Polytope of Type {214,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {214,4}*1712
Also Known As : {214,4|2}. if this polytope has another name.
Group : SmallGroup(1712,34)
Rank : 3
Schlafli Type : {214,4}
Number of vertices, edges, etc : 214, 428, 4
Order of s₀s₁s₂ : 428
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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 : {214,2}*856
   4-fold quotients : {107,2}*428
   107-fold quotients : {2,4}*16
   214-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,215)(  2,321)(  3,320)(  4,319)(  5,318)(  6,317)(  7,316)(  8,315)
(  9,314)( 10,313)( 11,312)( 12,311)( 13,310)( 14,309)( 15,308)( 16,307)
( 17,306)( 18,305)( 19,304)( 20,303)( 21,302)( 22,301)( 23,300)( 24,299)
( 25,298)( 26,297)( 27,296)( 28,295)( 29,294)( 30,293)( 31,292)( 32,291)
( 33,290)( 34,289)( 35,288)( 36,287)( 37,286)( 38,285)( 39,284)( 40,283)
( 41,282)( 42,281)( 43,280)( 44,279)( 45,278)( 46,277)( 47,276)( 48,275)
( 49,274)( 50,273)( 51,272)( 52,271)( 53,270)( 54,269)( 55,268)( 56,267)
( 57,266)( 58,265)( 59,264)( 60,263)( 61,262)( 62,261)( 63,260)( 64,259)
( 65,258)( 66,257)( 67,256)( 68,255)( 69,254)( 70,253)( 71,252)( 72,251)
( 73,250)( 74,249)( 75,248)( 76,247)( 77,246)( 78,245)( 79,244)( 80,243)
( 81,242)( 82,241)( 83,240)( 84,239)( 85,238)( 86,237)( 87,236)( 88,235)
( 89,234)( 90,233)( 91,232)( 92,231)( 93,230)( 94,229)( 95,228)( 96,227)
( 97,226)( 98,225)( 99,224)(100,223)(101,222)(102,221)(103,220)(104,219)
(105,218)(106,217)(107,216)(108,322)(109,428)(110,427)(111,426)(112,425)
(113,424)(114,423)(115,422)(116,421)(117,420)(118,419)(119,418)(120,417)
(121,416)(122,415)(123,414)(124,413)(125,412)(126,411)(127,410)(128,409)
(129,408)(130,407)(131,406)(132,405)(133,404)(134,403)(135,402)(136,401)
(137,400)(138,399)(139,398)(140,397)(141,396)(142,395)(143,394)(144,393)
(145,392)(146,391)(147,390)(148,389)(149,388)(150,387)(151,386)(152,385)
(153,384)(154,383)(155,382)(156,381)(157,380)(158,379)(159,378)(160,377)
(161,376)(162,375)(163,374)(164,373)(165,372)(166,371)(167,370)(168,369)
(169,368)(170,367)(171,366)(172,365)(173,364)(174,363)(175,362)(176,361)
(177,360)(178,359)(179,358)(180,357)(181,356)(182,355)(183,354)(184,353)
(185,352)(186,351)(187,350)(188,349)(189,348)(190,347)(191,346)(192,345)
(193,344)(194,343)(195,342)(196,341)(197,340)(198,339)(199,338)(200,337)
(201,336)(202,335)(203,334)(204,333)(205,332)(206,331)(207,330)(208,329)
(209,328)(210,327)(211,326)(212,325)(213,324)(214,323)(429,643)(430,749)
(431,748)(432,747)(433,746)(434,745)(435,744)(436,743)(437,742)(438,741)
(439,740)(440,739)(441,738)(442,737)(443,736)(444,735)(445,734)(446,733)
(447,732)(448,731)(449,730)(450,729)(451,728)(452,727)(453,726)(454,725)
(455,724)(456,723)(457,722)(458,721)(459,720)(460,719)(461,718)(462,717)
(463,716)(464,715)(465,714)(466,713)(467,712)(468,711)(469,710)(470,709)
(471,708)(472,707)(473,706)(474,705)(475,704)(476,703)(477,702)(478,701)
(479,700)(480,699)(481,698)(482,697)(483,696)(484,695)(485,694)(486,693)
(487,692)(488,691)(489,690)(490,689)(491,688)(492,687)(493,686)(494,685)
(495,684)(496,683)(497,682)(498,681)(499,680)(500,679)(501,678)(502,677)
(503,676)(504,675)(505,674)(506,673)(507,672)(508,671)(509,670)(510,669)
(511,668)(512,667)(513,666)(514,665)(515,664)(516,663)(517,662)(518,661)
(519,660)(520,659)(521,658)(522,657)(523,656)(524,655)(525,654)(526,653)
(527,652)(528,651)(529,650)(530,649)(531,648)(532,647)(533,646)(534,645)
(535,644)(536,750)(537,856)(538,855)(539,854)(540,853)(541,852)(542,851)
(543,850)(544,849)(545,848)(546,847)(547,846)(548,845)(549,844)(550,843)
(551,842)(552,841)(553,840)(554,839)(555,838)(556,837)(557,836)(558,835)
(559,834)(560,833)(561,832)(562,831)(563,830)(564,829)(565,828)(566,827)
(567,826)(568,825)(569,824)(570,823)(571,822)(572,821)(573,820)(574,819)
(575,818)(576,817)(577,816)(578,815)(579,814)(580,813)(581,812)(582,811)
(583,810)(584,809)(585,808)(586,807)(587,806)(588,805)(589,804)(590,803)
(591,802)(592,801)(593,800)(594,799)(595,798)(596,797)(597,796)(598,795)
(599,794)(600,793)(601,792)(602,791)(603,790)(604,789)(605,788)(606,787)
(607,786)(608,785)(609,784)(610,783)(611,782)(612,781)(613,780)(614,779)
(615,778)(616,777)(617,776)(618,775)(619,774)(620,773)(621,772)(622,771)
(623,770)(624,769)(625,768)(626,767)(627,766)(628,765)(629,764)(630,763)
(631,762)(632,761)(633,760)(634,759)(635,758)(636,757)(637,756)(638,755)
(639,754)(640,753)(641,752)(642,751);;
s1 := (  1,216)(  2,215)(  3,321)(  4,320)(  5,319)(  6,318)(  7,317)(  8,316)
(  9,315)( 10,314)( 11,313)( 12,312)( 13,311)( 14,310)( 15,309)( 16,308)
( 17,307)( 18,306)( 19,305)( 20,304)( 21,303)( 22,302)( 23,301)( 24,300)
( 25,299)( 26,298)( 27,297)( 28,296)( 29,295)( 30,294)( 31,293)( 32,292)
( 33,291)( 34,290)( 35,289)( 36,288)( 37,287)( 38,286)( 39,285)( 40,284)
( 41,283)( 42,282)( 43,281)( 44,280)( 45,279)( 46,278)( 47,277)( 48,276)
( 49,275)( 50,274)( 51,273)( 52,272)( 53,271)( 54,270)( 55,269)( 56,268)
( 57,267)( 58,266)( 59,265)( 60,264)( 61,263)( 62,262)( 63,261)( 64,260)
( 65,259)( 66,258)( 67,257)( 68,256)( 69,255)( 70,254)( 71,253)( 72,252)
( 73,251)( 74,250)( 75,249)( 76,248)( 77,247)( 78,246)( 79,245)( 80,244)
( 81,243)( 82,242)( 83,241)( 84,240)( 85,239)( 86,238)( 87,237)( 88,236)
( 89,235)( 90,234)( 91,233)( 92,232)( 93,231)( 94,230)( 95,229)( 96,228)
( 97,227)( 98,226)( 99,225)(100,224)(101,223)(102,222)(103,221)(104,220)
(105,219)(106,218)(107,217)(108,323)(109,322)(110,428)(111,427)(112,426)
(113,425)(114,424)(115,423)(116,422)(117,421)(118,420)(119,419)(120,418)
(121,417)(122,416)(123,415)(124,414)(125,413)(126,412)(127,411)(128,410)
(129,409)(130,408)(131,407)(132,406)(133,405)(134,404)(135,403)(136,402)
(137,401)(138,400)(139,399)(140,398)(141,397)(142,396)(143,395)(144,394)
(145,393)(146,392)(147,391)(148,390)(149,389)(150,388)(151,387)(152,386)
(153,385)(154,384)(155,383)(156,382)(157,381)(158,380)(159,379)(160,378)
(161,377)(162,376)(163,375)(164,374)(165,373)(166,372)(167,371)(168,370)
(169,369)(170,368)(171,367)(172,366)(173,365)(174,364)(175,363)(176,362)
(177,361)(178,360)(179,359)(180,358)(181,357)(182,356)(183,355)(184,354)
(185,353)(186,352)(187,351)(188,350)(189,349)(190,348)(191,347)(192,346)
(193,345)(194,344)(195,343)(196,342)(197,341)(198,340)(199,339)(200,338)
(201,337)(202,336)(203,335)(204,334)(205,333)(206,332)(207,331)(208,330)
(209,329)(210,328)(211,327)(212,326)(213,325)(214,324)(429,751)(430,750)
(431,856)(432,855)(433,854)(434,853)(435,852)(436,851)(437,850)(438,849)
(439,848)(440,847)(441,846)(442,845)(443,844)(444,843)(445,842)(446,841)
(447,840)(448,839)(449,838)(450,837)(451,836)(452,835)(453,834)(454,833)
(455,832)(456,831)(457,830)(458,829)(459,828)(460,827)(461,826)(462,825)
(463,824)(464,823)(465,822)(466,821)(467,820)(468,819)(469,818)(470,817)
(471,816)(472,815)(473,814)(474,813)(475,812)(476,811)(477,810)(478,809)
(479,808)(480,807)(481,806)(482,805)(483,804)(484,803)(485,802)(486,801)
(487,800)(488,799)(489,798)(490,797)(491,796)(492,795)(493,794)(494,793)
(495,792)(496,791)(497,790)(498,789)(499,788)(500,787)(501,786)(502,785)
(503,784)(504,783)(505,782)(506,781)(507,780)(508,779)(509,778)(510,777)
(511,776)(512,775)(513,774)(514,773)(515,772)(516,771)(517,770)(518,769)
(519,768)(520,767)(521,766)(522,765)(523,764)(524,763)(525,762)(526,761)
(527,760)(528,759)(529,758)(530,757)(531,756)(532,755)(533,754)(534,753)
(535,752)(536,644)(537,643)(538,749)(539,748)(540,747)(541,746)(542,745)
(543,744)(544,743)(545,742)(546,741)(547,740)(548,739)(549,738)(550,737)
(551,736)(552,735)(553,734)(554,733)(555,732)(556,731)(557,730)(558,729)
(559,728)(560,727)(561,726)(562,725)(563,724)(564,723)(565,722)(566,721)
(567,720)(568,719)(569,718)(570,717)(571,716)(572,715)(573,714)(574,713)
(575,712)(576,711)(577,710)(578,709)(579,708)(580,707)(581,706)(582,705)
(583,704)(584,703)(585,702)(586,701)(587,700)(588,699)(589,698)(590,697)
(591,696)(592,695)(593,694)(594,693)(595,692)(596,691)(597,690)(598,689)
(599,688)(600,687)(601,686)(602,685)(603,684)(604,683)(605,682)(606,681)
(607,680)(608,679)(609,678)(610,677)(611,676)(612,675)(613,674)(614,673)
(615,672)(616,671)(617,670)(618,669)(619,668)(620,667)(621,666)(622,665)
(623,664)(624,663)(625,662)(626,661)(627,660)(628,659)(629,658)(630,657)
(631,656)(632,655)(633,654)(634,653)(635,652)(636,651)(637,650)(638,649)
(639,648)(640,647)(641,646)(642,645);;
s2 := (  1,429)(  2,430)(  3,431)(  4,432)(  5,433)(  6,434)(  7,435)(  8,436)
(  9,437)( 10,438)( 11,439)( 12,440)( 13,441)( 14,442)( 15,443)( 16,444)
( 17,445)( 18,446)( 19,447)( 20,448)( 21,449)( 22,450)( 23,451)( 24,452)
( 25,453)( 26,454)( 27,455)( 28,456)( 29,457)( 30,458)( 31,459)( 32,460)
( 33,461)( 34,462)( 35,463)( 36,464)( 37,465)( 38,466)( 39,467)( 40,468)
( 41,469)( 42,470)( 43,471)( 44,472)( 45,473)( 46,474)( 47,475)( 48,476)
( 49,477)( 50,478)( 51,479)( 52,480)( 53,481)( 54,482)( 55,483)( 56,484)
( 57,485)( 58,486)( 59,487)( 60,488)( 61,489)( 62,490)( 63,491)( 64,492)
( 65,493)( 66,494)( 67,495)( 68,496)( 69,497)( 70,498)( 71,499)( 72,500)
( 73,501)( 74,502)( 75,503)( 76,504)( 77,505)( 78,506)( 79,507)( 80,508)
( 81,509)( 82,510)( 83,511)( 84,512)( 85,513)( 86,514)( 87,515)( 88,516)
( 89,517)( 90,518)( 91,519)( 92,520)( 93,521)( 94,522)( 95,523)( 96,524)
( 97,525)( 98,526)( 99,527)(100,528)(101,529)(102,530)(103,531)(104,532)
(105,533)(106,534)(107,535)(108,536)(109,537)(110,538)(111,539)(112,540)
(113,541)(114,542)(115,543)(116,544)(117,545)(118,546)(119,547)(120,548)
(121,549)(122,550)(123,551)(124,552)(125,553)(126,554)(127,555)(128,556)
(129,557)(130,558)(131,559)(132,560)(133,561)(134,562)(135,563)(136,564)
(137,565)(138,566)(139,567)(140,568)(141,569)(142,570)(143,571)(144,572)
(145,573)(146,574)(147,575)(148,576)(149,577)(150,578)(151,579)(152,580)
(153,581)(154,582)(155,583)(156,584)(157,585)(158,586)(159,587)(160,588)
(161,589)(162,590)(163,591)(164,592)(165,593)(166,594)(167,595)(168,596)
(169,597)(170,598)(171,599)(172,600)(173,601)(174,602)(175,603)(176,604)
(177,605)(178,606)(179,607)(180,608)(181,609)(182,610)(183,611)(184,612)
(185,613)(186,614)(187,615)(188,616)(189,617)(190,618)(191,619)(192,620)
(193,621)(194,622)(195,623)(196,624)(197,625)(198,626)(199,627)(200,628)
(201,629)(202,630)(203,631)(204,632)(205,633)(206,634)(207,635)(208,636)
(209,637)(210,638)(211,639)(212,640)(213,641)(214,642)(215,643)(216,644)
(217,645)(218,646)(219,647)(220,648)(221,649)(222,650)(223,651)(224,652)
(225,653)(226,654)(227,655)(228,656)(229,657)(230,658)(231,659)(232,660)
(233,661)(234,662)(235,663)(236,664)(237,665)(238,666)(239,667)(240,668)
(241,669)(242,670)(243,671)(244,672)(245,673)(246,674)(247,675)(248,676)
(249,677)(250,678)(251,679)(252,680)(253,681)(254,682)(255,683)(256,684)
(257,685)(258,686)(259,687)(260,688)(261,689)(262,690)(263,691)(264,692)
(265,693)(266,694)(267,695)(268,696)(269,697)(270,698)(271,699)(272,700)
(273,701)(274,702)(275,703)(276,704)(277,705)(278,706)(279,707)(280,708)
(281,709)(282,710)(283,711)(284,712)(285,713)(286,714)(287,715)(288,716)
(289,717)(290,718)(291,719)(292,720)(293,721)(294,722)(295,723)(296,724)
(297,725)(298,726)(299,727)(300,728)(301,729)(302,730)(303,731)(304,732)
(305,733)(306,734)(307,735)(308,736)(309,737)(310,738)(311,739)(312,740)
(313,741)(314,742)(315,743)(316,744)(317,745)(318,746)(319,747)(320,748)
(321,749)(322,750)(323,751)(324,752)(325,753)(326,754)(327,755)(328,756)
(329,757)(330,758)(331,759)(332,760)(333,761)(334,762)(335,763)(336,764)
(337,765)(338,766)(339,767)(340,768)(341,769)(342,770)(343,771)(344,772)
(345,773)(346,774)(347,775)(348,776)(349,777)(350,778)(351,779)(352,780)
(353,781)(354,782)(355,783)(356,784)(357,785)(358,786)(359,787)(360,788)
(361,789)(362,790)(363,791)(364,792)(365,793)(366,794)(367,795)(368,796)
(369,797)(370,798)(371,799)(372,800)(373,801)(374,802)(375,803)(376,804)
(377,805)(378,806)(379,807)(380,808)(381,809)(382,810)(383,811)(384,812)
(385,813)(386,814)(387,815)(388,816)(389,817)(390,818)(391,819)(392,820)
(393,821)(394,822)(395,823)(396,824)(397,825)(398,826)(399,827)(400,828)
(401,829)(402,830)(403,831)(404,832)(405,833)(406,834)(407,835)(408,836)
(409,837)(410,838)(411,839)(412,840)(413,841)(414,842)(415,843)(416,844)
(417,845)(418,846)(419,847)(420,848)(421,849)(422,850)(423,851)(424,852)
(425,853)(426,854)(427,855)(428,856);;
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*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(856)!(  1,215)(  2,321)(  3,320)(  4,319)(  5,318)(  6,317)(  7,316)
(  8,315)(  9,314)( 10,313)( 11,312)( 12,311)( 13,310)( 14,309)( 15,308)
( 16,307)( 17,306)( 18,305)( 19,304)( 20,303)( 21,302)( 22,301)( 23,300)
( 24,299)( 25,298)( 26,297)( 27,296)( 28,295)( 29,294)( 30,293)( 31,292)
( 32,291)( 33,290)( 34,289)( 35,288)( 36,287)( 37,286)( 38,285)( 39,284)
( 40,283)( 41,282)( 42,281)( 43,280)( 44,279)( 45,278)( 46,277)( 47,276)
( 48,275)( 49,274)( 50,273)( 51,272)( 52,271)( 53,270)( 54,269)( 55,268)
( 56,267)( 57,266)( 58,265)( 59,264)( 60,263)( 61,262)( 62,261)( 63,260)
( 64,259)( 65,258)( 66,257)( 67,256)( 68,255)( 69,254)( 70,253)( 71,252)
( 72,251)( 73,250)( 74,249)( 75,248)( 76,247)( 77,246)( 78,245)( 79,244)
( 80,243)( 81,242)( 82,241)( 83,240)( 84,239)( 85,238)( 86,237)( 87,236)
( 88,235)( 89,234)( 90,233)( 91,232)( 92,231)( 93,230)( 94,229)( 95,228)
( 96,227)( 97,226)( 98,225)( 99,224)(100,223)(101,222)(102,221)(103,220)
(104,219)(105,218)(106,217)(107,216)(108,322)(109,428)(110,427)(111,426)
(112,425)(113,424)(114,423)(115,422)(116,421)(117,420)(118,419)(119,418)
(120,417)(121,416)(122,415)(123,414)(124,413)(125,412)(126,411)(127,410)
(128,409)(129,408)(130,407)(131,406)(132,405)(133,404)(134,403)(135,402)
(136,401)(137,400)(138,399)(139,398)(140,397)(141,396)(142,395)(143,394)
(144,393)(145,392)(146,391)(147,390)(148,389)(149,388)(150,387)(151,386)
(152,385)(153,384)(154,383)(155,382)(156,381)(157,380)(158,379)(159,378)
(160,377)(161,376)(162,375)(163,374)(164,373)(165,372)(166,371)(167,370)
(168,369)(169,368)(170,367)(171,366)(172,365)(173,364)(174,363)(175,362)
(176,361)(177,360)(178,359)(179,358)(180,357)(181,356)(182,355)(183,354)
(184,353)(185,352)(186,351)(187,350)(188,349)(189,348)(190,347)(191,346)
(192,345)(193,344)(194,343)(195,342)(196,341)(197,340)(198,339)(199,338)
(200,337)(201,336)(202,335)(203,334)(204,333)(205,332)(206,331)(207,330)
(208,329)(209,328)(210,327)(211,326)(212,325)(213,324)(214,323)(429,643)
(430,749)(431,748)(432,747)(433,746)(434,745)(435,744)(436,743)(437,742)
(438,741)(439,740)(440,739)(441,738)(442,737)(443,736)(444,735)(445,734)
(446,733)(447,732)(448,731)(449,730)(450,729)(451,728)(452,727)(453,726)
(454,725)(455,724)(456,723)(457,722)(458,721)(459,720)(460,719)(461,718)
(462,717)(463,716)(464,715)(465,714)(466,713)(467,712)(468,711)(469,710)
(470,709)(471,708)(472,707)(473,706)(474,705)(475,704)(476,703)(477,702)
(478,701)(479,700)(480,699)(481,698)(482,697)(483,696)(484,695)(485,694)
(486,693)(487,692)(488,691)(489,690)(490,689)(491,688)(492,687)(493,686)
(494,685)(495,684)(496,683)(497,682)(498,681)(499,680)(500,679)(501,678)
(502,677)(503,676)(504,675)(505,674)(506,673)(507,672)(508,671)(509,670)
(510,669)(511,668)(512,667)(513,666)(514,665)(515,664)(516,663)(517,662)
(518,661)(519,660)(520,659)(521,658)(522,657)(523,656)(524,655)(525,654)
(526,653)(527,652)(528,651)(529,650)(530,649)(531,648)(532,647)(533,646)
(534,645)(535,644)(536,750)(537,856)(538,855)(539,854)(540,853)(541,852)
(542,851)(543,850)(544,849)(545,848)(546,847)(547,846)(548,845)(549,844)
(550,843)(551,842)(552,841)(553,840)(554,839)(555,838)(556,837)(557,836)
(558,835)(559,834)(560,833)(561,832)(562,831)(563,830)(564,829)(565,828)
(566,827)(567,826)(568,825)(569,824)(570,823)(571,822)(572,821)(573,820)
(574,819)(575,818)(576,817)(577,816)(578,815)(579,814)(580,813)(581,812)
(582,811)(583,810)(584,809)(585,808)(586,807)(587,806)(588,805)(589,804)
(590,803)(591,802)(592,801)(593,800)(594,799)(595,798)(596,797)(597,796)
(598,795)(599,794)(600,793)(601,792)(602,791)(603,790)(604,789)(605,788)
(606,787)(607,786)(608,785)(609,784)(610,783)(611,782)(612,781)(613,780)
(614,779)(615,778)(616,777)(617,776)(618,775)(619,774)(620,773)(621,772)
(622,771)(623,770)(624,769)(625,768)(626,767)(627,766)(628,765)(629,764)
(630,763)(631,762)(632,761)(633,760)(634,759)(635,758)(636,757)(637,756)
(638,755)(639,754)(640,753)(641,752)(642,751);
s1 := Sym(856)!(  1,216)(  2,215)(  3,321)(  4,320)(  5,319)(  6,318)(  7,317)
(  8,316)(  9,315)( 10,314)( 11,313)( 12,312)( 13,311)( 14,310)( 15,309)
( 16,308)( 17,307)( 18,306)( 19,305)( 20,304)( 21,303)( 22,302)( 23,301)
( 24,300)( 25,299)( 26,298)( 27,297)( 28,296)( 29,295)( 30,294)( 31,293)
( 32,292)( 33,291)( 34,290)( 35,289)( 36,288)( 37,287)( 38,286)( 39,285)
( 40,284)( 41,283)( 42,282)( 43,281)( 44,280)( 45,279)( 46,278)( 47,277)
( 48,276)( 49,275)( 50,274)( 51,273)( 52,272)( 53,271)( 54,270)( 55,269)
( 56,268)( 57,267)( 58,266)( 59,265)( 60,264)( 61,263)( 62,262)( 63,261)
( 64,260)( 65,259)( 66,258)( 67,257)( 68,256)( 69,255)( 70,254)( 71,253)
( 72,252)( 73,251)( 74,250)( 75,249)( 76,248)( 77,247)( 78,246)( 79,245)
( 80,244)( 81,243)( 82,242)( 83,241)( 84,240)( 85,239)( 86,238)( 87,237)
( 88,236)( 89,235)( 90,234)( 91,233)( 92,232)( 93,231)( 94,230)( 95,229)
( 96,228)( 97,227)( 98,226)( 99,225)(100,224)(101,223)(102,222)(103,221)
(104,220)(105,219)(106,218)(107,217)(108,323)(109,322)(110,428)(111,427)
(112,426)(113,425)(114,424)(115,423)(116,422)(117,421)(118,420)(119,419)
(120,418)(121,417)(122,416)(123,415)(124,414)(125,413)(126,412)(127,411)
(128,410)(129,409)(130,408)(131,407)(132,406)(133,405)(134,404)(135,403)
(136,402)(137,401)(138,400)(139,399)(140,398)(141,397)(142,396)(143,395)
(144,394)(145,393)(146,392)(147,391)(148,390)(149,389)(150,388)(151,387)
(152,386)(153,385)(154,384)(155,383)(156,382)(157,381)(158,380)(159,379)
(160,378)(161,377)(162,376)(163,375)(164,374)(165,373)(166,372)(167,371)
(168,370)(169,369)(170,368)(171,367)(172,366)(173,365)(174,364)(175,363)
(176,362)(177,361)(178,360)(179,359)(180,358)(181,357)(182,356)(183,355)
(184,354)(185,353)(186,352)(187,351)(188,350)(189,349)(190,348)(191,347)
(192,346)(193,345)(194,344)(195,343)(196,342)(197,341)(198,340)(199,339)
(200,338)(201,337)(202,336)(203,335)(204,334)(205,333)(206,332)(207,331)
(208,330)(209,329)(210,328)(211,327)(212,326)(213,325)(214,324)(429,751)
(430,750)(431,856)(432,855)(433,854)(434,853)(435,852)(436,851)(437,850)
(438,849)(439,848)(440,847)(441,846)(442,845)(443,844)(444,843)(445,842)
(446,841)(447,840)(448,839)(449,838)(450,837)(451,836)(452,835)(453,834)
(454,833)(455,832)(456,831)(457,830)(458,829)(459,828)(460,827)(461,826)
(462,825)(463,824)(464,823)(465,822)(466,821)(467,820)(468,819)(469,818)
(470,817)(471,816)(472,815)(473,814)(474,813)(475,812)(476,811)(477,810)
(478,809)(479,808)(480,807)(481,806)(482,805)(483,804)(484,803)(485,802)
(486,801)(487,800)(488,799)(489,798)(490,797)(491,796)(492,795)(493,794)
(494,793)(495,792)(496,791)(497,790)(498,789)(499,788)(500,787)(501,786)
(502,785)(503,784)(504,783)(505,782)(506,781)(507,780)(508,779)(509,778)
(510,777)(511,776)(512,775)(513,774)(514,773)(515,772)(516,771)(517,770)
(518,769)(519,768)(520,767)(521,766)(522,765)(523,764)(524,763)(525,762)
(526,761)(527,760)(528,759)(529,758)(530,757)(531,756)(532,755)(533,754)
(534,753)(535,752)(536,644)(537,643)(538,749)(539,748)(540,747)(541,746)
(542,745)(543,744)(544,743)(545,742)(546,741)(547,740)(548,739)(549,738)
(550,737)(551,736)(552,735)(553,734)(554,733)(555,732)(556,731)(557,730)
(558,729)(559,728)(560,727)(561,726)(562,725)(563,724)(564,723)(565,722)
(566,721)(567,720)(568,719)(569,718)(570,717)(571,716)(572,715)(573,714)
(574,713)(575,712)(576,711)(577,710)(578,709)(579,708)(580,707)(581,706)
(582,705)(583,704)(584,703)(585,702)(586,701)(587,700)(588,699)(589,698)
(590,697)(591,696)(592,695)(593,694)(594,693)(595,692)(596,691)(597,690)
(598,689)(599,688)(600,687)(601,686)(602,685)(603,684)(604,683)(605,682)
(606,681)(607,680)(608,679)(609,678)(610,677)(611,676)(612,675)(613,674)
(614,673)(615,672)(616,671)(617,670)(618,669)(619,668)(620,667)(621,666)
(622,665)(623,664)(624,663)(625,662)(626,661)(627,660)(628,659)(629,658)
(630,657)(631,656)(632,655)(633,654)(634,653)(635,652)(636,651)(637,650)
(638,649)(639,648)(640,647)(641,646)(642,645);
s2 := Sym(856)!(  1,429)(  2,430)(  3,431)(  4,432)(  5,433)(  6,434)(  7,435)
(  8,436)(  9,437)( 10,438)( 11,439)( 12,440)( 13,441)( 14,442)( 15,443)
( 16,444)( 17,445)( 18,446)( 19,447)( 20,448)( 21,449)( 22,450)( 23,451)
( 24,452)( 25,453)( 26,454)( 27,455)( 28,456)( 29,457)( 30,458)( 31,459)
( 32,460)( 33,461)( 34,462)( 35,463)( 36,464)( 37,465)( 38,466)( 39,467)
( 40,468)( 41,469)( 42,470)( 43,471)( 44,472)( 45,473)( 46,474)( 47,475)
( 48,476)( 49,477)( 50,478)( 51,479)( 52,480)( 53,481)( 54,482)( 55,483)
( 56,484)( 57,485)( 58,486)( 59,487)( 60,488)( 61,489)( 62,490)( 63,491)
( 64,492)( 65,493)( 66,494)( 67,495)( 68,496)( 69,497)( 70,498)( 71,499)
( 72,500)( 73,501)( 74,502)( 75,503)( 76,504)( 77,505)( 78,506)( 79,507)
( 80,508)( 81,509)( 82,510)( 83,511)( 84,512)( 85,513)( 86,514)( 87,515)
( 88,516)( 89,517)( 90,518)( 91,519)( 92,520)( 93,521)( 94,522)( 95,523)
( 96,524)( 97,525)( 98,526)( 99,527)(100,528)(101,529)(102,530)(103,531)
(104,532)(105,533)(106,534)(107,535)(108,536)(109,537)(110,538)(111,539)
(112,540)(113,541)(114,542)(115,543)(116,544)(117,545)(118,546)(119,547)
(120,548)(121,549)(122,550)(123,551)(124,552)(125,553)(126,554)(127,555)
(128,556)(129,557)(130,558)(131,559)(132,560)(133,561)(134,562)(135,563)
(136,564)(137,565)(138,566)(139,567)(140,568)(141,569)(142,570)(143,571)
(144,572)(145,573)(146,574)(147,575)(148,576)(149,577)(150,578)(151,579)
(152,580)(153,581)(154,582)(155,583)(156,584)(157,585)(158,586)(159,587)
(160,588)(161,589)(162,590)(163,591)(164,592)(165,593)(166,594)(167,595)
(168,596)(169,597)(170,598)(171,599)(172,600)(173,601)(174,602)(175,603)
(176,604)(177,605)(178,606)(179,607)(180,608)(181,609)(182,610)(183,611)
(184,612)(185,613)(186,614)(187,615)(188,616)(189,617)(190,618)(191,619)
(192,620)(193,621)(194,622)(195,623)(196,624)(197,625)(198,626)(199,627)
(200,628)(201,629)(202,630)(203,631)(204,632)(205,633)(206,634)(207,635)
(208,636)(209,637)(210,638)(211,639)(212,640)(213,641)(214,642)(215,643)
(216,644)(217,645)(218,646)(219,647)(220,648)(221,649)(222,650)(223,651)
(224,652)(225,653)(226,654)(227,655)(228,656)(229,657)(230,658)(231,659)
(232,660)(233,661)(234,662)(235,663)(236,664)(237,665)(238,666)(239,667)
(240,668)(241,669)(242,670)(243,671)(244,672)(245,673)(246,674)(247,675)
(248,676)(249,677)(250,678)(251,679)(252,680)(253,681)(254,682)(255,683)
(256,684)(257,685)(258,686)(259,687)(260,688)(261,689)(262,690)(263,691)
(264,692)(265,693)(266,694)(267,695)(268,696)(269,697)(270,698)(271,699)
(272,700)(273,701)(274,702)(275,703)(276,704)(277,705)(278,706)(279,707)
(280,708)(281,709)(282,710)(283,711)(284,712)(285,713)(286,714)(287,715)
(288,716)(289,717)(290,718)(291,719)(292,720)(293,721)(294,722)(295,723)
(296,724)(297,725)(298,726)(299,727)(300,728)(301,729)(302,730)(303,731)
(304,732)(305,733)(306,734)(307,735)(308,736)(309,737)(310,738)(311,739)
(312,740)(313,741)(314,742)(315,743)(316,744)(317,745)(318,746)(319,747)
(320,748)(321,749)(322,750)(323,751)(324,752)(325,753)(326,754)(327,755)
(328,756)(329,757)(330,758)(331,759)(332,760)(333,761)(334,762)(335,763)
(336,764)(337,765)(338,766)(339,767)(340,768)(341,769)(342,770)(343,771)
(344,772)(345,773)(346,774)(347,775)(348,776)(349,777)(350,778)(351,779)
(352,780)(353,781)(354,782)(355,783)(356,784)(357,785)(358,786)(359,787)
(360,788)(361,789)(362,790)(363,791)(364,792)(365,793)(366,794)(367,795)
(368,796)(369,797)(370,798)(371,799)(372,800)(373,801)(374,802)(375,803)
(376,804)(377,805)(378,806)(379,807)(380,808)(381,809)(382,810)(383,811)
(384,812)(385,813)(386,814)(387,815)(388,816)(389,817)(390,818)(391,819)
(392,820)(393,821)(394,822)(395,823)(396,824)(397,825)(398,826)(399,827)
(400,828)(401,829)(402,830)(403,831)(404,832)(405,833)(406,834)(407,835)
(408,836)(409,837)(410,838)(411,839)(412,840)(413,841)(414,842)(415,843)
(416,844)(417,845)(418,846)(419,847)(420,848)(421,849)(422,850)(423,851)
(424,852)(425,853)(426,854)(427,855)(428,856);
poly := sub<Sym(856)|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*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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