Polytope of Type {684}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {684}*1368
Also Known As : 684-gon, {684}. if this polytope has another name.
Group : SmallGroup(1368,36)
Rank : 2
Schlafli Type : {684}
Number of vertices, edges, etc : 684, 684
Order of s₀s₁ : 684
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 : {342}*684
   3-fold quotients : {228}*456
   4-fold quotients : {171}*342
   6-fold quotients : {114}*228
   9-fold quotients : {76}*152
   12-fold quotients : {57}*114
   18-fold quotients : {38}*76
   19-fold quotients : {36}*72
   36-fold quotients : {19}*38
   38-fold quotients : {18}*36
   57-fold quotients : {12}*24
   76-fold quotients : {9}*18
   114-fold quotients : {6}*12
   171-fold quotients : {4}*8
   228-fold quotients : {3}*6
   342-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope