Polytope of Type {208,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {208,4}*1664a
Also Known As : {208,4|2}. if this polytope has another name.
Group : SmallGroup(1664,6424)
Rank : 3
Schlafli Type : {208,4}
Number of vertices, edges, etc : 208, 416, 4
Order of s₀s₁s₂ : 208
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
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 : {104,4}*832a, {208,2}*832
   4-fold quotients : {52,4}*416, {104,2}*416
   8-fold quotients : {52,2}*208, {26,4}*208
   13-fold quotients : {16,4}*128a
   16-fold quotients : {26,2}*104
   26-fold quotients : {8,4}*64a, {16,2}*64
   32-fold quotients : {13,2}*52
   52-fold quotients : {4,4}*32, {8,2}*32
   104-fold quotients : {2,4}*16, {4,2}*16
   208-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope