Polytope of Type {672}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {672}*1344
Also Known As : 672-gon, {672}. if this polytope has another name.
Group : SmallGroup(1344,642)
Rank : 2
Schlafli Type : {672}
Number of vertices, edges, etc : 672, 672
Order of s₀s₁ : 672
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 : {336}*672
   3-fold quotients : {224}*448
   4-fold quotients : {168}*336
   6-fold quotients : {112}*224
   7-fold quotients : {96}*192
   8-fold quotients : {84}*168
   12-fold quotients : {56}*112
   14-fold quotients : {48}*96
   16-fold quotients : {42}*84
   21-fold quotients : {32}*64
   24-fold quotients : {28}*56
   28-fold quotients : {24}*48
   32-fold quotients : {21}*42
   42-fold quotients : {16}*32
   48-fold quotients : {14}*28
   56-fold quotients : {12}*24
   84-fold quotients : {8}*16
   96-fold quotients : {7}*14
   112-fold quotients : {6}*12
   168-fold quotients : {4}*8
   224-fold quotients : {3}*6
   336-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope