Polytope of Type {4,207}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,207}*1656
if this polytope has a name.
Group : SmallGroup(1656,38)
Rank : 3
Schlafli Type : {4,207}
Number of vertices, edges, etc : 4, 414, 207
Order of s₀s₁s₂ : 207
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) :
   3-fold quotients : {4,69}*552
   23-fold quotients : {4,9}*72
   69-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)
(817,819)(818,820)(821,823)(822,824)(825,827)(826,828);;
s1 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13,265)( 14,266)( 15,268)
( 16,267)( 17,273)( 18,274)( 19,276)( 20,275)( 21,269)( 22,270)( 23,272)
( 24,271)( 25,253)( 26,254)( 27,256)( 28,255)( 29,261)( 30,262)( 31,264)
( 32,263)( 33,257)( 34,258)( 35,260)( 36,259)( 37,241)( 38,242)( 39,244)
( 40,243)( 41,249)( 42,250)( 43,252)( 44,251)( 45,245)( 46,246)( 47,248)
( 48,247)( 49,229)( 50,230)( 51,232)( 52,231)( 53,237)( 54,238)( 55,240)
( 56,239)( 57,233)( 58,234)( 59,236)( 60,235)( 61,217)( 62,218)( 63,220)
( 64,219)( 65,225)( 66,226)( 67,228)( 68,227)( 69,221)( 70,222)( 71,224)
( 72,223)( 73,205)( 74,206)( 75,208)( 76,207)( 77,213)( 78,214)( 79,216)
( 80,215)( 81,209)( 82,210)( 83,212)( 84,211)( 85,193)( 86,194)( 87,196)
( 88,195)( 89,201)( 90,202)( 91,204)( 92,203)( 93,197)( 94,198)( 95,200)
( 96,199)( 97,181)( 98,182)( 99,184)(100,183)(101,189)(102,190)(103,192)
(104,191)(105,185)(106,186)(107,188)(108,187)(109,169)(110,170)(111,172)
(112,171)(113,177)(114,178)(115,180)(116,179)(117,173)(118,174)(119,176)
(120,175)(121,157)(122,158)(123,160)(124,159)(125,165)(126,166)(127,168)
(128,167)(129,161)(130,162)(131,164)(132,163)(133,145)(134,146)(135,148)
(136,147)(137,153)(138,154)(139,156)(140,155)(141,149)(142,150)(143,152)
(144,151)(277,561)(278,562)(279,564)(280,563)(281,557)(282,558)(283,560)
(284,559)(285,553)(286,554)(287,556)(288,555)(289,825)(290,826)(291,828)
(292,827)(293,821)(294,822)(295,824)(296,823)(297,817)(298,818)(299,820)
(300,819)(301,813)(302,814)(303,816)(304,815)(305,809)(306,810)(307,812)
(308,811)(309,805)(310,806)(311,808)(312,807)(313,801)(314,802)(315,804)
(316,803)(317,797)(318,798)(319,800)(320,799)(321,793)(322,794)(323,796)
(324,795)(325,789)(326,790)(327,792)(328,791)(329,785)(330,786)(331,788)
(332,787)(333,781)(334,782)(335,784)(336,783)(337,777)(338,778)(339,780)
(340,779)(341,773)(342,774)(343,776)(344,775)(345,769)(346,770)(347,772)
(348,771)(349,765)(350,766)(351,768)(352,767)(353,761)(354,762)(355,764)
(356,763)(357,757)(358,758)(359,760)(360,759)(361,753)(362,754)(363,756)
(364,755)(365,749)(366,750)(367,752)(368,751)(369,745)(370,746)(371,748)
(372,747)(373,741)(374,742)(375,744)(376,743)(377,737)(378,738)(379,740)
(380,739)(381,733)(382,734)(383,736)(384,735)(385,729)(386,730)(387,732)
(388,731)(389,725)(390,726)(391,728)(392,727)(393,721)(394,722)(395,724)
(396,723)(397,717)(398,718)(399,720)(400,719)(401,713)(402,714)(403,716)
(404,715)(405,709)(406,710)(407,712)(408,711)(409,705)(410,706)(411,708)
(412,707)(413,701)(414,702)(415,704)(416,703)(417,697)(418,698)(419,700)
(420,699)(421,693)(422,694)(423,696)(424,695)(425,689)(426,690)(427,692)
(428,691)(429,685)(430,686)(431,688)(432,687)(433,681)(434,682)(435,684)
(436,683)(437,677)(438,678)(439,680)(440,679)(441,673)(442,674)(443,676)
(444,675)(445,669)(446,670)(447,672)(448,671)(449,665)(450,666)(451,668)
(452,667)(453,661)(454,662)(455,664)(456,663)(457,657)(458,658)(459,660)
(460,659)(461,653)(462,654)(463,656)(464,655)(465,649)(466,650)(467,652)
(468,651)(469,645)(470,646)(471,648)(472,647)(473,641)(474,642)(475,644)
(476,643)(477,637)(478,638)(479,640)(480,639)(481,633)(482,634)(483,636)
(484,635)(485,629)(486,630)(487,632)(488,631)(489,625)(490,626)(491,628)
(492,627)(493,621)(494,622)(495,624)(496,623)(497,617)(498,618)(499,620)
(500,619)(501,613)(502,614)(503,616)(504,615)(505,609)(506,610)(507,612)
(508,611)(509,605)(510,606)(511,608)(512,607)(513,601)(514,602)(515,604)
(516,603)(517,597)(518,598)(519,600)(520,599)(521,593)(522,594)(523,596)
(524,595)(525,589)(526,590)(527,592)(528,591)(529,585)(530,586)(531,588)
(532,587)(533,581)(534,582)(535,584)(536,583)(537,577)(538,578)(539,580)
(540,579)(541,573)(542,574)(543,576)(544,575)(545,569)(546,570)(547,572)
(548,571)(549,565)(550,566)(551,568)(552,567);;
s2 := (  1,289)(  2,292)(  3,291)(  4,290)(  5,297)(  6,300)(  7,299)(  8,298)
(  9,293)( 10,296)( 11,295)( 12,294)( 13,277)( 14,280)( 15,279)( 16,278)
( 17,285)( 18,288)( 19,287)( 20,286)( 21,281)( 22,284)( 23,283)( 24,282)
( 25,541)( 26,544)( 27,543)( 28,542)( 29,549)( 30,552)( 31,551)( 32,550)
( 33,545)( 34,548)( 35,547)( 36,546)( 37,529)( 38,532)( 39,531)( 40,530)
( 41,537)( 42,540)( 43,539)( 44,538)( 45,533)( 46,536)( 47,535)( 48,534)
( 49,517)( 50,520)( 51,519)( 52,518)( 53,525)( 54,528)( 55,527)( 56,526)
( 57,521)( 58,524)( 59,523)( 60,522)( 61,505)( 62,508)( 63,507)( 64,506)
( 65,513)( 66,516)( 67,515)( 68,514)( 69,509)( 70,512)( 71,511)( 72,510)
( 73,493)( 74,496)( 75,495)( 76,494)( 77,501)( 78,504)( 79,503)( 80,502)
( 81,497)( 82,500)( 83,499)( 84,498)( 85,481)( 86,484)( 87,483)( 88,482)
( 89,489)( 90,492)( 91,491)( 92,490)( 93,485)( 94,488)( 95,487)( 96,486)
( 97,469)( 98,472)( 99,471)(100,470)(101,477)(102,480)(103,479)(104,478)
(105,473)(106,476)(107,475)(108,474)(109,457)(110,460)(111,459)(112,458)
(113,465)(114,468)(115,467)(116,466)(117,461)(118,464)(119,463)(120,462)
(121,445)(122,448)(123,447)(124,446)(125,453)(126,456)(127,455)(128,454)
(129,449)(130,452)(131,451)(132,450)(133,433)(134,436)(135,435)(136,434)
(137,441)(138,444)(139,443)(140,442)(141,437)(142,440)(143,439)(144,438)
(145,421)(146,424)(147,423)(148,422)(149,429)(150,432)(151,431)(152,430)
(153,425)(154,428)(155,427)(156,426)(157,409)(158,412)(159,411)(160,410)
(161,417)(162,420)(163,419)(164,418)(165,413)(166,416)(167,415)(168,414)
(169,397)(170,400)(171,399)(172,398)(173,405)(174,408)(175,407)(176,406)
(177,401)(178,404)(179,403)(180,402)(181,385)(182,388)(183,387)(184,386)
(185,393)(186,396)(187,395)(188,394)(189,389)(190,392)(191,391)(192,390)
(193,373)(194,376)(195,375)(196,374)(197,381)(198,384)(199,383)(200,382)
(201,377)(202,380)(203,379)(204,378)(205,361)(206,364)(207,363)(208,362)
(209,369)(210,372)(211,371)(212,370)(213,365)(214,368)(215,367)(216,366)
(217,349)(218,352)(219,351)(220,350)(221,357)(222,360)(223,359)(224,358)
(225,353)(226,356)(227,355)(228,354)(229,337)(230,340)(231,339)(232,338)
(233,345)(234,348)(235,347)(236,346)(237,341)(238,344)(239,343)(240,342)
(241,325)(242,328)(243,327)(244,326)(245,333)(246,336)(247,335)(248,334)
(249,329)(250,332)(251,331)(252,330)(253,313)(254,316)(255,315)(256,314)
(257,321)(258,324)(259,323)(260,322)(261,317)(262,320)(263,319)(264,318)
(265,301)(266,304)(267,303)(268,302)(269,309)(270,312)(271,311)(272,310)
(273,305)(274,308)(275,307)(276,306)(553,573)(554,576)(555,575)(556,574)
(557,569)(558,572)(559,571)(560,570)(561,565)(562,568)(563,567)(564,566)
(577,825)(578,828)(579,827)(580,826)(581,821)(582,824)(583,823)(584,822)
(585,817)(586,820)(587,819)(588,818)(589,813)(590,816)(591,815)(592,814)
(593,809)(594,812)(595,811)(596,810)(597,805)(598,808)(599,807)(600,806)
(601,801)(602,804)(603,803)(604,802)(605,797)(606,800)(607,799)(608,798)
(609,793)(610,796)(611,795)(612,794)(613,789)(614,792)(615,791)(616,790)
(617,785)(618,788)(619,787)(620,786)(621,781)(622,784)(623,783)(624,782)
(625,777)(626,780)(627,779)(628,778)(629,773)(630,776)(631,775)(632,774)
(633,769)(634,772)(635,771)(636,770)(637,765)(638,768)(639,767)(640,766)
(641,761)(642,764)(643,763)(644,762)(645,757)(646,760)(647,759)(648,758)
(649,753)(650,756)(651,755)(652,754)(653,749)(654,752)(655,751)(656,750)
(657,745)(658,748)(659,747)(660,746)(661,741)(662,744)(663,743)(664,742)
(665,737)(666,740)(667,739)(668,738)(669,733)(670,736)(671,735)(672,734)
(673,729)(674,732)(675,731)(676,730)(677,725)(678,728)(679,727)(680,726)
(681,721)(682,724)(683,723)(684,722)(685,717)(686,720)(687,719)(688,718)
(689,713)(690,716)(691,715)(692,714)(693,709)(694,712)(695,711)(696,710)
(697,705)(698,708)(699,707)(700,706)(702,704);;
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*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(828)!(  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)(817,819)(818,820)(821,823)(822,824)(825,827)(826,828);
s1 := Sym(828)!(  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13,265)( 14,266)
( 15,268)( 16,267)( 17,273)( 18,274)( 19,276)( 20,275)( 21,269)( 22,270)
( 23,272)( 24,271)( 25,253)( 26,254)( 27,256)( 28,255)( 29,261)( 30,262)
( 31,264)( 32,263)( 33,257)( 34,258)( 35,260)( 36,259)( 37,241)( 38,242)
( 39,244)( 40,243)( 41,249)( 42,250)( 43,252)( 44,251)( 45,245)( 46,246)
( 47,248)( 48,247)( 49,229)( 50,230)( 51,232)( 52,231)( 53,237)( 54,238)
( 55,240)( 56,239)( 57,233)( 58,234)( 59,236)( 60,235)( 61,217)( 62,218)
( 63,220)( 64,219)( 65,225)( 66,226)( 67,228)( 68,227)( 69,221)( 70,222)
( 71,224)( 72,223)( 73,205)( 74,206)( 75,208)( 76,207)( 77,213)( 78,214)
( 79,216)( 80,215)( 81,209)( 82,210)( 83,212)( 84,211)( 85,193)( 86,194)
( 87,196)( 88,195)( 89,201)( 90,202)( 91,204)( 92,203)( 93,197)( 94,198)
( 95,200)( 96,199)( 97,181)( 98,182)( 99,184)(100,183)(101,189)(102,190)
(103,192)(104,191)(105,185)(106,186)(107,188)(108,187)(109,169)(110,170)
(111,172)(112,171)(113,177)(114,178)(115,180)(116,179)(117,173)(118,174)
(119,176)(120,175)(121,157)(122,158)(123,160)(124,159)(125,165)(126,166)
(127,168)(128,167)(129,161)(130,162)(131,164)(132,163)(133,145)(134,146)
(135,148)(136,147)(137,153)(138,154)(139,156)(140,155)(141,149)(142,150)
(143,152)(144,151)(277,561)(278,562)(279,564)(280,563)(281,557)(282,558)
(283,560)(284,559)(285,553)(286,554)(287,556)(288,555)(289,825)(290,826)
(291,828)(292,827)(293,821)(294,822)(295,824)(296,823)(297,817)(298,818)
(299,820)(300,819)(301,813)(302,814)(303,816)(304,815)(305,809)(306,810)
(307,812)(308,811)(309,805)(310,806)(311,808)(312,807)(313,801)(314,802)
(315,804)(316,803)(317,797)(318,798)(319,800)(320,799)(321,793)(322,794)
(323,796)(324,795)(325,789)(326,790)(327,792)(328,791)(329,785)(330,786)
(331,788)(332,787)(333,781)(334,782)(335,784)(336,783)(337,777)(338,778)
(339,780)(340,779)(341,773)(342,774)(343,776)(344,775)(345,769)(346,770)
(347,772)(348,771)(349,765)(350,766)(351,768)(352,767)(353,761)(354,762)
(355,764)(356,763)(357,757)(358,758)(359,760)(360,759)(361,753)(362,754)
(363,756)(364,755)(365,749)(366,750)(367,752)(368,751)(369,745)(370,746)
(371,748)(372,747)(373,741)(374,742)(375,744)(376,743)(377,737)(378,738)
(379,740)(380,739)(381,733)(382,734)(383,736)(384,735)(385,729)(386,730)
(387,732)(388,731)(389,725)(390,726)(391,728)(392,727)(393,721)(394,722)
(395,724)(396,723)(397,717)(398,718)(399,720)(400,719)(401,713)(402,714)
(403,716)(404,715)(405,709)(406,710)(407,712)(408,711)(409,705)(410,706)
(411,708)(412,707)(413,701)(414,702)(415,704)(416,703)(417,697)(418,698)
(419,700)(420,699)(421,693)(422,694)(423,696)(424,695)(425,689)(426,690)
(427,692)(428,691)(429,685)(430,686)(431,688)(432,687)(433,681)(434,682)
(435,684)(436,683)(437,677)(438,678)(439,680)(440,679)(441,673)(442,674)
(443,676)(444,675)(445,669)(446,670)(447,672)(448,671)(449,665)(450,666)
(451,668)(452,667)(453,661)(454,662)(455,664)(456,663)(457,657)(458,658)
(459,660)(460,659)(461,653)(462,654)(463,656)(464,655)(465,649)(466,650)
(467,652)(468,651)(469,645)(470,646)(471,648)(472,647)(473,641)(474,642)
(475,644)(476,643)(477,637)(478,638)(479,640)(480,639)(481,633)(482,634)
(483,636)(484,635)(485,629)(486,630)(487,632)(488,631)(489,625)(490,626)
(491,628)(492,627)(493,621)(494,622)(495,624)(496,623)(497,617)(498,618)
(499,620)(500,619)(501,613)(502,614)(503,616)(504,615)(505,609)(506,610)
(507,612)(508,611)(509,605)(510,606)(511,608)(512,607)(513,601)(514,602)
(515,604)(516,603)(517,597)(518,598)(519,600)(520,599)(521,593)(522,594)
(523,596)(524,595)(525,589)(526,590)(527,592)(528,591)(529,585)(530,586)
(531,588)(532,587)(533,581)(534,582)(535,584)(536,583)(537,577)(538,578)
(539,580)(540,579)(541,573)(542,574)(543,576)(544,575)(545,569)(546,570)
(547,572)(548,571)(549,565)(550,566)(551,568)(552,567);
s2 := Sym(828)!(  1,289)(  2,292)(  3,291)(  4,290)(  5,297)(  6,300)(  7,299)
(  8,298)(  9,293)( 10,296)( 11,295)( 12,294)( 13,277)( 14,280)( 15,279)
( 16,278)( 17,285)( 18,288)( 19,287)( 20,286)( 21,281)( 22,284)( 23,283)
( 24,282)( 25,541)( 26,544)( 27,543)( 28,542)( 29,549)( 30,552)( 31,551)
( 32,550)( 33,545)( 34,548)( 35,547)( 36,546)( 37,529)( 38,532)( 39,531)
( 40,530)( 41,537)( 42,540)( 43,539)( 44,538)( 45,533)( 46,536)( 47,535)
( 48,534)( 49,517)( 50,520)( 51,519)( 52,518)( 53,525)( 54,528)( 55,527)
( 56,526)( 57,521)( 58,524)( 59,523)( 60,522)( 61,505)( 62,508)( 63,507)
( 64,506)( 65,513)( 66,516)( 67,515)( 68,514)( 69,509)( 70,512)( 71,511)
( 72,510)( 73,493)( 74,496)( 75,495)( 76,494)( 77,501)( 78,504)( 79,503)
