Polytope of Type {686}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {686}*1372
Also Known As : 686-gon, {686}. if this polytope has another name.
Group : SmallGroup(1372,3)
Rank : 2
Schlafli Type : {686}
Number of vertices, edges, etc : 686, 686
Order of s₀s₁ : 686
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
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 : {343}*686
   7-fold quotients : {98}*196
   14-fold quotients : {49}*98
   49-fold quotients : {14}*28
   98-fold quotients : {7}*14
   343-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1> := Group< s0,s1 | s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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