Polytope of Type {810}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {810}*1620
Also Known As : 810-gon, {810}. if this polytope has another name.
Group : SmallGroup(1620,11)
Rank : 2
Schlafli Type : {810}
Number of vertices, edges, etc : 810, 810
Order of s₀s₁ : 810
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 : {405}*810
   3-fold quotients : {270}*540
   5-fold quotients : {162}*324
   6-fold quotients : {135}*270
   9-fold quotients : {90}*180
   10-fold quotients : {81}*162
   15-fold quotients : {54}*108
   18-fold quotients : {45}*90
   27-fold quotients : {30}*60
   30-fold quotients : {27}*54
   45-fold quotients : {18}*36
   54-fold quotients : {15}*30
   81-fold quotients : {10}*20
   90-fold quotients : {9}*18
   135-fold quotients : {6}*12
   162-fold quotients : {5}*10
   270-fold quotients : {3}*6
   405-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2,  3)(  4,  8)(  5,  7)(  6,  9)( 10, 23)( 11, 22)( 12, 24)( 13, 20)
( 14, 19)( 15, 21)( 16, 27)( 17, 26)( 18, 25)( 28,109)( 29,111)( 30,110)
( 31,116)( 32,115)( 33,117)( 34,113)( 35,112)( 36,114)( 37,131)( 38,130)
( 39,132)( 40,128)( 41,127)( 42,129)( 43,135)( 44,134)( 45,133)( 46,122)
( 47,121)( 48,123)( 49,119)( 50,118)( 51,120)( 52,126)( 53,125)( 54,124)
( 55, 82)( 56, 84)( 57, 83)( 58, 89)( 59, 88)( 60, 90)( 61, 86)( 62, 85)
( 63, 87)( 64,104)( 65,103)( 66,105)( 67,101)( 68,100)( 69,102)( 70,108)
( 71,107)( 72,106)( 73, 95)( 74, 94)( 75, 96)( 76, 92)( 77, 91)( 78, 93)
( 79, 99)( 80, 98)( 81, 97)(136,284)(137,283)(138,285)(139,281)(140,280)
(141,282)(142,288)(143,287)(144,286)(145,275)(146,274)(147,276)(148,272)
(149,271)(150,273)(151,279)(152,278)(153,277)(154,297)(155,296)(156,295)
(157,294)(158,293)(159,292)(160,291)(161,290)(162,289)(163,392)(164,391)
(165,393)(166,389)(167,388)(168,390)(169,396)(170,395)(171,394)(172,383)
(173,382)(174,384)(175,380)(176,379)(177,381)(178,387)(179,386)(180,385)
(181,405)(182,404)(183,403)(184,402)(185,401)(186,400)(187,399)(188,398)
(189,397)(190,365)(191,364)(192,366)(193,362)(194,361)(195,363)(196,369)
(197,368)(198,367)(199,356)(200,355)(201,357)(202,353)(203,352)(204,354)
(205,360)(206,359)(207,358)(208,378)(209,377)(210,376)(211,375)(212,374)
(213,373)(214,372)(215,371)(216,370)(217,338)(218,337)(219,339)(220,335)
(221,334)(222,336)(223,342)(224,341)(225,340)(226,329)(227,328)(228,330)
(229,326)(230,325)(231,327)(232,333)(233,332)(234,331)(235,351)(236,350)
(237,349)(238,348)(239,347)(240,346)(241,345)(242,344)(243,343)(244,311)
(245,310)(246,312)(247,308)(248,307)(249,309)(250,315)(251,314)(252,313)
(253,302)(254,301)(255,303)(256,299)(257,298)(258,300)(259,306)(260,305)
(261,304)(262,324)(263,323)(264,322)(265,321)(266,320)(267,319)(268,318)
(269,317)(270,316)(407,408)(409,413)(410,412)(411,414)(415,428)(416,427)
(417,429)(418,425)(419,424)(420,426)(421,432)(422,431)(423,430)(433,514)
(434,516)(435,515)(436,521)(437,520)(438,522)(439,518)(440,517)(441,519)
(442,536)(443,535)(444,537)(445,533)(446,532)(447,534)(448,540)(449,539)
(450,538)(451,527)(452,526)(453,528)(454,524)(455,523)(456,525)(457,531)
(458,530)(459,529)(460,487)(461,489)(462,488)(463,494)(464,493)(465,495)
