Polytope of Type {4,210}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,210}*1680b
if this polytope has a name.
Group : SmallGroup(1680,971)
Rank : 3
Schlafli Type : {4,210}
Number of vertices, edges, etc : 4, 420, 210
Order of s₀s₁s₂ : 210
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
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 : {4,105}*840
   5-fold quotients : {4,42}*336b
   7-fold quotients : {4,30}*240b
   10-fold quotients : {4,21}*168
   14-fold quotients : {4,15}*120
   35-fold quotients : {4,6}*48c
   70-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)(270,272)
(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)(286,288)
(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)(302,304)
(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)(318,320)
(321,323)(322,324)(325,327)(326,328)(329,331)(330,332)(333,335)(334,336)
(337,339)(338,340)(341,343)(342,344)(345,347)(346,348)(349,351)(350,352)
(353,355)(354,356)(357,359)(358,360)(361,363)(362,364)(365,367)(366,368)
(369,371)(370,372)(373,375)(374,376)(377,379)(378,380)(381,383)(382,384)
(385,387)(386,388)(389,391)(390,392)(393,395)(394,396)(397,399)(398,400)
(401,403)(402,404)(405,407)(406,408)(409,411)(410,412)(413,415)(414,416)
(417,419)(418,420)(421,423)(422,424)(425,427)(426,428)(429,431)(430,432)
(433,435)(434,436)(437,439)(438,440)(441,443)(442,444)(445,447)(446,448)
(449,451)(450,452)(453,455)(454,456)(457,459)(458,460)(461,463)(462,464)
(465,467)(466,468)(469,471)(470,472)(473,475)(474,476)(477,479)(478,480)
(481,483)(482,484)(485,487)(486,488)(489,491)(490,492)(493,495)(494,496)
(497,499)(498,500)(501,503)(502,504)(505,507)(506,508)(509,511)(510,512)
(513,515)(514,516)(517,519)(518,520)(521,523)(522,524)(525,527)(526,528)
(529,531)(530,532)(533,535)(534,536)(537,539)(538,540)(541,543)(542,544)
(545,547)(546,548)(549,551)(550,552)(553,555)(554,556)(557,559)(558,560)
(561,563)(562,564)(565,567)(566,568)(569,571)(570,572)(573,575)(574,576)
(577,579)(578,580)(581,583)(582,584)(585,587)(586,588)(589,591)(590,592)
(593,595)(594,596)(597,599)(598,600)(601,603)(602,604)(605,607)(606,608)
(609,611)(610,612)(613,615)(614,616)(617,619)(618,620)(621,623)(622,624)
(625,627)(626,628)(629,631)(630,632)(633,635)(634,636)(637,639)(638,640)
(641,643)(642,644)(645,647)(646,648)(649,651)(650,652)(653,655)(654,656)
(657,659)(658,660)(661,663)(662,664)(665,667)(666,668)(669,671)(670,672)
(673,675)(674,676)(677,679)(678,680)(681,683)(682,684)(685,687)(686,688)
(689,691)(690,692)(693,695)(694,696)(697,699)(698,700)(701,703)(702,704)
(705,707)(706,708)(709,711)(710,712)(713,715)(714,716)(717,719)(718,720)
(721,723)(722,724)(725,727)(726,728)(729,731)(730,732)(733,735)(734,736)
(737,739)(738,740)(741,743)(742,744)(745,747)(746,748)(749,751)(750,752)
(753,755)(754,756)(757,759)(758,760)(761,763)(762,764)(765,767)(766,768)
(769,771)(770,772)(773,775)(774,776)(777,779)(778,780)(781,783)(782,784)
(785,787)(786,788)(789,791)(790,792)(793,795)(794,796)(797,799)(798,800)
(801,803)(802,804)(805,807)(806,808)(809,811)(810,812)(813,815)(814,816)
(817,819)(818,820)(821,823)(822,824)(825,827)(826,828)(829,831)(830,832)
(833,835)(834,836)(837,839)(838,840);;
s1 := (  3,  4)(  5, 25)(  6, 26)(  7, 28)(  8, 27)(  9, 21)( 10, 22)( 11, 24)
( 12, 23)( 13, 17)( 14, 18)( 15, 20)( 16, 19)( 29,113)( 30,114)( 31,116)
( 32,115)( 33,137)( 34,138)( 35,140)( 36,139)( 37,133)( 38,134)( 39,136)
( 40,135)( 41,129)( 42,130)( 43,132)( 44,131)( 45,125)( 46,126)( 47,128)
( 48,127)( 49,121)( 50,122)( 51,124)( 52,123)( 53,117)( 54,118)( 55,120)
( 56,119)( 57, 85)( 58, 86)( 59, 88)( 60, 87)( 61,109)( 62,110)( 63,112)
( 64,111)( 65,105)( 66,106)( 67,108)( 68,107)( 69,101)( 70,102)( 71,104)
( 72,103)( 73, 97)( 74, 98)( 75,100)( 76, 99)( 77, 93)( 78, 94)( 79, 96)
( 80, 95)( 81, 89)( 82, 90)( 83, 92)( 84, 91)(141,281)(142,282)(143,284)
(144,283)(145,305)(146,306)(147,308)(148,307)(149,301)(150,302)(151,304)
(152,303)(153,297)(154,298)(155,300)(156,299)(157,293)(158,294)(159,296)
(160,295)(161,289)(162,290)(163,292)(164,291)(165,285)(166,286)(167,288)
(168,287)(169,393)(170,394)(171,396)(172,395)(173,417)(174,418)(175,420)
(176,419)(177,413)(178,414)(179,416)(180,415)(181,409)(182,410)(183,412)
(184,411)(185,405)(186,406)(187,408)(188,407)(189,401)(190,402)(191,404)
(192,403)(193,397)(194,398)(195,400)(196,399)(197,365)(198,366)(199,368)
(200,367)(201,389)(202,390)(203,392)(204,391)(205,385)(206,386)(207,388)
(208,387)(209,381)(210,382)(211,384)(212,383)(213,377)(214,378)(215,380)
(216,379)(217,373)(218,374)(219,376)(220,375)(221,369)(222,370)(223,372)
(224,371)(225,337)(226,338)(227,340)(228,339)(229,361)(230,362)(231,364)
(232,363)(233,357)(234,358)(235,360)(236,359)(237,353)(238,354)(239,356)
(240,355)(241,349)(242,350)(243,352)(244,351)(245,345)(246,346)(247,348)
(248,347)(249,341)(250,342)(251,344)(252,343)(253,309)(254,310)(255,312)
(256,311)(257,333)(258,334)(259,336)(260,335)(261,329)(262,330)(263,332)
(264,331)(265,325)(266,326)(267,328)(268,327)(269,321)(270,322)(271,324)
(272,323)(273,317)(274,318)(275,320)(276,319)(277,313)(278,314)(279,316)
(280,315)(423,424)(425,445)(426,446)(427,448)(428,447)(429,441)(430,442)
(431,444)(432,443)(433,437)(434,438)(435,440)(436,439)(449,533)(450,534)
(451,536)(452,535)(453,557)(454,558)(455,560)(456,559)(457,553)(458,554)
(459,556)(460,555)(461,549)(462,550)(463,552)(464,551)(465,545)(466,546)
(467,548)(468,547)(469,541)(470,542)(471,544)(472,543)(473,537)(474,538)
(475,540)(476,539)(477,505)(478,506)(479,508)(480,507)(481,529)(482,530)
(483,532)(484,531)(485,525)(486,526)(487,528)(488,527)(489,521)(490,522)
(491,524)(492,523)(493,517)(494,518)(495,520)(496,519)(497,513)(498,514)
(499,516)(500,515)(501,509)(502,510)(503,512)(504,511)(561,701)(562,702)
(563,704)(564,703)(565,725)(566,726)(567,728)(568,727)(569,721)(570,722)
(571,724)(572,723)(573,717)(574,718)(575,720)(576,719)(577,713)(578,714)
(579,716)(580,715)(581,709)(582,710)(583,712)(584,711)(585,705)(586,706)
(587,708)(588,707)(589,813)(590,814)(591,816)(592,815)(593,837)(594,838)
(595,840)(596,839)(597,833)(598,834)(599,836)(600,835)(601,829)(602,830)
(603,832)(604,831)(605,825)(606,826)(607,828)(608,827)(609,821)(610,822)
(611,824)(612,823)(613,817)(614,818)(615,820)(616,819)(617,785)(618,786)
(619,788)(620,787)(621,809)(622,810)(623,812)(624,811)(625,805)(626,806)
(627,808)(628,807)(629,801)(630,802)(631,804)(632,803)(633,797)(634,798)
(635,800)(636,799)(637,793)(638,794)(639,796)(640,795)(641,789)(642,790)
(643,792)(644,791)(645,757)(646,758)(647,760)(648,759)(649,781)(650,782)
(651,784)(652,783)(653,777)(654,778)(655,780)(656,779)(657,773)(658,774)
(659,776)(660,775)(661,769)(662,770)(663,772)(664,771)(665,765)(666,766)
(667,768)(668,767)(669,761)(670,762)(671,764)(672,763)(673,729)(674,730)
(675,732)(676,731)(677,753)(678,754)(679,756)(680,755)(681,749)(682,750)
(683,752)(684,751)(685,745)(686,746)(687,748)(688,747)(689,741)(690,742)
(691,744)(692,743)(693,737)(694,738)(695,740)(696,739)(697,733)(698,734)
(699,736)(700,735);;
s2 := (  1,593)(  2,596)(  3,595)(  4,594)(  5,589)(  6,592)(  7,591)(  8,590)
(  9,613)( 10,616)( 11,615)( 12,614)( 13,609)( 14,612)( 15,611)( 16,610)
( 17,605)( 18,608)( 19,607)( 20,606)( 21,601)( 22,604)( 23,603)( 24,602)
( 25,597)( 26,600)( 27,599)( 28,598)( 29,565)( 30,568)( 31,567)( 32,566)
( 33,561)( 34,564)( 35,563)( 36,562)( 37,585)( 38,588)( 39,587)( 40,586)
( 41,581)( 42,584)( 43,583)( 44,582)( 45,577)( 46,580)( 47,579)( 48,578)
( 49,573)( 50,576)( 51,575)( 52,574)( 53,569)( 54,572)( 55,571)( 56,570)
( 57,677)( 58,680)( 59,679)( 60,678)( 61,673)( 62,676)( 63,675)( 64,674)
( 65,697)( 66,700)( 67,699)( 68,698)( 69,693)( 70,696)( 71,695)( 72,694)
( 73,689)( 74,692)( 75,691)( 76,690)( 77,685)( 78,688)( 79,687)( 80,686)
( 81,681)( 82,684)( 83,683)( 84,682)( 85,649)( 86,652)( 87,651)( 88,650)
( 89,645)( 90,648)( 91,647)( 92,646)( 93,669)( 94,672)( 95,671)( 96,670)
( 97,665)( 98,668)( 99,667)(100,666)(101,661)(102,664)(103,663)(104,662)
(105,657)(106,660)(107,659)(108,658)(109,653)(110,656)(111,655)(112,654)
(113,621)(114,624)(115,623)(116,622)(117,617)(118,620)(119,619)(120,618)
(121,641)(122,644)(123,643)(124,642)(125,637)(126,640)(127,639)(128,638)
(129,633)(130,636)(131,635)(132,634)(133,629)(134,632)(135,631)(136,630)
(137,625)(138,628)(139,627)(140,626)(141,453)(142,456)(143,455)(144,454)
(145,449)(146,452)(147,451)(148,450)(149,473)(150,476)(151,475)(152,474)
(153,469)(154,472)(155,471)(156,470)(157,465)(158,468)(159,467)(160,466)
(161,461)(162,464)(163,463)(164,462)(165,457)(166,460)(167,459)(168,458)
(169,425)(170,428)(171,427)(172,426)(173,421)(174,424)(175,423)(176,422)
(177,445)(178,448)(179,447)(180,446)(181,441)(182,444)(183,443)(184,442)
(185,437)(186,440)(187,439)(188,438)(189,433)(190,436)(191,435)(192,434)
(193,429)(194,432)(195,431)(196,430)(197,537)(198,540)(199,539)(200,538)
(201,533)(202,536)(203,535)(204,534)(205,557)(206,560)(207,559)(208,558)
(209,553)(210,556)(211,555)(212,554)(213,549)(214,552)(215,551)(216,550)
(217,545)(218,548)(219,547)(220,546)(221,541)(222,544)(223,543)(224,542)
(225,509)(226,512)(227,511)(228,510)(229,505)(230,508)(231,507)(232,506)
(233,529)(234,532)(235,531)(236,530)(237,525)(238,528)(239,527)(240,526)
(241,521)(242,524)(243,523)(244,522)(245,517)(246,520)(247,519)(248,518)
(249,513)(250,516)(251,515)(252,514)(253,481)(254,484)(255,483)(256,482)
(257,477)(258,480)(259,479)(260,478)(261,501)(262,504)(263,503)(264,502)
(265,497)(266,500)(267,499)(268,498)(269,493)(270,496)(271,495)(272,494)
(273,489)(274,492)(275,491)(276,490)(277,485)(278,488)(279,487)(280,486)
(281,733)(282,736)(283,735)(284,734)(285,729)(286,732)(287,731)(288,730)
(289,753)(290,756)(291,755)(292,754)(293,749)(294,752)(295,751)(296,750)
(297,745)(298,748)(299,747)(300,746)(301,741)(302,744)(303,743)(304,742)
(305,737)(306,740)(307,739)(308,738)(309,705)(310,708)(311,707)(312,706)
(313,701)(314,704)(315,703)(316,702)(317,725)(318,728)(319,727)(320,726)
(321,721)(322,724)(323,723)(324,722)(325,717)(326,720)(327,719)(328,718)
(329,713)(330,716)(331,715)(332,714)(333,709)(334,712)(335,711)(336,710)
(337,817)(338,820)(339,819)(340,818)(341,813)(342,816)(343,815)(344,814)
(345,837)(346,840)(347,839)(348,838)(349,833)(350,836)(351,835)(352,834)
(353,829)(354,832)(355,831)(356,830)(357,825)(358,828)(359,827)(360,826)
(361,821)(362,824)(363,823)(364,822)(365,789)(366,792)(367,791)(368,790)
(369,785)(370,788)(371,787)(372,786)(373,809)(374,812)(375,811)(376,810)
(377,805)(378,808)(379,807)(380,806)(381,801)(382,804)(383,803)(384,802)
(385,797)(386,800)(387,799)(388,798)(389,793)(390,796)(391,795)(392,794)
(393,761)(394,764)(395,763)(396,762)(397,757)(398,760)(399,759)(400,758)
(401,781)(402,784)(403,783)(404,782)(405,777)(406,780)(407,779)(408,778)
(409,773)(410,776)(411,775)(412,774)(413,769)(414,772)(415,771)(416,770)
(417,765)(418,768)(419,767)(420,766);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(840)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)
(238,240)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)
(270,272)(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)
(286,288)(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)
(302,304)(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)
(318,320)(321,323)(322,324)(325,327)(326,328)(329,331)(330,332)(333,335)
(334,336)(337,339)(338,340)(341,343)(342,344)(345,347)(346,348)(349,351)
(350,352)(353,355)(354,356)(357,359)(358,360)(361,363)(362,364)(365,367)
(366,368)(369,371)(370,372)(373,375)(374,376)(377,379)(378,380)(381,383)
(382,384)(385,387)(386,388)(389,391)(390,392)(393,395)(394,396)(397,399)
(398,400)(401,403)(402,404)(405,407)(406,408)(409,411)(410,412)(413,415)
(414,416)(417,419)(418,420)(421,423)(422,424)(425,427)(426,428)(429,431)
(430,432)(433,435)(434,436)(437,439)(438,440)(441,443)(442,444)(445,447)
(446,448)(449,451)(450,452)(453,455)(454,456)(457,459)(458,460)(461,463)
(462,464)(465,467)(466,468)(469,471)(470,472)(473,475)(474,476)(477,479)
(478,480)(481,483)(482,484)(485,487)(486,488)(489,491)(490,492)(493,495)
(494,496)(497,499)(498,500)(501,503)(502,504)(505,507)(506,508)(509,511)
(510,512)(513,515)(514,516)(517,519)(518,520)(521,523)(522,524)(525,527)
(526,528)(529,531)(530,532)(533,535)(534,536)(537,539)(538,540)(541,543)
(542,544)(545,547)(546,548)(549,551)(550,552)(553,555)(554,556)(557,559)
(558,560)(561,563)(562,564)(565,567)(566,568)(569,571)(570,572)(573,575)
(574,576)(577,579)(578,580)(581,583)(582,584)(585,587)(586,588)(589,591)
(590,592)(593,595)(594,596)(597,599)(598,600)(601,603)(602,604)(605,607)
(606,608)(609,611)(610,612)(613,615)(614,616)(617,619)(618,620)(621,623)
(622,624)(625,627)(626,628)(629,631)(630,632)(633,635)(634,636)(637,639)
(638,640)(641,643)(642,644)(645,647)(646,648)(649,651)(650,652)(653,655)
(654,656)(657,659)(658,660)(661,663)(662,664)(665,667)(666,668)(669,671)
(670,672)(673,675)(674,676)(677,679)(678,680)(681,683)(682,684)(685,687)
(686,688)(689,691)(690,692)(693,695)(694,696)(697,699)(698,700)(701,703)
(702,704)(705,707)(706,708)(709,711)(710,712)(713,715)(714,716)(717,719)
(718,720)(721,723)(722,724)(725,727)(726,728)(729,731)(730,732)(733,735)
(734,736)(737,739)(738,740)(741,743)(742,744)(745,747)(746,748)(749,751)
(750,752)(753,755)(754,756)(757,759)(758,760)(761,763)(762,764)(765,767)
(766,768)(769,771)(770,772)(773,775)(774,776)(777,779)(778,780)(781,783)
(782,784)(785,787)(786,788)(789,791)(790,792)(793,795)(794,796)(797,799)
(798,800)(801,803)(802,804)(805,807)(806,808)(809,811)(810,812)(813,815)
(814,816)(817,819)(818,820)(821,823)(822,824)(825,827)(826,828)(829,831)
(830,832)(833,835)(834,836)(837,839)(838,840);
s1 := Sym(840)!(  3,  4)(  5, 25)(  6, 26)(  7, 28)(  8, 27)(  9, 21)( 10, 22)
( 11, 24)( 12, 23)( 13, 17)( 14, 18)( 15, 20)( 16, 19)( 29,113)( 30,114)
( 31,116)( 32,115)( 33,137)( 34,138)( 35,140)( 36,139)( 37,133)( 38,134)
( 39,136)( 40,135)( 41,129)( 42,130)( 43,132)( 44,131)( 45,125)( 46,126)
( 47,128)( 48,127)( 49,121)( 50,122)( 51,124)( 52,123)( 53,117)( 54,118)
( 55,120)( 56,119)( 57, 85)( 58, 86)( 59, 88)( 60, 87)( 61,109)( 62,110)
( 63,112)( 64,111)( 65,105)( 66,106)( 67,108)( 68,107)( 69,101)( 70,102)
( 71,104)( 72,103)( 73, 97)( 74, 98)( 75,100)( 76, 99)( 77, 93)( 78, 94)
( 79, 96)( 80, 95)( 81, 89)( 82, 90)( 83, 92)( 84, 91)(141,281)(142,282)
(143,284)(144,283)(145,305)(146,306)(147,308)(148,307)(149,301)(150,302)
(151,304)(152,303)(153,297)(154,298)(155,300)(156,299)(157,293)(158,294)
(159,296)(160,295)(161,289)(162,290)(163,292)(164,291)(165,285)(166,286)
(167,288)(168,287)(169,393)(170,394)(171,396)(172,395)(173,417)(174,418)
(175,420)(176,419)(177,413)(178,414)(179,416)(180,415)(181,409)(182,410)
(183,412)(184,411)(185,405)(186,406)(187,408)(188,407)(189,401)(190,402)
(191,404)(192,403)(193,397)(194,398)(195,400)(196,399)(197,365)(198,366)
(199,368)(200,367)(201,389)(202,390)(203,392)(204,391)(205,385)(206,386)
(207,388)(208,387)(209,381)(210,382)(211,384)(212,383)(213,377)(214,378)
(215,380)(216,379)(217,373)(218,374)(219,376)(220,375)(221,369)(222,370)
(223,372)(224,371)(225,337)(226,338)(227,340)(228,339)(229,361)(230,362)
(231,364)(232,363)(233,357)(234,358)(235,360)(236,359)(237,353)(238,354)
(239,356)(240,355)(241,349)(242,350)(243,352)(244,351)(245,345)(246,346)
(247,348)(248,347)(249,341)(250,342)(251,344)(252,343)(253,309)(254,310)
(255,312)(256,311)(257,333)(258,334)(259,336)(260,335)(261,329)(262,330)
(263,332)(264,331)(265,325)(266,326)(267,328)(268,327)(269,321)(270,322)
(271,324)(272,323)(273,317)(274,318)(275,320)(276,319)(277,313)(278,314)
(279,316)(280,315)(423,424)(425,445)(426,446)(427,448)(428,447)(429,441)
(430,442)(431,444)(432,443)(433,437)(434,438)(435,440)(436,439)(449,533)
(450,534)(451,536)(452,535)(453,557)(454,558)(455,560)(456,559)(457,553)
(458,554)(459,556)(460,555)(461,549)(462,550)(463,552)(464,551)(465,545)
(466,546)(467,548)(468,547)(469,541)(470,542)(471,544)(472,543)(473,537)
(474,538)(475,540)(476,539)(477,505)(478,506)(479,508)(480,507)(481,529)
(482,530)(483,532)(484,531)(485,525)(486,526)(487,528)(488,527)(489,521)
(490,522)(491,524)(492,523)(493,517)(494,518)(495,520)(496,519)(497,513)
(498,514)(499,516)(500,515)(501,509)(502,510)(503,512)(504,511)(561,701)
(562,702)(563,704)(564,703)(565,725)(566,726)(567,728)(568,727)(569,721)
(570,722)(571,724)(572,723)(573,717)(574,718)(575,720)(576,719)(577,713)
(578,714)(579,716)(580,715)(581,709)(582,710)(583,712)(584,711)(585,705)
(586,706)(587,708)(588,707)(589,813)(590,814)(591,816)(592,815)(593,837)
(594,838)(595,840)(596,839)(597,833)(598,834)(599,836)(600,835)(601,829)
(602,830)(603,832)(604,831)(605,825)(606,826)(607,828)(608,827)(609,821)
(610,822)(611,824)(612,823)(613,817)(614,818)(615,820)(616,819)(617,785)
(618,786)(619,788)(620,787)(621,809)(622,810)(623,812)(624,811)(625,805)
(626,806)(627,808)(628,807)(629,801)(630,802)(631,804)(632,803)(633,797)
(634,798)(635,800)(636,799)(637,793)(638,794)(639,796)(640,795)(641,789)
(642,790)(643,792)(644,791)(645,757)(646,758)(647,760)(648,759)(649,781)
(650,782)(651,784)(652,783)(653,777)(654,778)(655,780)(656,779)(657,773)
(658,774)(659,776)(660,775)(661,769)(662,770)(663,772)(664,771)(665,765)
(666,766)(667,768)(668,767)(669,761)(670,762)(671,764)(672,763)(673,729)
(674,730)(675,732)(676,731)(677,753)(678,754)(679,756)(680,755)(681,749)
(682,750)(683,752)(684,751)(685,745)(686,746)(687,748)(688,747)(689,741)
(690,742)(691,744)(692,743)(693,737)(694,738)(695,740)(696,739)(697,733)
(698,734)(699,736)(700,735);
s2 := Sym(840)!(  1,593)(  2,596)(  3,595)(  4,594)(  5,589)(  6,592)(  7,591)
(  8,590)(  9,613)( 10,616)( 11,615)( 12,614)( 13,609)( 14,612)( 15,611)
( 16,610)( 17,605)( 18,608)( 19,607)( 20,606)( 21,601)( 22,604)( 23,603)
( 24,602)( 25,597)( 26,600)( 27,599)( 28,598)( 29,565)( 30,568)( 31,567)
( 32,566)( 33,561)( 34,564)( 35,563)( 36,562)( 37,585)( 38,588)( 39,587)
( 40,586)( 41,581)( 42,584)( 43,583)( 44,582)( 45,577)( 46,580)( 47,579)
( 48,578)( 49,573)( 50,576)( 51,575)( 52,574)( 53,569)( 54,572)( 55,571)
( 56,570)( 57,677)( 58,680)( 59,679)( 60,678)( 61,673)( 62,676)( 63,675)
( 64,674)( 65,697)( 66,700)( 67,699)( 68,698)( 69,693)( 70,696)( 71,695)
( 72,694)( 73,689)( 74,692)( 75,691)( 76,690)( 77,685)( 78,688)( 79,687)
( 80,686)( 81,681)( 82,684)( 83,683)( 84,682)( 85,649)( 86,652)( 87,651)
( 88,650)( 89,645)( 90,648)( 91,647)( 92,646)( 93,669)( 94,672)( 95,671)
( 96,670)( 97,665)( 98,668)( 99,667)(100,666)(101,661)(102,664)(103,663)
(104,662)(105,657)(106,660)(107,659)(108,658)(109,653)(110,656)(111,655)
(112,654)(113,621)(114,624)(115,623)(116,622)(117,617)(118,620)(119,619)
(120,618)(121,641)(122,644)(123,643)(124,642)(125,637)(126,640)(127,639)
(128,638)(129,633)(130,636)(131,635)(132,634)(133,629)(134,632)(135,631)
(136,630)(137,625)(138,628)(139,627)(140,626)(141,453)(142,456)(143,455)
(144,454)(145,449)(146,452)(147,451)(148,450)(149,473)(150,476)(151,475)
(152,474)(153,469)(154,472)(155,471)(156,470)(157,465)(158,468)(159,467)
(160,466)(161,461)(162,464)(163,463)(164,462)(165,457)(166,460)(167,459)
(168,458)(169,425)(170,428)(171,427)(172,426)(173,421)(174,424)(175,423)
(176,422)(177,445)(178,448)(179,447)(180,446)(181,441)(182,444)(183,443)
(184,442)(185,437)(186,440)(187,439)(188,438)(189,433)(190,436)(191,435)
(192,434)(193,429)(194,432)(195,431)(196,430)(197,537)(198,540)(199,539)
(200,538)(201,533)(202,536)(203,535)(204,534)(205,557)(206,560)(207,559)
(208,558)(209,553)(210,556)(211,555)(212,554)(213,549)(214,552)(215,551)
(216,550)(217,545)(218,548)(219,547)(220,546)(221,541)(222,544)(223,543)
(224,542)(225,509)(226,512)(227,511)(228,510)(229,505)(230,508)(231,507)
(232,506)(233,529)(234,532)(235,531)(236,530)(237,525)(238,528)(239,527)
(240,526)(241,521)(242,524)(243,523)(244,522)(245,517)(246,520)(247,519)
(248,518)(249,513)(250,516)(251,515)(252,514)(253,481)(254,484)(255,483)
(256,482)(257,477)(258,480)(259,479)(260,478)(261,501)(262,504)(263,503)
(264,502)(265,497)(266,500)(267,499)(268,498)(269,493)(270,496)(271,495)
(272,494)(273,489)(274,492)(275,491)(276,490)(277,485)(278,488)(279,487)
(280,486)(281,733)(282,736)(283,735)(284,734)(285,729)(286,732)(287,731)
(288,730)(289,753)(290,756)(291,755)(292,754)(293,749)(294,752)(295,751)
(296,750)(297,745)(298,748)(299,747)(300,746)(301,741)(302,744)(303,743)
(304,742)(305,737)(306,740)(307,739)(308,738)(309,705)(310,708)(311,707)
(312,706)(313,701)(314,704)(315,703)(316,702)(317,725)(318,728)(319,727)
(320,726)(321,721)(322,724)(323,723)(324,722)(325,717)(326,720)(327,719)
(328,718)(329,713)(330,716)(331,715)(332,714)(333,709)(334,712)(335,711)
(336,710)(337,817)(338,820)(339,819)(340,818)(341,813)(342,816)(343,815)
(344,814)(345,837)(346,840)(347,839)(348,838)(349,833)(350,836)(351,835)
(352,834)(353,829)(354,832)(355,831)(356,830)(357,825)(358,828)(359,827)
(360,826)(361,821)(362,824)(363,823)(364,822)(365,789)(366,792)(367,791)
(368,790)(369,785)(370,788)(371,787)(372,786)(373,809)(374,812)(375,811)
(376,810)(377,805)(378,808)(379,807)(380,806)(381,801)(382,804)(383,803)
(384,802)(385,797)(386,800)(387,799)(388,798)(389,793)(390,796)(391,795)
(392,794)(393,761)(394,764)(395,763)(396,762)(397,757)(398,760)(399,759)
(400,758)(401,781)(402,784)(403,783)(404,782)(405,777)(406,780)(407,779)
(408,778)(409,773)(410,776)(411,775)(412,774)(413,769)(414,772)(415,771)
(416,770)(417,765)(418,768)(419,767)(420,766);
poly := sub<Sym(840)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope