Polytope of Type {207,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {207,4}*1656
if this polytope has a name.
Group : SmallGroup(1656,38)
Rank : 3
Schlafli Type : {207,4}
Number of vertices, edges, etc : 207, 414, 4
Order of s₀s₁s₂ : 207
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) :
   3-fold quotients : {69,4}*552
   23-fold quotients : {9,4}*72
   69-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  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);;
s1 := (  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);;
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)
(817,818)(819,820)(821,822)(823,824)(825,826)(827,828);;
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*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := 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);
s1 := 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);
s2 := Sym(828)!(  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)(817,818)(819,820)(821,822)(823,824)(825,826)(827,828);
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, 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*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope