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Polytope of Type {105,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {105,4}*1680
if this polytope has a name.
Group : SmallGroup(1680,971)
Rank : 3
Schlafli Type : {105,4}
Number of vertices, edges, etc : 210, 420, 8
Order of s0s1s2 : 210
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {105,4}*840
   4-fold quotients : {105,2}*420
   5-fold quotients : {21,4}*336
   7-fold quotients : {15,4}*240
   10-fold quotients : {21,4}*168
   12-fold quotients : {35,2}*140
   14-fold quotients : {15,4}*120
   20-fold quotients : {21,2}*84
   28-fold quotients : {15,2}*60
   35-fold quotients : {3,4}*48
   60-fold quotients : {7,2}*28
   70-fold quotients : {3,4}*24
   84-fold quotients : {5,2}*20
   140-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  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);;
s1 := (  1,173)(  2,176)(  3,175)(  4,174)(  5,169)(  6,172)(  7,171)(  8,170)
(  9,193)( 10,196)( 11,195)( 12,194)( 13,189)( 14,192)( 15,191)( 16,190)
( 17,185)( 18,188)( 19,187)( 20,186)( 21,181)( 22,184)( 23,183)( 24,182)
( 25,177)( 26,180)( 27,179)( 28,178)( 29,145)( 30,148)( 31,147)( 32,146)
( 33,141)( 34,144)( 35,143)( 36,142)( 37,165)( 38,168)( 39,167)( 40,166)
( 41,161)( 42,164)( 43,163)( 44,162)( 45,157)( 46,160)( 47,159)( 48,158)
( 49,153)( 50,156)( 51,155)( 52,154)( 53,149)( 54,152)( 55,151)( 56,150)
( 57,257)( 58,260)( 59,259)( 60,258)( 61,253)( 62,256)( 63,255)( 64,254)
( 65,277)( 66,280)( 67,279)( 68,278)( 69,273)( 70,276)( 71,275)( 72,274)
( 73,269)( 74,272)( 75,271)( 76,270)( 77,265)( 78,268)( 79,267)( 80,266)
( 81,261)( 82,264)( 83,263)( 84,262)( 85,229)( 86,232)( 87,231)( 88,230)
( 89,225)( 90,228)( 91,227)( 92,226)( 93,249)( 94,252)( 95,251)( 96,250)
( 97,245)( 98,248)( 99,247)(100,246)(101,241)(102,244)(103,243)(104,242)
(105,237)(106,240)(107,239)(108,238)(109,233)(110,236)(111,235)(112,234)
(113,201)(114,204)(115,203)(116,202)(117,197)(118,200)(119,199)(120,198)
(121,221)(122,224)(123,223)(124,222)(125,217)(126,220)(127,219)(128,218)
(129,213)(130,216)(131,215)(132,214)(133,209)(134,212)(135,211)(136,210)
(137,205)(138,208)(139,207)(140,206)(281,313)(282,316)(283,315)(284,314)
(285,309)(286,312)(287,311)(288,310)(289,333)(290,336)(291,335)(292,334)
(293,329)(294,332)(295,331)(296,330)(297,325)(298,328)(299,327)(300,326)
(301,321)(302,324)(303,323)(304,322)(305,317)(306,320)(307,319)(308,318)
(337,397)(338,400)(339,399)(340,398)(341,393)(342,396)(343,395)(344,394)
(345,417)(346,420)(347,419)(348,418)(349,413)(350,416)(351,415)(352,414)
(353,409)(354,412)(355,411)(356,410)(357,405)(358,408)(359,407)(360,406)
(361,401)(362,404)(363,403)(364,402)(365,369)(366,372)(367,371)(368,370)
(373,389)(374,392)(375,391)(376,390)(377,385)(378,388)(379,387)(380,386)
(382,384)(421,593)(422,596)(423,595)(424,594)(425,589)(426,592)(427,591)
(428,590)(429,613)(430,616)(431,615)(432,614)(433,609)(434,612)(435,611)
(436,610)(437,605)(438,608)(439,607)(440,606)(441,601)(442,604)(443,603)
(444,602)(445,597)(446,600)(447,599)(448,598)(449,565)(450,568)(451,567)
(452,566)(453,561)(454,564)(455,563)(456,562)(457,585)(458,588)(459,587)
(460,586)(461,581)(462,584)(463,583)(464,582)(465,577)(466,580)(467,579)
(468,578)(469,573)(470,576)(471,575)(472,574)(473,569)(474,572)(475,571)
(476,570)(477,677)(478,680)(479,679)(480,678)(481,673)(482,676)(483,675)
(484,674)(485,697)(486,700)(487,699)(488,698)(489,693)(490,696)(491,695)
(492,694)(493,689)(494,692)(495,691)(496,690)(497,685)(498,688)(499,687)
(500,686)(501,681)(502,684)(503,683)(504,682)(505,649)(506,652)(507,651)
(508,650)(509,645)(510,648)(511,647)(512,646)(513,669)(514,672)(515,671)
(516,670)(517,665)(518,668)(519,667)(520,666)(521,661)(522,664)(523,663)
(524,662)(525,657)(526,660)(527,659)(528,658)(529,653)(530,656)(531,655)
(532,654)(533,621)(534,624)(535,623)(536,622)(537,617)(538,620)(539,619)
(540,618)(541,641)(542,644)(543,643)(544,642)(545,637)(546,640)(547,639)
(548,638)(549,633)(550,636)(551,635)(552,634)(553,629)(554,632)(555,631)
(556,630)(557,625)(558,628)(559,627)(560,626)(701,733)(702,736)(703,735)
(704,734)(705,729)(706,732)(707,731)(708,730)(709,753)(710,756)(711,755)
(712,754)(713,749)(714,752)(715,751)(716,750)(717,745)(718,748)(719,747)
(720,746)(721,741)(722,744)(723,743)(724,742)(725,737)(726,740)(727,739)
(728,738)(757,817)(758,820)(759,819)(760,818)(761,813)(762,816)(763,815)
(764,814)(765,837)(766,840)(767,839)(768,838)(769,833)(770,836)(771,835)
(772,834)(773,829)(774,832)(775,831)(776,830)(777,825)(778,828)(779,827)
(780,826)(781,821)(782,824)(783,823)(784,822)(785,789)(786,792)(787,791)
(788,790)(793,809)(794,812)(795,811)(796,810)(797,805)(798,808)(799,807)
(800,806)(802,804);;
s2 := (  1,422)(  2,421)(  3,424)(  4,423)(  5,426)(  6,425)(  7,428)(  8,427)
(  9,430)( 10,429)( 11,432)( 12,431)( 13,434)( 14,433)( 15,436)( 16,435)
( 17,438)( 18,437)( 19,440)( 20,439)( 21,442)( 22,441)( 23,444)( 24,443)
( 25,446)( 26,445)( 27,448)( 28,447)( 29,450)( 30,449)( 31,452)( 32,451)
( 33,454)( 34,453)( 35,456)( 36,455)( 37,458)( 38,457)( 39,460)( 40,459)
( 41,462)( 42,461)( 43,464)( 44,463)( 45,466)( 46,465)( 47,468)( 48,467)
( 49,470)( 50,469)( 51,472)( 52,471)( 53,474)( 54,473)( 55,476)( 56,475)
( 57,478)( 58,477)( 59,480)( 60,479)( 61,482)( 62,481)( 63,484)( 64,483)
( 65,486)( 66,485)( 67,488)( 68,487)( 69,490)( 70,489)( 71,492)( 72,491)
( 73,494)( 74,493)( 75,496)( 76,495)( 77,498)( 78,497)( 79,500)( 80,499)
( 81,502)( 82,501)( 83,504)( 84,503)( 85,506)( 86,505)( 87,508)( 88,507)
( 89,510)( 90,509)( 91,512)( 92,511)( 93,514)( 94,513)( 95,516)( 96,515)
( 97,518)( 98,517)( 99,520)(100,519)(101,522)(102,521)(103,524)(104,523)
(105,526)(106,525)(107,528)(108,527)(109,530)(110,529)(111,532)(112,531)
(113,534)(114,533)(115,536)(116,535)(117,538)(118,537)(119,540)(120,539)
(121,542)(122,541)(123,544)(124,543)(125,546)(126,545)(127,548)(128,547)
(129,550)(130,549)(131,552)(132,551)(133,554)(134,553)(135,556)(136,555)
(137,558)(138,557)(139,560)(140,559)(141,562)(142,561)(143,564)(144,563)
(145,566)(146,565)(147,568)(148,567)(149,570)(150,569)(151,572)(152,571)
(153,574)(154,573)(155,576)(156,575)(157,578)(158,577)(159,580)(160,579)
(161,582)(162,581)(163,584)(164,583)(165,586)(166,585)(167,588)(168,587)
(169,590)(170,589)(171,592)(172,591)(173,594)(174,593)(175,596)(176,595)
(177,598)(178,597)(179,600)(180,599)(181,602)(182,601)(183,604)(184,603)
(185,606)(186,605)(187,608)(188,607)(189,610)(190,609)(191,612)(192,611)
(193,614)(194,613)(195,616)(196,615)(197,618)(198,617)(199,620)(200,619)
(201,622)(202,621)(203,624)(204,623)(205,626)(206,625)(207,628)(208,627)
(209,630)(210,629)(211,632)(212,631)(213,634)(214,633)(215,636)(216,635)
(217,638)(218,637)(219,640)(220,639)(221,642)(222,641)(223,644)(224,643)
(225,646)(226,645)(227,648)(228,647)(229,650)(230,649)(231,652)(232,651)
(233,654)(234,653)(235,656)(236,655)(237,658)(238,657)(239,660)(240,659)
(241,662)(242,661)(243,664)(244,663)(245,666)(246,665)(247,668)(248,667)
(249,670)(250,669)(251,672)(252,671)(253,674)(254,673)(255,676)(256,675)
(257,678)(258,677)(259,680)(260,679)(261,682)(262,681)(263,684)(264,683)
(265,686)(266,685)(267,688)(268,687)(269,690)(270,689)(271,692)(272,691)
(273,694)(274,693)(275,696)(276,695)(277,698)(278,697)(279,700)(280,699)
(281,702)(282,701)(283,704)(284,703)(285,706)(286,705)(287,708)(288,707)
(289,710)(290,709)(291,712)(292,711)(293,714)(294,713)(295,716)(296,715)
(297,718)(298,717)(299,720)(300,719)(301,722)(302,721)(303,724)(304,723)
(305,726)(306,725)(307,728)(308,727)(309,730)(310,729)(311,732)(312,731)
(313,734)(314,733)(315,736)(316,735)(317,738)(318,737)(319,740)(320,739)
(321,742)(322,741)(323,744)(324,743)(325,746)(326,745)(327,748)(328,747)
(329,750)(330,749)(331,752)(332,751)(333,754)(334,753)(335,756)(336,755)
(337,758)(338,757)(339,760)(340,759)(341,762)(342,761)(343,764)(344,763)
(345,766)(346,765)(347,768)(348,767)(349,770)(350,769)(351,772)(352,771)
(353,774)(354,773)(355,776)(356,775)(357,778)(358,777)(359,780)(360,779)
(361,782)(362,781)(363,784)(364,783)(365,786)(366,785)(367,788)(368,787)
(369,790)(370,789)(371,792)(372,791)(373,794)(374,793)(375,796)(376,795)
(377,798)(378,797)(379,800)(380,799)(381,802)(382,801)(383,804)(384,803)
(385,806)(386,805)(387,808)(388,807)(389,810)(390,809)(391,812)(392,811)
(393,814)(394,813)(395,816)(396,815)(397,818)(398,817)(399,820)(400,819)
(401,822)(402,821)(403,824)(404,823)(405,826)(406,825)(407,828)(408,827)
(409,830)(410,829)(411,832)(412,831)(413,834)(414,833)(415,836)(416,835)
(417,838)(418,837)(419,840)(420,839);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(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);
s1 := Sym(840)!(  1,173)(  2,176)(  3,175)(  4,174)(  5,169)(  6,172)(  7,171)
(  8,170)(  9,193)( 10,196)( 11,195)( 12,194)( 13,189)( 14,192)( 15,191)
( 16,190)( 17,185)( 18,188)( 19,187)( 20,186)( 21,181)( 22,184)( 23,183)
( 24,182)( 25,177)( 26,180)( 27,179)( 28,178)( 29,145)( 30,148)( 31,147)
( 32,146)( 33,141)( 34,144)( 35,143)( 36,142)( 37,165)( 38,168)( 39,167)
( 40,166)( 41,161)( 42,164)( 43,163)( 44,162)( 45,157)( 46,160)( 47,159)
( 48,158)( 49,153)( 50,156)( 51,155)( 52,154)( 53,149)( 54,152)( 55,151)
( 56,150)( 57,257)( 58,260)( 59,259)( 60,258)( 61,253)( 62,256)( 63,255)
( 64,254)( 65,277)( 66,280)( 67,279)( 68,278)( 69,273)( 70,276)( 71,275)
( 72,274)( 73,269)( 74,272)( 75,271)( 76,270)( 77,265)( 78,268)( 79,267)
( 80,266)( 81,261)( 82,264)( 83,263)( 84,262)( 85,229)( 86,232)( 87,231)
( 88,230)( 89,225)( 90,228)( 91,227)( 92,226)( 93,249)( 94,252)( 95,251)
( 96,250)( 97,245)( 98,248)( 99,247)(100,246)(101,241)(102,244)(103,243)
(104,242)(105,237)(106,240)(107,239)(108,238)(109,233)(110,236)(111,235)
(112,234)(113,201)(114,204)(115,203)(116,202)(117,197)(118,200)(119,199)
(120,198)(121,221)(122,224)(123,223)(124,222)(125,217)(126,220)(127,219)
(128,218)(129,213)(130,216)(131,215)(132,214)(133,209)(134,212)(135,211)
(136,210)(137,205)(138,208)(139,207)(140,206)(281,313)(282,316)(283,315)
(284,314)(285,309)(286,312)(287,311)(288,310)(289,333)(290,336)(291,335)
(292,334)(293,329)(294,332)(295,331)(296,330)(297,325)(298,328)(299,327)
(300,326)(301,321)(302,324)(303,323)(304,322)(305,317)(306,320)(307,319)
(308,318)(337,397)(338,400)(339,399)(340,398)(341,393)(342,396)(343,395)
(344,394)(345,417)(346,420)(347,419)(348,418)(349,413)(350,416)(351,415)
(352,414)(353,409)(354,412)(355,411)(356,410)(357,405)(358,408)(359,407)
(360,406)(361,401)(362,404)(363,403)(364,402)(365,369)(366,372)(367,371)
(368,370)(373,389)(374,392)(375,391)(376,390)(377,385)(378,388)(379,387)
(380,386)(382,384)(421,593)(422,596)(423,595)(424,594)(425,589)(426,592)
(427,591)(428,590)(429,613)(430,616)(431,615)(432,614)(433,609)(434,612)
(435,611)(436,610)(437,605)(438,608)(439,607)(440,606)(441,601)(442,604)
(443,603)(444,602)(445,597)(446,600)(447,599)(448,598)(449,565)(450,568)
(451,567)(452,566)(453,561)(454,564)(455,563)(456,562)(457,585)(458,588)
(459,587)(460,586)(461,581)(462,584)(463,583)(464,582)(465,577)(466,580)
(467,579)(468,578)(469,573)(470,576)(471,575)(472,574)(473,569)(474,572)
(475,571)(476,570)(477,677)(478,680)(479,679)(480,678)(481,673)(482,676)
(483,675)(484,674)(485,697)(486,700)(487,699)(488,698)(489,693)(490,696)
(491,695)(492,694)(493,689)(494,692)(495,691)(496,690)(497,685)(498,688)
(499,687)(500,686)(501,681)(502,684)(503,683)(504,682)(505,649)(506,652)
(507,651)(508,650)(509,645)(510,648)(511,647)(512,646)(513,669)(514,672)
(515,671)(516,670)(517,665)(518,668)(519,667)(520,666)(521,661)(522,664)
(523,663)(524,662)(525,657)(526,660)(527,659)(528,658)(529,653)(530,656)
(531,655)(532,654)(533,621)(534,624)(535,623)(536,622)(537,617)(538,620)
(539,619)(540,618)(541,641)(542,644)(543,643)(544,642)(545,637)(546,640)
(547,639)(548,638)(549,633)(550,636)(551,635)(552,634)(553,629)(554,632)
(555,631)(556,630)(557,625)(558,628)(559,627)(560,626)(701,733)(702,736)
(703,735)(704,734)(705,729)(706,732)(707,731)(708,730)(709,753)(710,756)
(711,755)(712,754)(713,749)(714,752)(715,751)(716,750)(717,745)(718,748)
(719,747)(720,746)(721,741)(722,744)(723,743)(724,742)(725,737)(726,740)
(727,739)(728,738)(757,817)(758,820)(759,819)(760,818)(761,813)(762,816)
(763,815)(764,814)(765,837)(766,840)(767,839)(768,838)(769,833)(770,836)
(771,835)(772,834)(773,829)(774,832)(775,831)(776,830)(777,825)(778,828)
(779,827)(780,826)(781,821)(782,824)(783,823)(784,822)(785,789)(786,792)
(787,791)(788,790)(793,809)(794,812)(795,811)(796,810)(797,805)(798,808)
(799,807)(800,806)(802,804);
s2 := Sym(840)!(  1,422)(  2,421)(  3,424)(  4,423)(  5,426)(  6,425)(  7,428)
(  8,427)(  9,430)( 10,429)( 11,432)( 12,431)( 13,434)( 14,433)( 15,436)
( 16,435)( 17,438)( 18,437)( 19,440)( 20,439)( 21,442)( 22,441)( 23,444)
( 24,443)( 25,446)( 26,445)( 27,448)( 28,447)( 29,450)( 30,449)( 31,452)
( 32,451)( 33,454)( 34,453)( 35,456)( 36,455)( 37,458)( 38,457)( 39,460)
( 40,459)( 41,462)( 42,461)( 43,464)( 44,463)( 45,466)( 46,465)( 47,468)
( 48,467)( 49,470)( 50,469)( 51,472)( 52,471)( 53,474)( 54,473)( 55,476)
( 56,475)( 57,478)( 58,477)( 59,480)( 60,479)( 61,482)( 62,481)( 63,484)
( 64,483)( 65,486)( 66,485)( 67,488)( 68,487)( 69,490)( 70,489)( 71,492)
( 72,491)( 73,494)( 74,493)( 75,496)( 76,495)( 77,498)( 78,497)( 79,500)
( 80,499)( 81,502)( 82,501)( 83,504)( 84,503)( 85,506)( 86,505)( 87,508)
( 88,507)( 89,510)( 90,509)( 91,512)( 92,511)( 93,514)( 94,513)( 95,516)
( 96,515)( 97,518)( 98,517)( 99,520)(100,519)(101,522)(102,521)(103,524)
(104,523)(105,526)(106,525)(107,528)(108,527)(109,530)(110,529)(111,532)
(112,531)(113,534)(114,533)(115,536)(116,535)(117,538)(118,537)(119,540)
(120,539)(121,542)(122,541)(123,544)(124,543)(125,546)(126,545)(127,548)
(128,547)(129,550)(130,549)(131,552)(132,551)(133,554)(134,553)(135,556)
(136,555)(137,558)(138,557)(139,560)(140,559)(141,562)(142,561)(143,564)
(144,563)(145,566)(146,565)(147,568)(148,567)(149,570)(150,569)(151,572)
(152,571)(153,574)(154,573)(155,576)(156,575)(157,578)(158,577)(159,580)
(160,579)(161,582)(162,581)(163,584)(164,583)(165,586)(166,585)(167,588)
(168,587)(169,590)(170,589)(171,592)(172,591)(173,594)(174,593)(175,596)
(176,595)(177,598)(178,597)(179,600)(180,599)(181,602)(182,601)(183,604)
(184,603)(185,606)(186,605)(187,608)(188,607)(189,610)(190,609)(191,612)
(192,611)(193,614)(194,613)(195,616)(196,615)(197,618)(198,617)(199,620)
(200,619)(201,622)(202,621)(203,624)(204,623)(205,626)(206,625)(207,628)
(208,627)(209,630)(210,629)(211,632)(212,631)(213,634)(214,633)(215,636)
(216,635)(217,638)(218,637)(219,640)(220,639)(221,642)(222,641)(223,644)
(224,643)(225,646)(226,645)(227,648)(228,647)(229,650)(230,649)(231,652)
(232,651)(233,654)(234,653)(235,656)(236,655)(237,658)(238,657)(239,660)
(240,659)(241,662)(242,661)(243,664)(244,663)(245,666)(246,665)(247,668)
(248,667)(249,670)(250,669)(251,672)(252,671)(253,674)(254,673)(255,676)
(256,675)(257,678)(258,677)(259,680)(260,679)(261,682)(262,681)(263,684)
(264,683)(265,686)(266,685)(267,688)(268,687)(269,690)(270,689)(271,692)
(272,691)(273,694)(274,693)(275,696)(276,695)(277,698)(278,697)(279,700)
(280,699)(281,702)(282,701)(283,704)(284,703)(285,706)(286,705)(287,708)
(288,707)(289,710)(290,709)(291,712)(292,711)(293,714)(294,713)(295,716)
(296,715)(297,718)(298,717)(299,720)(300,719)(301,722)(302,721)(303,724)
(304,723)(305,726)(306,725)(307,728)(308,727)(309,730)(310,729)(311,732)
(312,731)(313,734)(314,733)(315,736)(316,735)(317,738)(318,737)(319,740)
(320,739)(321,742)(322,741)(323,744)(324,743)(325,746)(326,745)(327,748)
(328,747)(329,750)(330,749)(331,752)(332,751)(333,754)(334,753)(335,756)
(336,755)(337,758)(338,757)(339,760)(340,759)(341,762)(342,761)(343,764)
(344,763)(345,766)(346,765)(347,768)(348,767)(349,770)(350,769)(351,772)
(352,771)(353,774)(354,773)(355,776)(356,775)(357,778)(358,777)(359,780)
(360,779)(361,782)(362,781)(363,784)(364,783)(365,786)(366,785)(367,788)
(368,787)(369,790)(370,789)(371,792)(372,791)(373,794)(374,793)(375,796)
(376,795)(377,798)(378,797)(379,800)(380,799)(381,802)(382,801)(383,804)
(384,803)(385,806)(386,805)(387,808)(388,807)(389,810)(390,809)(391,812)
(392,811)(393,814)(394,813)(395,816)(396,815)(397,818)(398,817)(399,820)
(400,819)(401,822)(402,821)(403,824)(404,823)(405,826)(406,825)(407,828)
(408,827)(409,830)(410,829)(411,832)(412,831)(413,834)(414,833)(415,836)
(416,835)(417,838)(418,837)(419,840)(420,839);
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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