Polytope of Type {678}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {678}*1356
Also Known As : 678-gon, {678}. if this polytope has another name.
Group : SmallGroup(1356,11)
Rank : 2
Schlafli Type : {678}
Number of vertices, edges, etc : 678, 678
Order of s₀s₁ : 678
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 : {339}*678
   3-fold quotients : {226}*452
   6-fold quotients : {113}*226
   113-fold quotients : {6}*12
   226-fold quotients : {3}*6
   339-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope