Polytope of Type {204,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {204,4}*1632b
if this polytope has a name.
Group : SmallGroup(1632,1027)
Rank : 3
Schlafli Type : {204,4}
Number of vertices, edges, etc : 204, 408, 4
Order of s₀s₁s₂ : 204
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {102,4}*816b
   4-fold quotients : {51,4}*408
   17-fold quotients : {12,4}*96b
   34-fold quotients : {6,4}*48c
   68-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  3,  4)(  5, 65)(  6, 66)(  7, 68)(  8, 67)(  9, 61)( 10, 62)( 11, 64)
( 12, 63)( 13, 57)( 14, 58)( 15, 60)( 16, 59)( 17, 53)( 18, 54)( 19, 56)
( 20, 55)( 21, 49)( 22, 50)( 23, 52)( 24, 51)( 25, 45)( 26, 46)( 27, 48)
( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 33, 37)( 34, 38)( 35, 40)
( 36, 39)( 69,137)( 70,138)( 71,140)( 72,139)( 73,201)( 74,202)( 75,204)
( 76,203)( 77,197)( 78,198)( 79,200)( 80,199)( 81,193)( 82,194)( 83,196)
( 84,195)( 85,189)( 86,190)( 87,192)( 88,191)( 89,185)( 90,186)( 91,188)
( 92,187)( 93,181)( 94,182)( 95,184)( 96,183)( 97,177)( 98,178)( 99,180)
(100,179)(101,173)(102,174)(103,176)(104,175)(105,169)(106,170)(107,172)
(108,171)(109,165)(110,166)(111,168)(112,167)(113,161)(114,162)(115,164)
(116,163)(117,157)(118,158)(119,160)(120,159)(121,153)(122,154)(123,156)
(124,155)(125,149)(126,150)(127,152)(128,151)(129,145)(130,146)(131,148)
(132,147)(133,141)(134,142)(135,144)(136,143)(207,208)(209,269)(210,270)
(211,272)(212,271)(213,265)(214,266)(215,268)(216,267)(217,261)(218,262)
(219,264)(220,263)(221,257)(222,258)(223,260)(224,259)(225,253)(226,254)
(227,256)(228,255)(229,249)(230,250)(231,252)(232,251)(233,245)(234,246)
(235,248)(236,247)(237,241)(238,242)(239,244)(240,243)(273,341)(274,342)
(275,344)(276,343)(277,405)(278,406)(279,408)(280,407)(281,401)(282,402)
(283,404)(284,403)(285,397)(286,398)(287,400)(288,399)(289,393)(290,394)
(291,396)(292,395)(293,389)(294,390)(295,392)(296,391)(297,385)(298,386)
(299,388)(300,387)(301,381)(302,382)(303,384)(304,383)(305,377)(306,378)
(307,380)(308,379)(309,373)(310,374)(311,376)(312,375)(313,369)(314,370)
(315,372)(316,371)(317,365)(318,366)(319,368)(320,367)(321,361)(322,362)
(323,364)(324,363)(325,357)(326,358)(327,360)(328,359)(329,353)(330,354)
(331,356)(332,355)(333,349)(334,350)(335,352)(336,351)(337,345)(338,346)
(339,348)(340,347)(409,613)(410,614)(411,616)(412,615)(413,677)(414,678)
(415,680)(416,679)(417,673)(418,674)(419,676)(420,675)(421,669)(422,670)
(423,672)(424,671)(425,665)(426,666)(427,668)(428,667)(429,661)(430,662)
(431,664)(432,663)(433,657)(434,658)(435,660)(436,659)(437,653)(438,654)
(439,656)(440,655)(441,649)(442,650)(443,652)(444,651)(445,645)(446,646)
(447,648)(448,647)(449,641)(450,642)(451,644)(452,643)(453,637)(454,638)
(455,640)(456,639)(457,633)(458,634)(459,636)(460,635)(461,629)(462,630)
(463,632)(464,631)(465,625)(466,626)(467,628)(468,627)(469,621)(470,622)
(471,624)(472,623)(473,617)(474,618)(475,620)(476,619)(477,749)(478,750)
(479,752)(480,751)(481,813)(482,814)(483,816)(484,815)(485,809)(486,810)
(487,812)(488,811)(489,805)(490,806)(491,808)(492,807)(493,801)(494,802)
(495,804)(496,803)(497,797)(498,798)(499,800)(500,799)(501,793)(502,794)
(503,796)(504,795)(505,789)(506,790)(507,792)(508,791)(509,785)(510,786)
(511,788)(512,787)(513,781)(514,782)(515,784)(516,783)(517,777)(518,778)
(519,780)(520,779)(521,773)(522,774)(523,776)(524,775)(525,769)(526,770)
(527,772)(528,771)(529,765)(530,766)(531,768)(532,767)(533,761)(534,762)
(535,764)(536,763)(537,757)(538,758)(539,760)(540,759)(541,753)(542,754)
(543,756)(544,755)(545,681)(546,682)(547,684)(548,683)(549,745)(550,746)
(551,748)(552,747)(553,741)(554,742)(555,744)(556,743)(557,737)(558,738)
(559,740)(560,739)(561,733)(562,734)(563,736)(564,735)(565,729)(566,730)
(567,732)(568,731)(569,725)(570,726)(571,728)(572,727)(573,721)(574,722)
(575,724)(576,723)(577,717)(578,718)(579,720)(580,719)(581,713)(582,714)
(583,716)(584,715)(585,709)(586,710)(587,712)(588,711)(589,705)(590,706)
(591,708)(592,707)(593,701)(594,702)(595,704)(596,703)(597,697)(598,698)
(599,700)(600,699)(601,693)(602,694)(603,696)(604,695)(605,689)(606,690)
(607,692)(608,691)(609,685)(610,686)(611,688)(612,687);;
s1 := (  1,481)(  2,484)(  3,483)(  4,482)(  5,477)(  6,480)(  7,479)(  8,478)
(  9,541)( 10,544)( 11,543)( 12,542)( 13,537)( 14,540)( 15,539)( 16,538)
( 17,533)( 18,536)( 19,535)( 20,534)( 21,529)( 22,532)( 23,531)( 24,530)
( 25,525)( 26,528)( 27,527)( 28,526)( 29,521)( 30,524)( 31,523)( 32,522)
( 33,517)( 34,520)( 35,519)( 36,518)( 37,513)( 38,516)( 39,515)( 40,514)
( 41,509)( 42,512)( 43,511)( 44,510)( 45,505)( 46,508)( 47,507)( 48,506)
( 49,501)( 50,504)( 51,503)( 52,502)( 53,497)( 54,500)( 55,499)( 56,498)
( 57,493)( 58,496)( 59,495)( 60,494)( 61,489)( 62,492)( 63,491)( 64,490)
( 65,485)( 66,488)( 67,487)( 68,486)( 69,413)( 70,416)( 71,415)( 72,414)
( 73,409)( 74,412)( 75,411)( 76,410)( 77,473)( 78,476)( 79,475)( 80,474)
( 81,469)( 82,472)( 83,471)( 84,470)( 85,465)( 86,468)( 87,467)( 88,466)
( 89,461)( 90,464)( 91,463)( 92,462)( 93,457)( 94,460)( 95,459)( 96,458)
( 97,453)( 98,456)( 99,455)(100,454)(101,449)(102,452)(103,451)(104,450)
(105,445)(106,448)(107,447)(108,446)(109,441)(110,444)(111,443)(112,442)
(113,437)(114,440)(115,439)(116,438)(117,433)(118,436)(119,435)(120,434)
(121,429)(122,432)(123,431)(124,430)(125,425)(126,428)(127,427)(128,426)
(129,421)(130,424)(131,423)(132,422)(133,417)(134,420)(135,419)(136,418)
(137,549)(138,552)(139,551)(140,550)(141,545)(142,548)(143,547)(144,546)
(145,609)(146,612)(147,611)(148,610)(149,605)(150,608)(151,607)(152,606)
(153,601)(154,604)(155,603)(156,602)(157,597)(158,600)(159,599)(160,598)
(161,593)(162,596)(163,595)(164,594)(165,589)(166,592)(167,591)(168,590)
(169,585)(170,588)(171,587)(172,586)(173,581)(174,584)(175,583)(176,582)
(177,577)(178,580)(179,579)(180,578)(181,573)(182,576)(183,575)(184,574)
(185,569)(186,572)(187,571)(188,570)(189,565)(190,568)(191,567)(192,566)
(193,561)(194,564)(195,563)(196,562)(197,557)(198,560)(199,559)(200,558)
(201,553)(202,556)(203,555)(204,554)(205,685)(206,688)(207,687)(208,686)
(209,681)(210,684)(211,683)(212,682)(213,745)(214,748)(215,747)(216,746)
(217,741)(218,744)(219,743)(220,742)(221,737)(222,740)(223,739)(224,738)
(225,733)(226,736)(227,735)(228,734)(229,729)(230,732)(231,731)(232,730)
(233,725)(234,728)(235,727)(236,726)(237,721)(238,724)(239,723)(240,722)
