Polytope of Type {228,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {228,4}*1824b
if this polytope has a name.
Group : SmallGroup(1824,1071)
Rank : 3
Schlafli Type : {228,4}
Number of vertices, edges, etc : 228, 456, 4
Order of s₀s₁s₂ : 228
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 : {114,4}*912b
   4-fold quotients : {57,4}*456
   19-fold quotients : {12,4}*96b
   38-fold quotients : {6,4}*48c
   76-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  3,  4)(  5, 73)(  6, 74)(  7, 76)(  8, 75)(  9, 69)( 10, 70)( 11, 72)
( 12, 71)( 13, 65)( 14, 66)( 15, 68)( 16, 67)( 17, 61)( 18, 62)( 19, 64)
( 20, 63)( 21, 57)( 22, 58)( 23, 60)( 24, 59)( 25, 53)( 26, 54)( 27, 56)
( 28, 55)( 29, 49)( 30, 50)( 31, 52)( 32, 51)( 33, 45)( 34, 46)( 35, 48)
( 36, 47)( 37, 41)( 38, 42)( 39, 44)( 40, 43)( 77,153)( 78,154)( 79,156)
( 80,155)( 81,225)( 82,226)( 83,228)( 84,227)( 85,221)( 86,222)( 87,224)
( 88,223)( 89,217)( 90,218)( 91,220)( 92,219)( 93,213)( 94,214)( 95,216)
( 96,215)( 97,209)( 98,210)( 99,212)(100,211)(101,205)(102,206)(103,208)
(104,207)(105,201)(106,202)(107,204)(108,203)(109,197)(110,198)(111,200)
(112,199)(113,193)(114,194)(115,196)(116,195)(117,189)(118,190)(119,192)
(120,191)(121,185)(122,186)(123,188)(124,187)(125,181)(126,182)(127,184)
(128,183)(129,177)(130,178)(131,180)(132,179)(133,173)(134,174)(135,176)
(136,175)(137,169)(138,170)(139,172)(140,171)(141,165)(142,166)(143,168)
(144,167)(145,161)(146,162)(147,164)(148,163)(149,157)(150,158)(151,160)
(152,159)(231,232)(233,301)(234,302)(235,304)(236,303)(237,297)(238,298)
(239,300)(240,299)(241,293)(242,294)(243,296)(244,295)(245,289)(246,290)
(247,292)(248,291)(249,285)(250,286)(251,288)(252,287)(253,281)(254,282)
(255,284)(256,283)(257,277)(258,278)(259,280)(260,279)(261,273)(262,274)
(263,276)(264,275)(265,269)(266,270)(267,272)(268,271)(305,381)(306,382)
(307,384)(308,383)(309,453)(310,454)(311,456)(312,455)(313,449)(314,450)
(315,452)(316,451)(317,445)(318,446)(319,448)(320,447)(321,441)(322,442)
(323,444)(324,443)(325,437)(326,438)(327,440)(328,439)(329,433)(330,434)
(331,436)(332,435)(333,429)(334,430)(335,432)(336,431)(337,425)(338,426)
(339,428)(340,427)(341,421)(342,422)(343,424)(344,423)(345,417)(346,418)
(347,420)(348,419)(349,413)(350,414)(351,416)(352,415)(353,409)(354,410)
(355,412)(356,411)(357,405)(358,406)(359,408)(360,407)(361,401)(362,402)
(363,404)(364,403)(365,397)(366,398)(367,400)(368,399)(369,393)(370,394)
(371,396)(372,395)(373,389)(374,390)(375,392)(376,391)(377,385)(378,386)
(379,388)(380,387)(457,685)(458,686)(459,688)(460,687)(461,757)(462,758)
(463,760)(464,759)(465,753)(466,754)(467,756)(468,755)(469,749)(470,750)
(471,752)(472,751)(473,745)(474,746)(475,748)(476,747)(477,741)(478,742)
(479,744)(480,743)(481,737)(482,738)(483,740)(484,739)(485,733)(486,734)
(487,736)(488,735)(489,729)(490,730)(491,732)(492,731)(493,725)(494,726)
(495,728)(496,727)(497,721)(498,722)(499,724)(500,723)(501,717)(502,718)
(503,720)(504,719)(505,713)(506,714)(507,716)(508,715)(509,709)(510,710)
(511,712)(512,711)(513,705)(514,706)(515,708)(516,707)(517,701)(518,702)
(519,704)(520,703)(521,697)(522,698)(523,700)(524,699)(525,693)(526,694)
(527,696)(528,695)(529,689)(530,690)(531,692)(532,691)(533,837)(534,838)
(535,840)(536,839)(537,909)(538,910)(539,912)(540,911)(541,905)(542,906)
