,537)(210,538)(211,539)(212,540)(213,541)(214,542)(215,543)
(216,544)(217,545)(218,546)(219,547)(220,548)(221,549)(222,550)(223,551)
(224,552)(225,553)(226,554)(227,555)(228,556)(229,557)(230,558)(231,559)
(232,560)(233,561)(234,562)(235,563)(236,564)(237,565)(238,566)(239,567)
(240,568)(241,569)(242,570)(243,571)(244,572)(245,573)(246,574)(247,575)
(248,576)(249,577)(250,578)(251,579)(252,580)(253,581)(254,582)(255,583)
(256,584)(257,585)(258,586)(259,587)(260,588)(261,589)(262,590)(263,591)
(264,592)(265,593)(266,594)(267,595)(268,596)(269,597)(270,598)(271,599)
(272,600)(273,601)(274,602)(275,603)(276,604)(277,605)(278,606)(279,607)
(280,608)(281,609)(282,610)(283,611)(284,612)(285,613)(286,614)(287,615)
(288,616)(289,617)(290,618)(291,619)(292,620)(293,621)(294,622)(295,623)
(296,624)(297,625)(298,626)(299,627)(300,628)(301,629)(302,630)(303,631)
(304,632)(305,633)(306,634)(307,635)(308,636)(309,637)(310,638)(311,639)
(312,640)(313,641)(314,642)(315,643)(316,644)(317,645)(318,646)(319,647)
(320,648)(321,649)(322,650)(323,651)(324,652)(325,653)(326,654)(327,655)
(328,656);
s1 := Sym(656)!(  2, 41)(  3, 40)(  4, 39)(  5, 38)(  6, 37)(  7, 36)(  8, 35)
(  9, 34)( 10, 33)( 11, 32)( 12, 31)( 13, 30)( 14, 29)( 15, 28)( 16, 27)
( 17, 26)( 18, 25)( 19, 24)( 20, 23)( 21, 22)( 43, 82)( 44, 81)( 45, 80)
( 46, 79)( 47, 78)( 48, 77)( 49, 76)( 50, 75)( 51, 74)( 52, 73)( 53, 72)
( 54, 71)( 55, 70)( 56, 69)( 57, 68)( 58, 67)( 59, 66)( 60, 65)( 61, 64)
( 62, 63)( 84,123)( 85,122)( 86,121)( 87,120)( 88,119)( 89,118)( 90,117)
( 91,116)( 92,115)( 93,114)( 94,113)( 95,112)( 96,111)( 97,110)( 98,109)
( 99,108)(100,107)(101,106)(102,105)(103,104)(125,164)(126,163)(127,162)
(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)
(136,153)(137,152)(138,151)(139,150)(140,149)(141,148)(142,147)(143,146)
(144,145)(165,206)(166,246)(167,245)(168,244)(169,243)(170,242)(171,241)
(172,240)(173,239)(174,238)(175,237)(176,236)(177,235)(178,234)(179,233)
(180,232)(181,231)(182,230)(183,229)(184,228)(185,227)(186,226)(187,225)
(188,224)(189,223)(190,222)(191,221)(192,220)(193,219)(194,218)(195,217)
(196,216)(197,215)(198,214)(199,213)(200,212)(201,211)(202,210)(203,209)
(204,208)(205,207)(247,288)(248,328)(249,327)(250,326)(251,325)(252,324)
(253,323)(254,322)(255,321)(256,320)(257,319)(258,318)(259,317)(260,316)
(261,315)(262,314)(263,313)(264,312)(265,311)(266,310)(267,309)(268,308)
(269,307)(270,306)(271,305)(272,304)(273,303)(274,302)(275,301)(276,300)
(277,299)(278,298)(279,297)(280,296)(281,295)(282,294)(283,293)(284,292)
(285,291)(286,290)(287,289)(329,411)(330,451)(331,450)(332,449)(333,448)
(334,447)(335,446)(336,445)(337,444)(338,443)(339,442)(340,441)(341,440)
(342,439)(343,438)(344,437)(345,436)(346,435)(347,434)(348,433)(349,432)
(350,431)(351,430)(352,429)(353,428)(354,427)(355,426)(356,425)(357,424)
(358,423)(359,422)(360,421)(361,420)(362,419)(363,418)(364,417)(365,416)
(366,415)(367,414)(368,413)(369,412)(370,452)(371,492)(372,491)(373,490)
(374,489)(375,488)(376,487)(377,486)(378,485)(379,484)(380,483)(381,482)
(382,481)(383,480)(384,479)(385,478)(386,477)(387,476)(388,475)(389,474)
(390,473)(391,472)(392,471)(393,470)(394,469)(395,468)(396,467)(397,466)
(398,465)(399,464)(400,463)(401,462)(402,461)(403,460)(404,459)(405,458)
(406,457)(407,456)(408,455)(409,454)(410,453)(493,616)(494,656)(495,655)
(496,654)(497,653)(498,652)(499,651)(500,650)(501,649)(502,648)(503,647)
(504,646)(505,645)(506,644)(507,643)(508,642)(509,641)(510,640)(511,639)
(512,638)(513,637)(514,636)(515,635)(516,634)(517,633)(518,632)(519,631)
(520,630)(521,629)(522,628)(523,627)(524,626)(525,625)(526,624)(527,623)
(528,622)(529,621)(530,620)(531,619)(532,618)(533,617)(534,575)(535,615)
(536,614)(537,613)(538,612)(539,611)(540,610)(541,609)(542,608)(543,607)
(544,606)(545,605)(546,604)(547,603)(548,602)(549,601)(550,600)(551,599)
(552,598)(553,597)(554,596)(555,595)(556,594)(557,593)(558,592)(559,591)
(560,590)(561,589)(562,588)(563,587)(564,586)(565,585)(566,584)(567,583)
(568,582)(569,581)(570,580)(571,579)(572,578)(573,577)(574,576);
s2 := Sym(656)!(  1,166)(  2,165)(  3,205)(  4,204)(  5,203)(  6,202)(  7,201)
(  8,200)(  9,199)( 10,198)( 11,197)( 12,196)( 13,195)( 14,194)( 15,193)
( 16,192)( 17,191)( 18,190)( 19,189)( 20,188)( 21,187)( 22,186)( 23,185)
( 24,184)( 25,183)( 26,182)( 27,181)( 28,180)( 29,179)( 30,178)( 31,177)
( 32,176)( 33,175)( 34,174)( 35,173)( 36,172)( 37,171)( 38,170)( 39,169)
( 40,168)( 41,167)( 42,207)( 43,206)( 44,246)( 45,245)( 46,244)( 47,243)
( 48,242)( 49,241)( 50,240)( 51,239)( 52,238)( 53,237)( 54,236)( 55,235)
( 56,234)( 57,233)( 58,232)( 59,231)( 60,230)( 61,229)( 62,228)( 63,227)
( 64,226)( 65,225)( 66,224)( 67,223)( 68,222)( 69,221)( 70,220)( 71,219)
( 72,218)( 73,217)( 74,216)( 75,215)( 76,214)( 77,213)( 78,212)( 79,211)
( 80,210)( 81,209)( 82,208)( 83,248)( 84,247)( 85,287)( 86,286)( 87,285)
( 88,284)( 89,283)( 90,282)( 91,281)( 92,280)( 93,279)( 94,278)( 95,277)
( 96,276)( 97,275)( 98,274)( 99,273)(100,272)(101,271)(102,270)(103,269)
(104,268)(105,267)(106,266)(107,265)(108,264)(109,263)(110,262)(111,261)
(112,260)(113,259)(114,258)(115,257)(116,256)(117,255)(118,254)(119,253)
(120,252)(121,251)(122,250)(123,249)(124,289)(125,288)(126,328)(127,327)
(128,326)(129,325)(130,324)(131,323)(132,322)(133,321)(134,320)(135,319)
(136,318)(137,317)(138,316)(139,315)(140,314)(141,313)(142,312)(143,311)
(144,310)(145,309)(146,308)(147,307)(148,306)(149,305)(150,304)(151,303)
(152,302)(153,301)(154,300)(155,299)(156,298)(157,297)(158,296)(159,295)
(160,294)(161,293)(162,292)(163,291)(164,290)(329,494)(330,493)(331,533)
(332,532)(333,531)(334,530)(335,529)(336,528)(337,527)(338,526)(339,525)
(340,524)(341,523)(342,522)(343,521)(344,520)(345,519)(346,518)(347,517)
(348,516)(349,515)(350,514)(351,513)(352,512)(353,511)(354,510)(355,509)
(356,508)(357,507)(358,506)(359,505)(360,504)(361,503)(362,502)(363,501)
(364,500)(365,499)(366,498)(367,497)(368,496)(369,495)(370,535)(371,534)
(372,574)(373,573)(374,572)(375,571)(376,570)(377,569)(378,568)(379,567)
(380,566)(381,565)(382,564)(383,563)(384,562)(385,561)(386,560)(387,559)
(388,558)(389,557)(390,556)(391,555)(392,554)(393,553)(394,552)(395,551)
(396,550)(397,549)(398,548)(399,547)(400,546)(401,545)(402,544)(403,543)
(404,542)(405,541)(406,540)(407,539)(408,538)(409,537)(410,536)(411,576)
(412,575)(413,615)(414,614)(415,613)(416,612)(417,611)(418,610)(419,609)
(420,608)(421,607)(422,606)(423,605)(424,604)(425,603)(426,602)(427,601)
(428,600)(429,599)(430,598)(431,597)(432,596)(433,595)(434,594)(435,593)
(436,592)(437,591)(438,590)(439,589)(440,588)(441,587)(442,586)(443,585)
(444,584)(445,583)(446,582)(447,581)(448,580)(449,579)(450,578)(451,577)
(452,617)(453,616)(454,656)(455,655)(456,654)(457,653)(458,652)(459,651)
(460,650)(461,649)(462,648)(463,647)(464,646)(465,645)(466,644)(467,643)
(468,642)(469,641)(470,640)(471,639)(472,638)(473,637)(474,636)(475,635)
(476,634)(477,633)(478,632)(479,631)(480,630)(481,629)(482,628)(483,627)
(484,626)(485,625)(486,624)(487,623)(488,622)(489,621)(490,620)(491,619)
(492,618);
poly := sub<Sym(656)|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*s2*s1*s0*s1*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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