(466,491)(467,490)(468,492)(469,509)(470,508)(471,510)(472,506)(473,505)
(474,507)(475,513)(476,512)(477,511)(478,500)(479,499)(480,501)(481,497)
(482,496)(483,498)(484,504)(485,503)(486,502)(541,689)(542,688)(543,690)
(544,686)(545,685)(546,687)(547,693)(548,692)(549,691)(550,680)(551,679)
(552,681)(553,677)(554,676)(555,678)(556,684)(557,683)(558,682)(559,702)
(560,701)(561,700)(562,699)(563,698)(564,697)(565,696)(566,695)(567,694)
(568,797)(569,796)(570,798)(571,794)(572,793)(573,795)(574,801)(575,800)
(576,799)(577,788)(578,787)(579,789)(580,785)(581,784)(582,786)(583,792)
(584,791)(585,790)(586,810)(587,809)(588,808)(589,807)(590,806)(591,805)
(592,804)(593,803)(594,802)(595,770)(596,769)(597,771)(598,767)(599,766)
(600,768)(601,774)(602,773)(603,772)(604,761)(605,760)(606,762)(607,758)
(608,757)(609,759)(610,765)(611,764)(612,763)(613,783)(614,782)(615,781)
(616,780)(617,779)(618,778)(619,777)(620,776)(621,775)(622,743)(623,742)
(624,744)(625,740)(626,739)(627,741)(628,747)(629,746)(630,745)(631,734)
(632,733)(633,735)(634,731)(635,730)(636,732)(637,738)(638,737)(639,736)
(640,756)(641,755)(642,754)(643,753)(644,752)(645,751)(646,750)(647,749)
(648,748)(649,716)(650,715)(651,717)(652,713)(653,712)(654,714)(655,720)
(656,719)(657,718)(658,707)(659,706)(660,708)(661,704)(662,703)(663,705)
(664,711)(665,710)(666,709)(667,729)(668,728)(669,727)(670,726)(671,725)
(672,724)(673,723)(674,722)(675,721);;
s1 := (  1,568)(  2,570)(  3,569)(  4,575)(  5,574)(  6,576)(  7,572)(  8,571)
(  9,573)( 10,590)( 11,589)( 12,591)( 13,587)( 14,586)( 15,588)( 16,594)
( 17,593)( 18,592)( 19,581)( 20,580)( 21,582)( 22,578)( 23,577)( 24,579)
( 25,585)( 26,584)( 27,583)( 28,541)( 29,543)( 30,542)( 31,548)( 32,547)
( 33,549)( 34,545)( 35,544)( 36,546)( 37,563)( 38,562)( 39,564)( 40,560)
( 41,559)( 42,561)( 43,567)( 44,566)( 45,565)( 46,554)( 47,553)( 48,555)
( 49,551)( 50,550)( 51,552)( 52,558)( 53,557)( 54,556)( 55,649)( 56,651)
( 57,650)( 58,656)( 59,655)( 60,657)( 61,653)( 62,652)( 63,654)( 64,671)
( 65,670)( 66,672)( 67,668)( 68,667)( 69,669)( 70,675)( 71,674)( 72,673)
( 73,662)( 74,661)( 75,663)( 76,659)( 77,658)( 78,660)( 79,666)( 80,665)
( 81,664)( 82,622)( 83,624)( 84,623)( 85,629)( 86,628)( 87,630)( 88,626)
( 89,625)( 90,627)( 91,644)( 92,643)( 93,645)( 94,641)( 95,640)( 96,642)
( 97,648)( 98,647)( 99,646)(100,635)(101,634)(102,636)(103,632)(104,631)
(105,633)(106,639)(107,638)(108,637)(109,595)(110,597)(111,596)(112,602)
(113,601)(114,603)(115,599)(116,598)(117,600)(118,617)(119,616)(120,618)
(121,614)(122,613)(123,615)(124,621)(125,620)(126,619)(127,608)(128,607)
(129,609)(130,605)(131,604)(132,606)(133,612)(134,611)(135,610)(136,433)
(137,435)(138,434)(139,440)(140,439)(141,441)(142,437)(143,436)(144,438)
(145,455)(146,454)(147,456)(148,452)(149,451)(150,453)(151,459)(152,458)
(153,457)(154,446)(155,445)(156,447)(157,443)(158,442)(159,444)(160,450)
(161,449)(162,448)(163,406)(164,408)(165,407)(166,413)(167,412)(168,414)
(169,410)(170,409)(171,411)(172,428)(173,427)(174,429)(175,425)(176,424)
(177,426)(178,432)(179,431)(180,430)(181,419)(182,418)(183,420)(184,416)
(185,415)(186,417)(187,423)(188,422)(189,421)(190,514)(191,516)(192,515)
(193,521)(194,520)(195,522)(196,518)(197,517)(198,519)(199,536)(200,535)
(201,537)(202,533)(203,532)(204,534)(205,540)(206,539)(207,538)(208,527)
(209,526)(210,528)(211,524)(212,523)(213,525)(214,531)(215,530)(216,529)
(217,487)(218,489)(219,488)(220,494)(221,493)(222,495)(223,491)(224,490)
(225,492)(226,509)(227,508)(228,510)(229,506)(230,505)(231,507)(232,513)
(233,512)(234,511)(235,500)(236,499)(237,501)(238,497)(239,496)(240,498)
(241,504)(242,503)(243,502)(244,460)(245,462)(246,461)(247,467)(248,466)
(249,468)(250,464)(251,463)(252,465)(253,482)(254,481)(255,483)(256,479)
(257,478)(258,480)(259,486)(260,485)(261,484)(262,473)(263,472)(264,474)
(265,470)(266,469)(267,471)(268,477)(269,476)(270,475)(271,716)(272,715)
(273,717)(274,713)(275,712)(276,714)(277,720)(278,719)(279,718)(280,707)
(281,706)(282,708)(283,704)(284,703)(285,705)(286,711)(287,710)(288,709)
(289,729)(290,728)(291,727)(292,726)(293,725)(294,724)(295,723)(296,722)
(297,721)(298,689)(299,688)(300,690)(301,686)(302,685)(303,687)(304,693)
(305,692)(306,691)(307,680)(308,679)(309,681)(310,677)(311,676)(312,678)
(313,684)(314,683)(315,682)(316,702)(317,701)(318,700)(319,699)(320,698)
(321,697)(322,696)(323,695)(324,694)(325,797)(326,796)(327,798)(328,794)
(329,793)(330,795)(331,801)(332,800)(333,799)(334,788)(335,787)(336,789)
(337,785)(338,784)(339,786)(340,792)(341,791)(342,790)(343,810)(344,809)
(345,808)(346,807)(347,806)(348,805)(349,804)(350,803)(351,802)(352,770)
(353,769)(354,771)(355,767)(356,766)(357,768)(358,774)(359,773)(360,772)
(361,761)(362,760)(363,762)(364,758)(365,757)(366,759)(367,765)(368,764)
(369,763)(370,783)(371,782)(372,781)(373,780)(374,779)(375,778)(376,777)
(377,776)(378,775)(379,743)(380,742)(381,744)(382,740)(383,739)(384,741)
(385,747)(386,746)(387,745)(388,734)(389,733)(390,735)(391,731)(392,730)
(393,732)(394,738)(395,737)(396,736)(397,756)(398,755)(399,754)(400,753)
(401,752)(402,751)(403,750)(404,749)(405,748);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(810)!(  2,  3)(  4,  8)(  5,  7)(  6,  9)( 10, 23)( 11, 22)( 12, 24)
( 13, 20)( 14, 19)( 15, 21)( 16, 27)( 17, 26)( 18, 25)( 28,109)( 29,111)
( 30,110)( 31,116)( 32,115)( 33,117)( 34,113)( 35,112)( 36,114)( 37,131)
( 38,130)( 39,132)( 40,128)( 41,127)( 42,129)( 43,135)( 44,134)( 45,133)
( 46,122)( 47,121)( 48,123)( 49,119)( 50,118)( 51,120)( 52,126)( 53,125)
( 54,124)( 55, 82)( 56, 84)( 57, 83)( 58, 89)( 59, 88)( 60, 90)( 61, 86)
( 62, 85)( 63, 87)( 64,104)( 65,103)( 66,105)( 67,101)( 68,100)( 69,102)
( 70,108)( 71,107)( 72,106)( 73, 95)( 74, 94)( 75, 96)( 76, 92)( 77, 91)
( 78, 93)( 79, 99)( 80, 98)( 81, 97)(136,284)(137,283)(138,285)(139,281)
(140,280)(141,282)(142,288)(143,287)(144,286)(145,275)(146,274)(147,276)
(148,272)(149,271)(150,273)(151,279)(152,278)(153,277)(154,297)(155,296)
(156,295)(157,294)(158,293)(159,292)(160,291)(161,290)(162,289)(163,392)
(164,391)(165,393)(166,389)(167,388)(168,390)(169,396)(170,395)(171,394)
(172,383)(173,382)(174,384)(175,380)(176,379)(177,381)(178,387)(179,386)
(180,385)(181,405)(182,404)(183,403)(184,402)(185,401)(186,400)(187,399)
(188,398)(189,397)(190,365)(191,364)(192,366)(193,362)(194,361)(195,363)
(196,369)(197,368)(198,367)(199,356)(200,355)(201,357)(202,353)(203,352)
(204,354)(205,360)(206,359)(207,358)(208,378)(209,377)(210,376)(211,375)
(212,374)(213,373)(214,372)(215,371)(216,370)(217,338)(218,337)(219,339)
(220,335)(221,334)(222,336)(223,342)(224,341)(225,340)(226,329)(227,328)
(228,330)(229,326)(230,325)(231,327)(232,333)(233,332)(234,331)(235,351)
(236,350)(237,349)(238,348)(239,347)(240,346)(241,345)(242,344)(243,343)
(244,311)(245,310)(246,312)(247,308)(248,307)(249,309)(250,315)(251,314)
(252,313)(253,302)(254,301)(255,303)(256,299)(257,298)(258,300)(259,306)
(260,305)(261,304)(262,324)(263,323)(264,322)(265,321)(266,320)(267,319)
(268,318)(269,317)(270,316)(407,408)(409,413)(410,412)(411,414)(415,428)
(416,427)(417,429)(418,425)(419,424)(420,426)(421,432)(422,431)(423,430)
(433,514)(434,516)(435,515)(436,521)(437,520)(438,522)(439,518)(440,517)
(441,519)(442,536)(443,535)(444,537)(445,533)(446,532)(447,534)(448,540)
(449,539)(450,538)(451,527)(452,526)(453,528)(454,524)(455,523)(456,525)
(457,531)(458,530)(459,529)(460,487)(461,489)(462,488)(463,494)(464,493)
(465,495)(466,491)(467,490)(468,492)(469,509)(470,508)(471,510)(472,506)
(473,505)(474,507)(475,513)(476,512)(477,511)(478,500)(479,499)(480,501)
(481,497)(482,496)(483,498)(484,504)(485,503)(486,502)(541,689)(542,688)
(543,690)(544,686)(545,685)(546,687)(547,693)(548,692)(549,691)(550,680)
(551,679)(552,681)(553,677)(554,676)(555,678)(556,684)(557,683)(558,682)
(559,702)(560,701)(561,700)(562,699)(563,698)(564,697)(565,696)(566,695)
(567,694)(568,797)(569,796)(570,798)(571,794)(572,793)(573,795)(574,801)
(575,800)(576,799)(577,788)(578,787)(579,789)(580,785)(581,784)(582,786)
(583,792)(584,791)(585,790)(586,810)(587,809)(588,808)(589,807)(590,806)
(591,805)(592,804)(593,803)(594,802)(595,770)(596,769)(597,771)(598,767)
(599,766)(600,768)(601,774)(602,773)(603,772)(604,761)(605,760)(606,762)
(607,758)(608,757)(609,759)(610,765)(611,764)(612,763)(613,783)(614,782)
(615,781)(616,780)(617,779)(618,778)(619,777)(620,776)(621,775)(622,743)
(623,742)(624,744)(625,740)(626,739)(627,741)(628,747)(629,746)(630,745)
(631,734)(632,733)(633,735)(634,731)(635,730)(636,732)(637,738)(638,737)
(639,736)(640,756)(641,755)(642,754)(643,753)(644,752)(645,751)(646,750)
(647,749)(648,748)(649,716)(650,715)(651,717)(652,713)(653,712)(654,714)
(655,720)(656,719)(657,718)(658,707)(659,706)(660,708)(661,704)(662,703)
(663,705)(664,711)(665,710)(666,709)(667,729)(668,728)(669,727)(670,726)
(671,725)(672,724)(673,723)(674,722)(675,721);
s1 := Sym(810)!(  1,568)(  2,570)(  3,569)(  4,575)(  5,574)(  6,576)(  7,572)
(  8,571)(  9,573)( 10,590)( 11,589)( 12,591)( 13,587)( 14,586)( 15,588)
( 16,594)( 17,593)( 18,592)( 19,581)( 20,580)( 21,582)( 22,578)( 23,577)
( 24,579)( 25,585)( 26,584)( 27,583)( 28,541)( 29,543)( 30,542)( 31,548)
( 32,547)( 33,549)( 34,545)( 35,544)( 36,546)( 37,563)( 38,562)( 39,564)
( 40,560)( 41,559)( 42,561)( 43,567)( 44,566)( 45,565)( 46,554)( 47,553)
( 48,555)( 49,551)( 50,550)( 51,552)( 52,558)( 53,557)( 54,556)( 55,649)
( 56,651)( 57,650)( 58,656)( 59,655)( 60,657)( 61,653)( 62,652)( 63,654)
( 64,671)( 65,670)( 66,672)( 67,668)( 68,667)( 69,669)( 70,675)( 71,674)
( 72,673)( 73,662)( 74,661)( 75,663)( 76,659)( 77,658)( 78,660)( 79,666)
( 80,665)( 81,664)( 82,622)( 83,624)( 84,623)( 85,629)( 86,628)( 87,630)
( 88,626)( 89,625)( 90,627)( 91,644)( 92,643)( 93,645)( 94,641)( 95,640)
( 96,642)( 97,648)( 98,647)( 99,646)(100,635)(101,634)(102,636)(103,632)
(104,631)(105,633)(106,639)(107,638)(108,637)(109,595)(110,597)(111,596)
(112,602)(113,601)(114,603)(115,599)(116,598)(117,600)(118,617)(119,616)
(120,618)(121,614)(122,613)(123,615)(124,621)(125,620)(126,619)(127,608)
(128,607)(129,609)(130,605)(131,604)(132,606)(133,612)(134,611)(135,610)
(136,433)(137,435)(138,434)(139,440)(140,439)(141,441)(142,437)(143,436)
(144,438)(145,455)(146,454)(147,456)(148,452)(149,451)(150,453)(151,459)
(152,458)(153,457)(154,446)(155,445)(156,447)(157,443)(158,442)(159,444)
(160,450)(161,449)(162,448)(163,406)(164,408)(165,407)(166,413)(167,412)
(168,414)(169,410)(170,409)(171,411)(172,428)(173,427)(174,429)(175,425)
(176,424)(177,426)(178,432)(179,431)(180,430)(181,419)(182,418)(183,420)
(184,416)(185,415)(186,417)(187,423)(188,422)(189,421)(190,514)(191,516)
(192,515)(193,521)(194,520)(195,522)(196,518)(197,517)(198,519)(199,536)
(200,535)(201,537)(202,533)(203,532)(204,534)(205,540)(206,539)(207,538)
(208,527)(209,526)(210,528)(211,524)(212,523)(213,525)(214,531)(215,530)
(216,529)(217,487)(218,489)(219,488)(220,494)(221,493)(222,495)(223,491)
(224,490)(225,492)(226,509)(227,508)(228,510)(229,506)(230,505)(231,507)
(232,513)(233,512)(234,511)(235,500)(236,499)(237,501)(238,497)(239,496)
(240,498)(241,504)(242,503)(243,502)(244,460)(245,462)(246,461)(247,467)
(248,466)(249,468)(250,464)(251,463)(252,465)(253,482)(254,481)(255,483)
(256,479)(257,478)(258,480)(259,486)(260,485)(261,484)(262,473)(263,472)
(264,474)(265,470)(266,469)(267,471)(268,477)(269,476)(270,475)(271,716)
(272,715)(273,717)(274,713)(275,712)(276,714)(277,720)(278,719)(279,718)
(280,707)(281,706)(282,708)(283,704)(284,703)(285,705)(286,711)(287,710)
(288,709)(289,729)(290,728)(291,727)(292,726)(293,725)(294,724)(295,723)
(296,722)(297,721)(298,689)(299,688)(300,690)(301,686)(302,685)(303,687)
(304,693)(305,692)(306,691)(307,680)(308,679)(309,681)(310,677)(311,676)
(312,678)(313,684)(314,683)(315,682)(316,702)(317,701)(318,700)(319,699)
(320,698)(321,697)(322,696)(323,695)(324,694)(325,797)(326,796)(327,798)
(328,794)(329,793)(330,795)(331,801)(332,800)(333,799)(334,788)(335,787)
(336,789)(337,785)(338,784)(339,786)(340,792)(341,791)(342,790)(343,810)
(344,809)(345,808)(346,807)(347,806)(348,805)(349,804)(350,803)(351,802)
(352,770)(353,769)(354,771)(355,767)(356,766)(357,768)(358,774)(359,773)
(360,772)(361,761)(362,760)(363,762)(364,758)(365,757)(366,759)(367,765)
(368,764)(369,763)(370,783)(371,782)(372,781)(373,780)(374,779)(375,778)
(376,777)(377,776)(378,775)(379,743)(380,742)(381,744)(382,740)(383,739)
(384,741)(385,747)(386,746)(387,745)(388,734)(389,733)(390,735)(391,731)
(392,730)(393,732)(394,738)(395,737)(396,736)(397,756)(398,755)(399,754)
(400,753)(401,752)(402,751)(403,750)(404,749)(405,748);
poly := sub<Sym(810)|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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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