Polytope of Type {206,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {206,4}*1648
Also Known As : {206,4|2}. if this polytope has another name.
Group : SmallGroup(1648,34)
Rank : 3
Schlafli Type : {206,4}
Number of vertices, edges, etc : 206, 412, 4
Order of s₀s₁s₂ : 412
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 : {206,2}*824
   4-fold quotients : {103,2}*412
   103-fold quotients : {2,4}*16
   206-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope