Polytope of Type {682}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {682}*1364
Also Known As : 682-gon, {682}. if this polytope has another name.
Group : SmallGroup(1364,8)
Rank : 2
Schlafli Type : {682}
Number of vertices, edges, etc : 682, 682
Order of s₀s₁ : 682
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 : {341}*682
   11-fold quotients : {62}*124
   22-fold quotients : {31}*62
   31-fold quotients : {22}*44
   62-fold quotients : {11}*22
   341-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope