Polytope of Type {4,204}

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

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