( 80,502)( 81,497)( 82,500)( 83,499)( 84,498)( 85,481)( 86,484)( 87,483)
( 88,482)( 89,489)( 90,492)( 91,491)( 92,490)( 93,485)( 94,488)( 95,487)
( 96,486)( 97,469)( 98,472)( 99,471)(100,470)(101,477)(102,480)(103,479)
(104,478)(105,473)(106,476)(107,475)(108,474)(109,457)(110,460)(111,459)
(112,458)(113,465)(114,468)(115,467)(116,466)(117,461)(118,464)(119,463)
(120,462)(121,445)(122,448)(123,447)(124,446)(125,453)(126,456)(127,455)
(128,454)(129,449)(130,452)(131,451)(132,450)(133,433)(134,436)(135,435)
(136,434)(137,441)(138,444)(139,443)(140,442)(141,437)(142,440)(143,439)
(144,438)(145,421)(146,424)(147,423)(148,422)(149,429)(150,432)(151,431)
(152,430)(153,425)(154,428)(155,427)(156,426)(157,409)(158,412)(159,411)
(160,410)(161,417)(162,420)(163,419)(164,418)(165,413)(166,416)(167,415)
(168,414)(169,397)(170,400)(171,399)(172,398)(173,405)(174,408)(175,407)
(176,406)(177,401)(178,404)(179,403)(180,402)(181,385)(182,388)(183,387)
(184,386)(185,393)(186,396)(187,395)(188,394)(189,389)(190,392)(191,391)
(192,390)(193,373)(194,376)(195,375)(196,374)(197,381)(198,384)(199,383)
(200,382)(201,377)(202,380)(203,379)(204,378)(205,361)(206,364)(207,363)
(208,362)(209,369)(210,372)(211,371)(212,370)(213,365)(214,368)(215,367)
(216,366)(217,349)(218,352)(219,351)(220,350)(221,357)(222,360)(223,359)
(224,358)(225,353)(226,356)(227,355)(228,354)(229,337)(230,340)(231,339)
(232,338)(233,345)(234,348)(235,347)(236,346)(237,341)(238,344)(239,343)
(240,342)(241,325)(242,328)(243,327)(244,326)(245,333)(246,336)(247,335)
(248,334)(249,329)(250,332)(251,331)(252,330)(253,313)(254,316)(255,315)
(256,314)(257,321)(258,324)(259,323)(260,322)(261,317)(262,320)(263,319)
(264,318)(265,301)(266,304)(267,303)(268,302)(269,309)(270,312)(271,311)
(272,310)(273,305)(274,308)(275,307)(276,306)(553,573)(554,576)(555,575)
(556,574)(557,569)(558,572)(559,571)(560,570)(561,565)(562,568)(563,567)
(564,566)(577,825)(578,828)(579,827)(580,826)(581,821)(582,824)(583,823)
(584,822)(585,817)(586,820)(587,819)(588,818)(589,813)(590,816)(591,815)
(592,814)(593,809)(594,812)(595,811)(596,810)(597,805)(598,808)(599,807)
(600,806)(601,801)(602,804)(603,803)(604,802)(605,797)(606,800)(607,799)
(608,798)(609,793)(610,796)(611,795)(612,794)(613,789)(614,792)(615,791)
(616,790)(617,785)(618,788)(619,787)(620,786)(621,781)(622,784)(623,783)
(624,782)(625,777)(626,780)(627,779)(628,778)(629,773)(630,776)(631,775)
(632,774)(633,769)(634,772)(635,771)(636,770)(637,765)(638,768)(639,767)
(640,766)(641,761)(642,764)(643,763)(644,762)(645,757)(646,760)(647,759)
(648,758)(649,753)(650,756)(651,755)(652,754)(653,749)(654,752)(655,751)
(656,750)(657,745)(658,748)(659,747)(660,746)(661,741)(662,744)(663,743)
(664,742)(665,737)(666,740)(667,739)(668,738)(669,733)(670,736)(671,735)
(672,734)(673,729)(674,732)(675,731)(676,730)(677,725)(678,728)(679,727)
(680,726)(681,721)(682,724)(683,723)(684,722)(685,717)(686,720)(687,719)
(688,718)(689,713)(690,716)(691,715)(692,714)(693,709)(694,712)(695,711)
(696,710)(697,705)(698,708)(699,707)(700,706)(702,704);
poly := sub<Sym(828)|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*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope