Polytope of Type {216,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {216,4}*1728c
if this polytope has a name.
Group : SmallGroup(1728,2203)
Rank : 3
Schlafli Type : {216,4}
Number of vertices, edges, etc : 216, 432, 4
Order of s₀s₁s₂ : 216
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-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 : {108,4}*864b
   3-fold quotients : {72,4}*576c
   4-fold quotients : {54,4}*432b
   6-fold quotients : {36,4}*288b
   8-fold quotients : {27,4}*216
   9-fold quotients : {24,4}*192c
   12-fold quotients : {18,4}*144b
   18-fold quotients : {12,4}*96b
   24-fold quotients : {9,4}*72
   36-fold quotients : {6,4}*48c
   72-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 33)( 14, 34)( 15, 36)
( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)( 23, 28)
( 24, 27)( 37,105)( 38,106)( 39,108)( 40,107)( 41,101)( 42,102)( 43,104)
( 44,103)( 45, 97)( 46, 98)( 47,100)( 48, 99)( 49, 93)( 50, 94)( 51, 96)
( 52, 95)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 85)( 58, 86)( 59, 88)
( 60, 87)( 61, 81)( 62, 82)( 63, 84)( 64, 83)( 65, 77)( 66, 78)( 67, 80)
( 68, 79)( 69, 73)( 70, 74)( 71, 76)( 72, 75)(111,112)(113,117)(114,118)
(115,120)(116,119)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)
(127,140)(128,139)(129,133)(130,134)(131,136)(132,135)(145,213)(146,214)
(147,216)(148,215)(149,209)(150,210)(151,212)(152,211)(153,205)(154,206)
(155,208)(156,207)(157,201)(158,202)(159,204)(160,203)(161,197)(162,198)
(163,200)(164,199)(165,193)(166,194)(167,196)(168,195)(169,189)(170,190)
(171,192)(172,191)(173,185)(174,186)(175,188)(176,187)(177,181)(178,182)
(179,184)(180,183)(217,325)(218,326)(219,328)(220,327)(221,333)(222,334)
(223,336)(224,335)(225,329)(226,330)(227,332)(228,331)(229,357)(230,358)
(231,360)(232,359)(233,353)(234,354)(235,356)(236,355)(237,349)(238,350)
(239,352)(240,351)(241,345)(242,346)(243,348)(244,347)(245,341)(246,342)
(247,344)(248,343)(249,337)(250,338)(251,340)(252,339)(253,429)(254,430)
(255,432)(256,431)(257,425)(258,426)(259,428)(260,427)(261,421)(262,422)
(263,424)(264,423)(265,417)(266,418)(267,420)(268,419)(269,413)(270,414)
(271,416)(272,415)(273,409)(274,410)(275,412)(276,411)(277,405)(278,406)
(279,408)(280,407)(281,401)(282,402)(283,404)(284,403)(285,397)(286,398)
(287,400)(288,399)(289,393)(290,394)(291,396)(292,395)(293,389)(294,390)
(295,392)(296,391)(297,385)(298,386)(299,388)(300,387)(301,381)(302,382)
(303,384)(304,383)(305,377)(306,378)(307,380)(308,379)(309,373)(310,374)
(311,376)(312,375)(313,369)(314,370)(315,372)(316,371)(317,365)(318,366)
(319,368)(320,367)(321,361)(322,362)(323,364)(324,363)(433,649)(434,650)
(435,652)(436,651)(437,657)(438,658)(439,660)(440,659)(441,653)(442,654)
(443,656)(444,655)(445,681)(446,682)(447,684)(448,683)(449,677)(450,678)
(451,680)(452,679)(453,673)(454,674)(455,676)(456,675)(457,669)(458,670)
(459,672)(460,671)(461,665)(462,666)(463,668)(464,667)(465,661)(466,662)
(467,664)(468,663)(469,753)(470,754)(471,756)(472,755)(473,749)(474,750)
(475,752)(476,751)(477,745)(478,746)(479,748)(480,747)(481,741)(482,742)
(483,744)(484,743)(485,737)(486,738)(487,740)(488,739)(489,733)(490,734)
(491,736)(492,735)(493,729)(494,730)(495,732)(496,731)(497,725)(498,726)
(499,728)(500,727)(501,721)(502,722)(503,724)(504,723)(505,717)(506,718)
(507,720)(508,719)(509,713)(510,714)(511,716)(512,715)(513,709)(514,710)
(515,712)(516,711)(517,705)(518,706)(519,708)(520,707)(521,701)(522,702)
(523,704)(524,703)(525,697)(526,698)(527,700)(528,699)(529,693)(530,694)
(531,696)(532,695)(533,689)(534,690)(535,692)(536,691)(537,685)(538,686)
(539,688)(540,687)(541,757)(542,758)(543,760)(544,759)(545,765)(546,766)
(547,768)(548,767)(549,761)(550,762)(551,764)(552,763)(553,789)(554,790)
(555,792)(556,791)(557,785)(558,786)(559,788)(560,787)(561,781)(562,782)
(563,784)(564,783)(565,777)(566,778)(567,780)(568,779)(569,773)(570,774)
(571,776)(572,775)(573,769)(574,770)(575,772)(576,771)(577,861)(578,862)
(579,864)(580,863)(581,857)(582,858)(583,860)(584,859)(585,853)(586,854)
(587,856)(588,855)(589,849)(590,850)(591,852)(592,851)(593,845)(594,846)
(595,848)(596,847)(597,841)(598,842)(599,844)(600,843)(601,837)(602,838)
(603,840)(604,839)(605,833)(606,834)(607,836)(608,835)(609,829)(610,830)
(611,832)(612,831)(613,825)(614,826)(615,828)(616,827)(617,821)(618,822)
(619,824)(620,823)(621,817)(622,818)(623,820)(624,819)(625,813)(626,814)
(627,816)(628,815)(629,809)(630,810)(631,812)(632,811)(633,805)(634,806)
(635,808)(636,807)(637,801)(638,802)(639,804)(640,803)(641,797)(642,798)
(643,800)(644,799)(645,793)(646,794)(647,796)(648,795);;
s1 := (  1,469)(  2,472)(  3,471)(  4,470)(  5,477)(  6,480)(  7,479)(  8,478)
(  9,473)( 10,476)( 11,475)( 12,474)( 13,501)( 14,504)( 15,503)( 16,502)
( 17,497)( 18,500)( 19,499)( 20,498)( 21,493)( 22,496)( 23,495)( 24,494)
( 25,489)( 26,492)( 27,491)( 28,490)( 29,485)( 30,488)( 31,487)( 32,486)
( 33,481)( 34,484)( 35,483)( 36,482)( 37,433)( 38,436)( 39,435)( 40,434)
( 41,441)( 42,444)( 43,443)( 44,442)( 45,437)( 46,440)( 47,439)( 48,438)
( 49,465)( 50,468)( 51,467)( 52,466)( 53,461)( 54,464)( 55,463)( 56,462)
( 57,457)( 58,460)( 59,459)( 60,458)( 61,453)( 62,456)( 63,455)( 64,454)
( 65,449)( 66,452)( 67,451)( 68,450)( 69,445)( 70,448)( 71,447)( 72,446)
( 73,537)( 74,540)( 75,539)( 76,538)( 77,533)( 78,536)( 79,535)( 80,534)
( 81,529)( 82,532)( 83,531)( 84,530)( 85,525)( 86,528)( 87,527)( 88,526)
( 89,521)( 90,524)( 91,523)( 92,522)( 93,517)( 94,520)( 95,519)( 96,518)
( 97,513)( 98,516)( 99,515)(100,514)(101,509)(102,512)(103,511)(104,510)
(105,505)(106,508)(107,507)(108,506)(109,577)(110,580)(111,579)(112,578)
(113,585)(114,588)(115,587)(116,586)(117,581)(118,584)(119,583)(120,582)
(121,609)(122,612)(123,611)(124,610)(125,605)(126,608)(127,607)(128,606)
(129,601)(130,604)(131,603)(132,602)(133,597)(134,600)(135,599)(136,598)
(137,593)(138,596)(139,595)(140,594)(141,589)(142,592)(143,591)(144,590)
(145,541)(146,544)(147,543)(148,542)(149,549)(150,552)(151,551)(152,550)
(153,545)(154,548)(155,547)(156,546)(157,573)(158,576)(159,575)(160,574)
(161,569)(162,572)(163,571)(164,570)(165,565)(166,568)(167,567)(168,566)
(169,561)(170,564)(171,563)(172,562)(173,557)(174,560)(175,559)(176,558)
(177,553)(178,556)(179,555)(180,554)(181,645)(182,648)(183,647)(184,646)
(185,641)(186,644)(187,643)(188,642)(189,637)(190,640)(191,639)(192,638)
(193,633)(194,636)(195,635)(196,634)(197,629)(198,632)(199,631)(200,630)
(201,625)(202,628)(203,627)(204,626)(205,621)(206,624)(207,623)(208,622)
(209,617)(210,620)(211,619)(212,618)(213,613)(214,616)(215,615)(216,614)
(217,793)(218,796)(219,795)(220,794)(221,801)(222,804)(223,803)(224,802)
(225,797)(226,800)(227,799)(228,798)(229,825)(230,828)(231,827)(232,826)
(233,821)(234,824)(235,823)(236,822)(237,817)(238,820)(239,819)(240,818)
(241,813)(242,816)(243,815)(244,814)(245,809)(246,812)(247,811)(248,810)
(249,805)(250,808)(251,807)(252,806)(253,757)(254,760)(255,759)(256,758)
(257,765)(258,768)(259,767)(260,766)(261,761)(262,764)(263,763)(264,762)
(265,789)(266,792)(267,791)(268,790)(269,785)(270,788)(271,787)(272,786)
(273,781)(274,784)(275,783)(276,782)(277,777)(278,780)(279,779)(280,778)
(281,773)(282,776)(283,775)(284,774)(285,769)(286,772)(287,771)(288,770)
(289,861)(290,864)(291,863)(292,862)(293,857)(294,860)(295,859)(296,858)
(297,853)(298,856)(299,855)(300,854)(301,849)(302,852)(303,851)(304,850)
(305,845)(306,848)(307,847)(308,846)(309,841)(310,844)(311,843)(312,842)
(313,837)(314,840)(315,839)(316,838)(317,833)(318,836)(319,835)(320,834)
(321,829)(322,832)(323,831)(324,830)(325,685)(326,688)(327,687)(328,686)
(329,693)(330,696)(331,695)(332,694)(333,689)(334,692)(335,691)(336,690)
(337,717)(338,720)(339,719)(340,718)(341,713)(342,716)(343,715)(344,714)
(345,709)(346,712)(347,711)(348,710)(349,705)(350,708)(351,707)(352,706)
(353,701)(354,704)(355,703)(356,702)(357,697)(358,700)(359,699)(360,698)
(361,649)(362,652)(363,651)(364,650)(365,657)(366,660)(367,659)(368,658)
(369,653)(370,656)(371,655)(372,654)(373,681)(374,684)(375,683)(376,682)
(377,677)(378,680)(379,679)(380,678)(381,673)(382,676)(383,675)(384,674)
(385,669)(386,672)(387,671)(388,670)(389,665)(390,668)(391,667)(392,666)
(393,661)(394,664)(395,663)(396,662)(397,753)(398,756)(399,755)(400,754)
(401,749)(402,752)(403,751)(404,750)(405,745)(406,748)(407,747)(408,746)
(409,741)(410,744)(411,743)(412,742)(413,737)(414,740)(415,739)(416,738)
(417,733)(418,736)(419,735)(420,734)(421,729)(422,732)(423,731)(424,730)
(425,725)(426,728)(427,727)(428,726)(429,721)(430,724)(431,723)(432,722);;
s2 := (  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)
( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)
( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)
(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)
(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)
(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)
(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)
(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)
(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)(223,224)
(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)
(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)
(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)(271,272)
(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)(287,288)
(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)(303,304)
(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)(319,320)
(321,322)(323,324)(325,326)(327,328)(329,330)(331,332)(333,334)(335,336)
(337,338)(339,340)(341,342)(343,344)(345,346)(347,348)(349,350)(351,352)
(353,354)(355,356)(357,358)(359,360)(361,362)(363,364)(365,366)(367,368)
(369,370)(371,372)(373,374)(375,376)(377,378)(379,380)(381,382)(383,384)
(385,386)(387,388)(389,390)(391,392)(393,394)(395,396)(397,398)(399,400)
(401,402)(403,404)(405,406)(407,408)(409,410)(411,412)(413,414)(415,416)
(417,418)(419,420)(421,422)(423,424)(425,426)(427,428)(429,430)(431,432)
(433,434)(435,436)(437,438)(439,440)(441,442)(443,444)(445,446)(447,448)
(449,450)(451,452)(453,454)(455,456)(457,458)(459,460)(461,462)(463,464)
(465,466)(467,468)(469,470)(471,472)(473,474)(475,476)(477,478)(479,480)
(481,482)(483,484)(485,486)(487,488)(489,490)(491,492)(493,494)(495,496)
(497,498)(499,500)(501,502)(503,504)(505,506)(507,508)(509,510)(511,512)
(513,514)(515,516)(517,518)(519,520)(521,522)(523,524)(525,526)(527,528)
(529,530)(531,532)(533,534)(535,536)(537,538)(539,540)(541,542)(543,544)
(545,546)(547,548)(549,550)(551,552)(553,554)(555,556)(557,558)(559,560)
(561,562)(563,564)(565,566)(567,568)(569,570)(571,572)(573,574)(575,576)
(577,578)(579,580)(581,582)(583,584)(585,586)(587,588)(589,590)(591,592)
(593,594)(595,596)(597,598)(599,600)(601,602)(603,604)(605,606)(607,608)
(609,610)(611,612)(613,614)(615,616)(617,618)(619,620)(621,622)(623,624)
(625,626)(627,628)(629,630)(631,632)(633,634)(635,636)(637,638)(639,640)
(641,642)(643,644)(645,646)(647,648)(649,650)(651,652)(653,654)(655,656)
(657,658)(659,660)(661,662)(663,664)(665,666)(667,668)(669,670)(671,672)
(673,674)(675,676)(677,678)(679,680)(681,682)(683,684)(685,686)(687,688)
(689,690)(691,692)(693,694)(695,696)(697,698)(699,700)(701,702)(703,704)
(705,706)(707,708)(709,710)(711,712)(713,714)(715,716)(717,718)(719,720)
(721,722)(723,724)(725,726)(727,728)(729,730)(731,732)(733,734)(735,736)
(737,738)(739,740)(741,742)(743,744)(745,746)(747,748)(749,750)(751,752)
(753,754)(755,756)(757,758)(759,760)(761,762)(763,764)(765,766)(767,768)
(769,770)(771,772)(773,774)(775,776)(777,778)(779,780)(781,782)(783,784)
(785,786)(787,788)(789,790)(791,792)(793,794)(795,796)(797,798)(799,800)
(801,802)(803,804)(805,806)(807,808)(809,810)(811,812)(813,814)(815,816)
(817,818)(819,820)(821,822)(823,824)(825,826)(827,828)(829,830)(831,832)
(833,834)(835,836)(837,838)(839,840)(841,842)(843,844)(845,846)(847,848)
(849,850)(851,852)(853,854)(855,856)(857,858)(859,860)(861,862)(863,864);;
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, 
s2*s1*s0*s2*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(864)!(  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 33)( 14, 34)
( 15, 36)( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)
( 23, 28)( 24, 27)( 37,105)( 38,106)( 39,108)( 40,107)( 41,101)( 42,102)
( 43,104)( 44,103)( 45, 97)( 46, 98)( 47,100)( 48, 99)( 49, 93)( 50, 94)
( 51, 96)( 52, 95)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 85)( 58, 86)
( 59, 88)( 60, 87)( 61, 81)( 62, 82)( 63, 84)( 64, 83)( 65, 77)( 66, 78)
( 67, 80)( 68, 79)( 69, 73)( 70, 74)( 71, 76)( 72, 75)(111,112)(113,117)
(114,118)(115,120)(116,119)(121,141)(122,142)(123,144)(124,143)(125,137)
(126,138)(127,140)(128,139)(129,133)(130,134)(131,136)(132,135)(145,213)
(146,214)(147,216)(148,215)(149,209)(150,210)(151,212)(152,211)(153,205)
(154,206)(155,208)(156,207)(157,201)(158,202)(159,204)(160,203)(161,197)
(162,198)(163,200)(164,199)(165,193)(166,194)(167,196)(168,195)(169,189)
(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)(176,187)(177,181)
(178,182)(179,184)(180,183)(217,325)(218,326)(219,328)(220,327)(221,333)
(222,334)(223,336)(224,335)(225,329)(226,330)(227,332)(228,331)(229,357)
(230,358)(231,360)(232,359)(233,353)(234,354)(235,356)(236,355)(237,349)
(238,350)(239,352)(240,351)(241,345)(242,346)(243,348)(244,347)(245,341)
(246,342)(247,344)(248,343)(249,337)(250,338)(251,340)(252,339)(253,429)
(254,430)(255,432)(256,431)(257,425)(258,426)(259,428)(260,427)(261,421)
(262,422)(263,424)(264,423)(265,417)(266,418)(267,420)(268,419)(269,413)
(270,414)(271,416)(272,415)(273,409)(274,410)(275,412)(276,411)(277,405)
(278,406)(279,408)(280,407)(281,401)(282,402)(283,404)(284,403)(285,397)
(286,398)(287,400)(288,399)(289,393)(290,394)(291,396)(292,395)(293,389)
(294,390)(295,392)(296,391)(297,385)(298,386)(299,388)(300,387)(301,381)
(302,382)(303,384)(304,383)(305,377)(306,378)(307,380)(308,379)(309,373)
(310,374)(311,376)(312,375)(313,369)(314,370)(315,372)(316,371)(317,365)
(318,366)(319,368)(320,367)(321,361)(322,362)(323,364)(324,363)(433,649)
(434,650)(435,652)(436,651)(437,657)(438,658)(439,660)(440,659)(441,653)
(442,654)(443,656)(444,655)(445,681)(446,682)(447,684)(448,683)(449,677)
(450,678)(451,680)(452,679)(453,673)(454,674)(455,676)(456,675)(457,669)
(458,670)(459,672)(460,671)(461,665)(462,666)(463,668)(464,667)(465,661)
(466,662)(467,664)(468,663)(469,753)(470,754)(471,756)(472,755)(473,749)
(474,750)(475,752)(476,751)(477,745)(478,746)(479,748)(480,747)(481,741)
(482,742)(483,744)(484,743)(485,737)(486,738)(487,740)(488,739)(489,733)
(490,734)(491,736)(492,735)(493,729)(494,730)(495,732)(496,731)(497,725)
(498,726)(499,728)(500,727)(501,721)(502,722)(503,724)(504,723)(505,717)
(506,718)(507,720)(508,719)(509,713)(510,714)(511,716)(512,715)(513,709)
(514,710)(515,712)(516,711)(517,705)(518,706)(519,708)(520,707)(521,701)
(522,702)(523,704)(524,703)(525,697)(526,698)(527,700)(528,699)(529,693)
(530,694)(531,696)(532,695)(533,689)(534,690)(535,692)(536,691)(537,685)
(538,686)(539,688)(540,687)(541,757)(542,758)(543,760)(544,759)(545,765)
(546,766)(547,768)(548,767)(549,761)(550,762)(551,764)(552,763)(553,789)
(554,790)(555,792)(556,791)(557,785)(558,786)(559,788)(560,787)(561,781)
(562,782)(563,784)(564,783)(565,777)(566,778)(567,780)(568,779)(569,773)
(570,774)(571,776)(572,775)(573,769)(574,770)(575,772)(576,771)(577,861)
(578,862)(579,864)(580,863)(581,857)(582,858)(583,860)(584,859)(585,853)
(586,854)(587,856)(588,855)(589,849)(590,850)(591,852)(592,851)(593,845)
(594,846)(595,848)(596,847)(597,841)(598,842)(599,844)(600,843)(601,837)
(602,838)(603,840)(604,839)(605,833)(606,834)(607,836)(608,835)(609,829)
(610,830)(611,832)(612,831)(613,825)(614,826)(615,828)(616,827)(617,821)
(618,822)(619,824)(620,823)(621,817)(622,818)(623,820)(624,819)(625,813)
(626,814)(627,816)(628,815)(629,809)(630,810)(631,812)(632,811)(633,805)
(634,806)(635,808)(636,807)(637,801)(638,802)(639,804)(640,803)(641,797)
(642,798)(643,800)(644,799)(645,793)(646,794)(647,796)(648,795);
s1 := Sym(864)!(  1,469)(  2,472)(  3,471)(  4,470)(  5,477)(  6,480)(  7,479)
(  8,478)(  9,473)( 10,476)( 11,475)( 12,474)( 13,501)( 14,504)( 15,503)
( 16,502)( 17,497)( 18,500)( 19,499)( 20,498)( 21,493)( 22,496)( 23,495)
( 24,494)( 25,489)( 26,492)( 27,491)( 28,490)( 29,485)( 30,488)( 31,487)
( 32,486)( 33,481)( 34,484)( 35,483)( 36,482)( 37,433)( 38,436)( 39,435)
( 40,434)( 41,441)( 42,444)( 43,443)( 44,442)( 45,437)( 46,440)( 47,439)
( 48,438)( 49,465)( 50,468)( 51,467)( 52,466)( 53,461)( 54,464)( 55,463)
( 56,462)( 57,457)( 58,460)( 59,459)( 60,458)( 61,453)( 62,456)( 63,455)
( 64,454)( 65,449)( 66,452)( 67,451)( 68,450)( 69,445)( 70,448)( 71,447)
( 72,446)( 73,537)( 74,540)( 75,539)( 76,538)( 77,533)( 78,536)( 79,535)
( 80,534)( 81,529)( 82,532)( 83,531)( 84,530)( 85,525)( 86,528)( 87,527)
( 88,526)( 89,521)( 90,524)( 91,523)( 92,522)( 93,517)( 94,520)( 95,519)
( 96,518)( 97,513)( 98,516)( 99,515)(100,514)(101,509)(102,512)(103,511)
(104,510)(105,505)(106,508)(107,507)(108,506)(109,577)(110,580)(111,579)
(112,578)(113,585)(114,588)(115,587)(116,586)(117,581)(118,584)(119,583)
(120,582)(121,609)(122,612)(123,611)(124,610)(125,605)(126,608)(127,607)
(128,606)(129,601)(130,604)(131,603)(132,602)(133,597)(134,600)(135,599)
(136,598)(137,593)(138,596)(139,595)(140,594)(141,589)(142,592)(143,591)
(144,590)(145,541)(146,544)(147,543)(148,542)(149,549)(150,552)(151,551)
(152,550)(153,545)(154,548)(155,547)(156,546)(157,573)(158,576)(159,575)
(160,574)(161,569)(162,572)(163,571)(164,570)(165,565)(166,568)(167,567)
(168,566)(169,561)(170,564)(171,563)(172,562)(173,557)(174,560)(175,559)
(176,558)(177,553)(178,556)(179,555)(180,554)(181,645)(182,648)(183,647)
(184,646)(185,641)(186,644)(187,643)(188,642)(189,637)(190,640)(191,639)
(192,638)(193,633)(194,636)(195,635)(196,634)(197,629)(198,632)(199,631)
(200,630)(201,625)(202,628)(203,627)(204,626)(205,621)(206,624)(207,623)
(208,622)(209,617)(210,620)(211,619)(212,618)(213,613)(214,616)(215,615)
(216,614)(217,793)(218,796)(219,795)(220,794)(221,801)(222,804)(223,803)
(224,802)(225,797)(226,800)(227,799)(228,798)(229,825)(230,828)(231,827)
(232,826)(233,821)(234,824)(235,823)(236,822)(237,817)(238,820)(239,819)
(240,818)(241,813)(242,816)(243,815)(244,814)(245,809)(246,812)(247,811)
(248,810)(249,805)(250,808)(251,807)(252,806)(253,757)(254,760)(255,759)
(256,758)(257,765)(258,768)(259,767)(260,766)(261,761)(262,764)(263,763)
(264,762)(265,789)(266,792)(267,791)(268,790)(269,785)(270,788)(271,787)
(272,786)(273,781)(274,784)(275,783)(276,782)(277,777)(278,780)(279,779)
(280,778)(281,773)(282,776)(283,775)(284,774)(285,769)(286,772)(287,771)
(288,770)(289,861)(290,864)(291,863)(292,862)(293,857)(294,860)(295,859)
(296,858)(297,853)(298,856)(299,855)(300,854)(301,849)(302,852)(303,851)
(304,850)(305,845)(306,848)(307,847)(308,846)(309,841)(310,844)(311,843)
(312,842)(313,837)(314,840)(315,839)(316,838)(317,833)(318,836)(319,835)
(320,834)(321,829)(322,832)(323,831)(324,830)(325,685)(326,688)(327,687)
(328,686)(329,693)(330,696)(331,695)(332,694)(333,689)(334,692)(335,691)
(336,690)(337,717)(338,720)(339,719)(340,718)(341,713)(342,716)(343,715)
(344,714)(345,709)(346,712)(347,711)(348,710)(349,705)(350,708)(351,707)
(352,706)(353,701)(354,704)(355,703)(356,702)(357,697)(358,700)(359,699)
(360,698)(361,649)(362,652)(363,651)(364,650)(365,657)(366,660)(367,659)
(368,658)(369,653)(370,656)(371,655)(372,654)(373,681)(374,684)(375,683)
(376,682)(377,677)(378,680)(379,679)(380,678)(381,673)(382,676)(383,675)
(384,674)(385,669)(386,672)(387,671)(388,670)(389,665)(390,668)(391,667)
(392,666)(393,661)(394,664)(395,663)(396,662)(397,753)(398,756)(399,755)
(400,754)(401,749)(402,752)(403,751)(404,750)(405,745)(406,748)(407,747)
(408,746)(409,741)(410,744)(411,743)(412,742)(413,737)(414,740)(415,739)
(416,738)(417,733)(418,736)(419,735)(420,734)(421,729)(422,732)(423,731)
(424,730)(425,725)(426,728)(427,727)(428,726)(429,721)(430,724)(431,723)
(432,722);
s2 := Sym(864)!(  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)
( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)
( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)
( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)
( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)
( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)
( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)
(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)
(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)
(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)
(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)
(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)
(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)
(207,208)(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)
(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)
(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)
(255,256)(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)
(271,272)(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)
(287,288)(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)
(303,304)(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)
(319,320)(321,322)(323,324)(325,326)(327,328)(329,330)(331,332)(333,334)
(335,336)(337,338)(339,340)(341,342)(343,344)(345,346)(347,348)(349,350)
(351,352)(353,354)(355,356)(357,358)(359,360)(361,362)(363,364)(365,366)
(367,368)(369,370)(371,372)(373,374)(375,376)(377,378)(379,380)(381,382)
(383,384)(385,386)(387,388)(389,390)(391,392)(393,394)(395,396)(397,398)
(399,400)(401,402)(403,404)(405,406)(407,408)(409,410)(411,412)(413,414)
(415,416)(417,418)(419,420)(421,422)(423,424)(425,426)(427,428)(429,430)
(431,432)(433,434)(435,436)(437,438)(439,440)(441,442)(443,444)(445,446)
(447,448)(449,450)(451,452)(453,454)(455,456)(457,458)(459,460)(461,462)
(463,464)(465,466)(467,468)(469,470)(471,472)(473,474)(475,476)(477,478)
(479,480)(481,482)(483,484)(485,486)(487,488)(489,490)(491,492)(493,494)
(495,496)(497,498)(499,500)(501,502)(503,504)(505,506)(507,508)(509,510)
(511,512)(513,514)(515,516)(517,518)(519,520)(521,522)(523,524)(525,526)
(527,528)(529,530)(531,532)(533,534)(535,536)(537,538)(539,540)(541,542)
(543,544)(545,546)(547,548)(549,550)(551,552)(553,554)(555,556)(557,558)
(559,560)(561,562)(563,564)(565,566)(567,568)(569,570)(571,572)(573,574)
(575,576)(577,578)(579,580)(581,582)(583,584)(585,586)(587,588)(589,590)
(591,592)(593,594)(595,596)(597,598)(599,600)(601,602)(603,604)(605,606)
(607,608)(609,610)(611,612)(613,614)(615,616)(617,618)(619,620)(621,622)
(623,624)(625,626)(627,628)(629,630)(631,632)(633,634)(635,636)(637,638)
(639,640)(641,642)(643,644)(645,646)(647,648)(649,650)(651,652)(653,654)
(655,656)(657,658)(659,660)(661,662)(663,664)(665,666)(667,668)(669,670)
(671,672)(673,674)(675,676)(677,678)(679,680)(681,682)(683,684)(685,686)
(687,688)(689,690)(691,692)(693,694)(695,696)(697,698)(699,700)(701,702)
(703,704)(705,706)(707,708)(709,710)(711,712)(713,714)(715,716)(717,718)
(719,720)(721,722)(723,724)(725,726)(727,728)(729,730)(731,732)(733,734)
(735,736)(737,738)(739,740)(741,742)(743,744)(745,746)(747,748)(749,750)
(751,752)(753,754)(755,756)(757,758)(759,760)(761,762)(763,764)(765,766)
(767,768)(769,770)(771,772)(773,774)(775,776)(777,778)(779,780)(781,782)
(783,784)(785,786)(787,788)(789,790)(791,792)(793,794)(795,796)(797,798)
(799,800)(801,802)(803,804)(805,806)(807,808)(809,810)(811,812)(813,814)
(815,816)(817,818)(819,820)(821,822)(823,824)(825,826)(827,828)(829,830)
(831,832)(833,834)(835,836)(837,838)(839,840)(841,842)(843,844)(845,846)
(847,848)(849,850)(851,852)(853,854)(855,856)(857,858)(859,860)(861,862)
(863,864);
poly := sub<Sym(864)|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, 
s2*s1*s0*s2*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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