(241,717)(242,720)(243,719)(244,718)(245,713)(246,716)(247,715)(248,714)
(249,709)(250,712)(251,711)(252,710)(253,705)(254,708)(255,707)(256,706)
(257,701)(258,704)(259,703)(260,702)(261,697)(262,700)(263,699)(264,698)
(265,693)(266,696)(267,695)(268,694)(269,689)(270,692)(271,691)(272,690)
(273,617)(274,620)(275,619)(276,618)(277,613)(278,616)(279,615)(280,614)
(281,677)(282,680)(283,679)(284,678)(285,673)(286,676)(287,675)(288,674)
(289,669)(290,672)(291,671)(292,670)(293,665)(294,668)(295,667)(296,666)
(297,661)(298,664)(299,663)(300,662)(301,657)(302,660)(303,659)(304,658)
(305,653)(306,656)(307,655)(308,654)(309,649)(310,652)(311,651)(312,650)
(313,645)(314,648)(315,647)(316,646)(317,641)(318,644)(319,643)(320,642)
(321,637)(322,640)(323,639)(324,638)(325,633)(326,636)(327,635)(328,634)
(329,629)(330,632)(331,631)(332,630)(333,625)(334,628)(335,627)(336,626)
(337,621)(338,624)(339,623)(340,622)(341,753)(342,756)(343,755)(344,754)
(345,749)(346,752)(347,751)(348,750)(349,813)(350,816)(351,815)(352,814)
(353,809)(354,812)(355,811)(356,810)(357,805)(358,808)(359,807)(360,806)
(361,801)(362,804)(363,803)(364,802)(365,797)(366,800)(367,799)(368,798)
(369,793)(370,796)(371,795)(372,794)(373,789)(374,792)(375,791)(376,790)
(377,785)(378,788)(379,787)(380,786)(381,781)(382,784)(383,783)(384,782)
(385,777)(386,780)(387,779)(388,778)(389,773)(390,776)(391,775)(392,774)
(393,769)(394,772)(395,771)(396,770)(397,765)(398,768)(399,767)(400,766)
(401,761)(402,764)(403,763)(404,762)(405,757)(406,760)(407,759)(408,758);;
s2 := (  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)
( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)
( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)
(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)
(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)
(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)
(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)
(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)
(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)(223,224)
(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)
(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)
(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)(271,272)
(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)(287,288)
(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)(303,304)
(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)(319,320)
(321,322)(323,324)(325,326)(327,328)(329,330)(331,332)(333,334)(335,336)
(337,338)(339,340)(341,342)(343,344)(345,346)(347,348)(349,350)(351,352)
(353,354)(355,356)(357,358)(359,360)(361,362)(363,364)(365,366)(367,368)
(369,370)(371,372)(373,374)(375,376)(377,378)(379,380)(381,382)(383,384)
(385,386)(387,388)(389,390)(391,392)(393,394)(395,396)(397,398)(399,400)
(401,402)(403,404)(405,406)(407,408)(409,410)(411,412)(413,414)(415,416)
(417,418)(419,420)(421,422)(423,424)(425,426)(427,428)(429,430)(431,432)
(433,434)(435,436)(437,438)(439,440)(441,442)(443,444)(445,446)(447,448)
(449,450)(451,452)(453,454)(455,456)(457,458)(459,460)(461,462)(463,464)
(465,466)(467,468)(469,470)(471,472)(473,474)(475,476)(477,478)(479,480)
(481,482)(483,484)(485,486)(487,488)(489,490)(491,492)(493,494)(495,496)
(497,498)(499,500)(501,502)(503,504)(505,506)(507,508)(509,510)(511,512)
(513,514)(515,516)(517,518)(519,520)(521,522)(523,524)(525,526)(527,528)
(529,530)(531,532)(533,534)(535,536)(537,538)(539,540)(541,542)(543,544)
(545,546)(547,548)(549,550)(551,552)(553,554)(555,556)(557,558)(559,560)
(561,562)(563,564)(565,566)(567,568)(569,570)(571,572)(573,574)(575,576)
(577,578)(579,580)(581,582)(583,584)(585,586)(587,588)(589,590)(591,592)
(593,594)(595,596)(597,598)(599,600)(601,602)(603,604)(605,606)(607,608)
(609,610)(611,612)(613,614)(615,616)(617,618)(619,620)(621,622)(623,624)
(625,626)(627,628)(629,630)(631,632)(633,634)(635,636)(637,638)(639,640)
(641,642)(643,644)(645,646)(647,648)(649,650)(651,652)(653,654)(655,656)
(657,658)(659,660)(661,662)(663,664)(665,666)(667,668)(669,670)(671,672)
(673,674)(675,676)(677,678)(679,680)(681,682)(683,684)(685,686)(687,688)
(689,690)(691,692)(693,694)(695,696)(697,698)(699,700)(701,702)(703,704)
(705,706)(707,708)(709,710)(711,712)(713,714)(715,716)(717,718)(719,720)
(721,722)(723,724)(725,726)(727,728)(729,730)(731,732)(733,734)(735,736)
(737,738)(739,740)(741,742)(743,744)(745,746)(747,748)(749,750)(751,752)
(753,754)(755,756)(757,758)(759,760)(761,762)(763,764)(765,766)(767,768)
(769,770)(771,772)(773,774)(775,776)(777,778)(779,780)(781,782)(783,784)
(785,786)(787,788)(789,790)(791,792)(793,794)(795,796)(797,798)(799,800)
(801,802)(803,804)(805,806)(807,808)(809,810)(811,812)(813,814)(815,816);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s1*s0*s2*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(816)!(  3,  4)(  5, 65)(  6, 66)(  7, 68)(  8, 67)(  9, 61)( 10, 62)
( 11, 64)( 12, 63)( 13, 57)( 14, 58)( 15, 60)( 16, 59)( 17, 53)( 18, 54)
( 19, 56)( 20, 55)( 21, 49)( 22, 50)( 23, 52)( 24, 51)( 25, 45)( 26, 46)
( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 33, 37)( 34, 38)
( 35, 40)( 36, 39)( 69,137)( 70,138)( 71,140)( 72,139)( 73,201)( 74,202)
( 75,204)( 76,203)( 77,197)( 78,198)( 79,200)( 80,199)( 81,193)( 82,194)
( 83,196)( 84,195)( 85,189)( 86,190)( 87,192)( 88,191)( 89,185)( 90,186)
( 91,188)( 92,187)( 93,181)( 94,182)( 95,184)( 96,183)( 97,177)( 98,178)
( 99,180)(100,179)(101,173)(102,174)(103,176)(104,175)(105,169)(106,170)
(107,172)(108,171)(109,165)(110,166)(111,168)(112,167)(113,161)(114,162)
(115,164)(116,163)(117,157)(118,158)(119,160)(120,159)(121,153)(122,154)
(123,156)(124,155)(125,149)(126,150)(127,152)(128,151)(129,145)(130,146)
(131,148)(132,147)(133,141)(134,142)(135,144)(136,143)(207,208)(209,269)
(210,270)(211,272)(212,271)(213,265)(214,266)(215,268)(216,267)(217,261)
(218,262)(219,264)(220,263)(221,257)(222,258)(223,260)(224,259)(225,253)
(226,254)(227,256)(228,255)(229,249)(230,250)(231,252)(232,251)(233,245)
(234,246)(235,248)(236,247)(237,241)(238,242)(239,244)(240,243)(273,341)
(274,342)(275,344)(276,343)(277,405)(278,406)(279,408)(280,407)(281,401)
(282,402)(283,404)(284,403)(285,397)(286,398)(287,400)(288,399)(289,393)
(290,394)(291,396)(292,395)(293,389)(294,390)(295,392)(296,391)(297,385)
(298,386)(299,388)(300,387)(301,381)(302,382)(303,384)(304,383)(305,377)
(306,378)(307,380)(308,379)(309,373)(310,374)(311,376)(312,375)(313,369)
(314,370)(315,372)(316,371)(317,365)(318,366)(319,368)(320,367)(321,361)
(322,362)(323,364)(324,363)(325,357)(326,358)(327,360)(328,359)(329,353)
(330,354)(331,356)(332,355)(333,349)(334,350)(335,352)(336,351)(337,345)
(338,346)(339,348)(340,347)(409,613)(410,614)(411,616)(412,615)(413,677)
(414,678)(415,680)(416,679)(417,673)(418,674)(419,676)(420,675)(421,669)
(422,670)(423,672)(424,671)(425,665)(426,666)(427,668)(428,667)(429,661)
(430,662)(431,664)(432,663)(433,657)(434,658)(435,660)(436,659)(437,653)
(438,654)(439,656)(440,655)(441,649)(442,650)(443,652)(444,651)(445,645)
(446,646)(447,648)(448,647)(449,641)(450,642)(451,644)(452,643)(453,637)
(454,638)(455,640)(456,639)(457,633)(458,634)(459,636)(460,635)(461,629)
(462,630)(463,632)(464,631)(465,625)(466,626)(467,628)(468,627)(469,621)
(470,622)(471,624)(472,623)(473,617)(474,618)(475,620)(476,619)(477,749)
(478,750)(479,752)(480,751)(481,813)(482,814)(483,816)(484,815)(485,809)
(486,810)(487,812)(488,811)(489,805)(490,806)(491,808)(492,807)(493,801)
(494,802)(495,804)(496,803)(497,797)(498,798)(499,800)(500,799)(501,793)
(502,794)(503,796)(504,795)(505,789)(506,790)(507,792)(508,791)(509,785)
(510,786)(511,788)(512,787)(513,781)(514,782)(515,784)(516,783)(517,777)
(518,778)(519,780)(520,779)(521,773)(522,774)(523,776)(524,775)(525,769)
(526,770)(527,772)(528,771)(529,765)(530,766)(531,768)(532,767)(533,761)
(534,762)(535,764)(536,763)(537,757)(538,758)(539,760)(540,759)(541,753)
(542,754)(543,756)(544,755)(545,681)(546,682)(547,684)(548,683)(549,745)
(550,746)(551,748)(552,747)(553,741)(554,742)(555,744)(556,743)(557,737)
(558,738)(559,740)(560,739)(561,733)(562,734)(563,736)(564,735)(565,729)
(566,730)(567,732)(568,731)(569,725)(570,726)(571,728)(572,727)(573,721)
(574,722)(575,724)(576,723)(577,717)(578,718)(579,720)(580,719)(581,713)
(582,714)(583,716)(584,715)(585,709)(586,710)(587,712)(588,711)(589,705)
(590,706)(591,708)(592,707)(593,701)(594,702)(595,704)(596,703)(597,697)
(598,698)(599,700)(600,699)(601,693)(602,694)(603,696)(604,695)(605,689)
(606,690)(607,692)(608,691)(609,685)(610,686)(611,688)(612,687);
s1 := Sym(816)!(  1,481)(  2,484)(  3,483)(  4,482)(  5,477)(  6,480)(  7,479)
(  8,478)(  9,541)( 10,544)( 11,543)( 12,542)( 13,537)( 14,540)( 15,539)
( 16,538)( 17,533)( 18,536)( 19,535)( 20,534)( 21,529)( 22,532)( 23,531)
( 24,530)( 25,525)( 26,528)( 27,527)( 28,526)( 29,521)( 30,524)( 31,523)
( 32,522)( 33,517)( 34,520)( 35,519)( 36,518)( 37,513)( 38,516)( 39,515)
( 40,514)( 41,509)( 42,512)( 43,511)( 44,510)( 45,505)( 46,508)( 47,507)
( 48,506)( 49,501)( 50,504)( 51,503)( 52,502)( 53,497)( 54,500)( 55,499)
( 56,498)( 57,493)( 58,496)( 59,495)( 60,494)( 61,489)( 62,492)( 63,491)
( 64,490)( 65,485)( 66,488)( 67,487)( 68,486)( 69,413)( 70,416)( 71,415)
( 72,414)( 73,409)( 74,412)( 75,411)( 76,410)( 77,473)( 78,476)( 79,475)
( 80,474)( 81,469)( 82,472)( 83,471)( 84,470)( 85,465)( 86,468)( 87,467)
( 88,466)( 89,461)( 90,464)( 91,463)( 92,462)( 93,457)( 94,460)( 95,459)
( 96,458)( 97,453)( 98,456)( 99,455)(100,454)(101,449)(102,452)(103,451)
(104,450)(105,445)(106,448)(107,447)(108,446)(109,441)(110,444)(111,443)
(112,442)(113,437)(114,440)(115,439)(116,438)(117,433)(118,436)(119,435)
(120,434)(121,429)(122,432)(123,431)(124,430)(125,425)(126,428)(127,427)
(128,426)(129,421)(130,424)(131,423)(132,422)(133,417)(134,420)(135,419)
(136,418)(137,549)(138,552)(139,551)(140,550)(141,545)(142,548)(143,547)
(144,546)(145,609)(146,612)(147,611)(148,610)(149,605)(150,608)(151,607)
(152,606)(153,601)(154,604)(155,603)(156,602)(157,597)(158,600)(159,599)
(160,598)(161,593)(162,596)(163,595)(164,594)(165,589)(166,592)(167,591)
(168,590)(169,585)(170,588)(171,587)(172,586)(173,581)(174,584)(175,583)
(176,582)(177,577)(178,580)(179,579)(180,578)(181,573)(182,576)(183,575)
(184,574)(185,569)(186,572)(187,571)(188,570)(189,565)(190,568)(191,567)
(192,566)(193,561)(194,564)(195,563)(196,562)(197,557)(198,560)(199,559)
(200,558)(201,553)(202,556)(203,555)(204,554)(205,685)(206,688)(207,687)
(208,686)(209,681)(210,684)(211,683)(212,682)(213,745)(214,748)(215,747)
(216,746)(217,741)(218,744)(219,743)(220,742)(221,737)(222,740)(223,739)
(224,738)(225,733)(226,736)(227,735)(228,734)(229,729)(230,732)(231,731)
(232,730)(233,725)(234,728)(235,727)(236,726)(237,721)(238,724)(239,723)
(240,722)(241,717)(242,720)(243,719)(244,718)(245,713)(246,716)(247,715)
(248,714)(249,709)(250,712)(251,711)(252,710)(253,705)(254,708)(255,707)
(256,706)(257,701)(258,704)(259,703)(260,702)(261,697)(262,700)(263,699)
(264,698)(265,693)(266,696)(267,695)(268,694)(269,689)(270,692)(271,691)
(272,690)(273,617)(274,620)(275,619)(276,618)(277,613)(278,616)(279,615)
(280,614)(281,677)(282,680)(283,679)(284,678)(285,673)(286,676)(287,675)
(288,674)(289,669)(290,672)(291,671)(292,670)(293,665)(294,668)(295,667)
(296,666)(297,661)(298,664)(299,663)(300,662)(301,657)(302,660)(303,659)
(304,658)(305,653)(306,656)(307,655)(308,654)(309,649)(310,652)(311,651)
(312,650)(313,645)(314,648)(315,647)(316,646)(317,641)(318,644)(319,643)
(320,642)(321,637)(322,640)(323,639)(324,638)(325,633)(326,636)(327,635)
(328,634)(329,629)(330,632)(331,631)(332,630)(333,625)(334,628)(335,627)
(336,626)(337,621)(338,624)(339,623)(340,622)(341,753)(342,756)(343,755)
(344,754)(345,749)(346,752)(347,751)(348,750)(349,813)(350,816)(351,815)
(352,814)(353,809)(354,812)(355,811)(356,810)(357,805)(358,808)(359,807)
(360,806)(361,801)(362,804)(363,803)(364,802)(365,797)(366,800)(367,799)
(368,798)(369,793)(370,796)(371,795)(372,794)(373,789)(374,792)(375,791)
(376,790)(377,785)(378,788)(379,787)(380,786)(381,781)(382,784)(383,783)
(384,782)(385,777)(386,780)(387,779)(388,778)(389,773)(390,776)(391,775)
(392,774)(393,769)(394,772)(395,771)(396,770)(397,765)(398,768)(399,767)
(400,766)(401,761)(402,764)(403,763)(404,762)(405,757)(406,760)(407,759)
(408,758);
s2 := Sym(816)!(  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)
( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)
( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)
( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)
( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)
( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)
( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)
(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)
(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)
(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)
(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)
(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)
(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)
(207,208)(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)
(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)
(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)
(255,256)(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)
(271,272)(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)
(287,288)(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)
(303,304)(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)
(319,320)(321,322)(323,324)(325,326)(327,328)(329,330)(331,332)(333,334)
(335,336)(337,338)(339,340)(341,342)(343,344)(345,346)(347,348)(349,350)
(351,352)(353,354)(355,356)(357,358)(359,360)(361,362)(363,364)(365,366)
(367,368)(369,370)(371,372)(373,374)(375,376)(377,378)(379,380)(381,382)
(383,384)(385,386)(387,388)(389,390)(391,392)(393,394)(395,396)(397,398)
(399,400)(401,402)(403,404)(405,406)(407,408)(409,410)(411,412)(413,414)
(415,416)(417,418)(419,420)(421,422)(423,424)(425,426)(427,428)(429,430)
(431,432)(433,434)(435,436)(437,438)(439,440)(441,442)(443,444)(445,446)
(447,448)(449,450)(451,452)(453,454)(455,456)(457,458)(459,460)(461,462)
(463,464)(465,466)(467,468)(469,470)(471,472)(473,474)(475,476)(477,478)
(479,480)(481,482)(483,484)(485,486)(487,488)(489,490)(491,492)(493,494)
(495,496)(497,498)(499,500)(501,502)(503,504)(505,506)(507,508)(509,510)
(511,512)(513,514)(515,516)(517,518)(519,520)(521,522)(523,524)(525,526)
(527,528)(529,530)(531,532)(533,534)(535,536)(537,538)(539,540)(541,542)
(543,544)(545,546)(547,548)(549,550)(551,552)(553,554)(555,556)(557,558)
(559,560)(561,562)(563,564)(565,566)(567,568)(569,570)(571,572)(573,574)
(575,576)(577,578)(579,580)(581,582)(583,584)(585,586)(587,588)(589,590)
(591,592)(593,594)(595,596)(597,598)(599,600)(601,602)(603,604)(605,606)
(607,608)(609,610)(611,612)(613,614)(615,616)(617,618)(619,620)(621,622)
(623,624)(625,626)(627,628)(629,630)(631,632)(633,634)(635,636)(637,638)
(639,640)(641,642)(643,644)(645,646)(647,648)(649,650)(651,652)(653,654)
(655,656)(657,658)(659,660)(661,662)(663,664)(665,666)(667,668)(669,670)
(671,672)(673,674)(675,676)(677,678)(679,680)(681,682)(683,684)(685,686)
(687,688)(689,690)(691,692)(693,694)(695,696)(697,698)(699,700)(701,702)
(703,704)(705,706)(707,708)(709,710)(711,712)(713,714)(715,716)(717,718)
(719,720)(721,722)(723,724)(725,726)(727,728)(729,730)(731,732)(733,734)
(735,736)(737,738)(739,740)(741,742)(743,744)(745,746)(747,748)(749,750)
(751,752)(753,754)(755,756)(757,758)(759,760)(761,762)(763,764)(765,766)
(767,768)(769,770)(771,772)(773,774)(775,776)(777,778)(779,780)(781,782)
(783,784)(785,786)(787,788)(789,790)(791,792)(793,794)(795,796)(797,798)
(799,800)(801,802)(803,804)(805,806)(807,808)(809,810)(811,812)(813,814)
(815,816);
poly := sub<Sym(816)|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, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s1*s0*s2*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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