(543,908)(544,907)(545,901)(546,902)(547,904)(548,903)(549,897)(550,898)
(551,900)(552,899)(553,893)(554,894)(555,896)(556,895)(557,889)(558,890)
(559,892)(560,891)(561,885)(562,886)(563,888)(564,887)(565,881)(566,882)
(567,884)(568,883)(569,877)(570,878)(571,880)(572,879)(573,873)(574,874)
(575,876)(576,875)(577,869)(578,870)(579,872)(580,871)(581,865)(582,866)
(583,868)(584,867)(585,861)(586,862)(587,864)(588,863)(589,857)(590,858)
(591,860)(592,859)(593,853)(594,854)(595,856)(596,855)(597,849)(598,850)
(599,852)(600,851)(601,845)(602,846)(603,848)(604,847)(605,841)(606,842)
(607,844)(608,843)(609,761)(610,762)(611,764)(612,763)(613,833)(614,834)
(615,836)(616,835)(617,829)(618,830)(619,832)(620,831)(621,825)(622,826)
(623,828)(624,827)(625,821)(626,822)(627,824)(628,823)(629,817)(630,818)
(631,820)(632,819)(633,813)(634,814)(635,816)(636,815)(637,809)(638,810)
(639,812)(640,811)(641,805)(642,806)(643,808)(644,807)(645,801)(646,802)
(647,804)(648,803)(649,797)(650,798)(651,800)(652,799)(653,793)(654,794)
(655,796)(656,795)(657,789)(658,790)(659,792)(660,791)(661,785)(662,786)
(663,788)(664,787)(665,781)(666,782)(667,784)(668,783)(669,777)(670,778)
(671,780)(672,779)(673,773)(674,774)(675,776)(676,775)(677,769)(678,770)
(679,772)(680,771)(681,765)(682,766)(683,768)(684,767);;
s1 := (  1,537)(  2,540)(  3,539)(  4,538)(  5,533)(  6,536)(  7,535)(  8,534)
(  9,605)( 10,608)( 11,607)( 12,606)( 13,601)( 14,604)( 15,603)( 16,602)
( 17,597)( 18,600)( 19,599)( 20,598)( 21,593)( 22,596)( 23,595)( 24,594)
( 25,589)( 26,592)( 27,591)( 28,590)( 29,585)( 30,588)( 31,587)( 32,586)
( 33,581)( 34,584)( 35,583)( 36,582)( 37,577)( 38,580)( 39,579)( 40,578)
( 41,573)( 42,576)( 43,575)( 44,574)( 45,569)( 46,572)( 47,571)( 48,570)
( 49,565)( 50,568)( 51,567)( 52,566)( 53,561)( 54,564)( 55,563)( 56,562)
( 57,557)( 58,560)( 59,559)( 60,558)( 61,553)( 62,556)( 63,555)( 64,554)
( 65,549)( 66,552)( 67,551)( 68,550)( 69,545)( 70,548)( 71,547)( 72,546)
( 73,541)( 74,544)( 75,543)( 76,542)( 77,461)( 78,464)( 79,463)( 80,462)
( 81,457)( 82,460)( 83,459)( 84,458)( 85,529)( 86,532)( 87,531)( 88,530)
( 89,525)( 90,528)( 91,527)( 92,526)( 93,521)( 94,524)( 95,523)( 96,522)
( 97,517)( 98,520)( 99,519)(100,518)(101,513)(102,516)(103,515)(104,514)
(105,509)(106,512)(107,511)(108,510)(109,505)(110,508)(111,507)(112,506)
(113,501)(114,504)(115,503)(116,502)(117,497)(118,500)(119,499)(120,498)
(121,493)(122,496)(123,495)(124,494)(125,489)(126,492)(127,491)(128,490)
(129,485)(130,488)(131,487)(132,486)(133,481)(134,484)(135,483)(136,482)
(137,477)(138,480)(139,479)(140,478)(141,473)(142,476)(143,475)(144,474)
(145,469)(146,472)(147,471)(148,470)(149,465)(150,468)(151,467)(152,466)
(153,613)(154,616)(155,615)(156,614)(157,609)(158,612)(159,611)(160,610)
(161,681)(162,684)(163,683)(164,682)(165,677)(166,680)(167,679)(168,678)
(169,673)(170,676)(171,675)(172,674)(173,669)(174,672)(175,671)(176,670)
(177,665)(178,668)(179,667)(180,666)(181,661)(182,664)(183,663)(184,662)
(185,657)(186,660)(187,659)(188,658)(189,653)(190,656)(191,655)(192,654)
(193,649)(194,652)(195,651)(196,650)(197,645)(198,648)(199,647)(200,646)
(201,641)(202,644)(203,643)(204,642)(205,637)(206,640)(207,639)(208,638)
(209,633)(210,636)(211,635)(212,634)(213,629)(214,632)(215,631)(216,630)
(217,625)(218,628)(219,627)(220,626)(221,621)(222,624)(223,623)(224,622)
(225,617)(226,620)(227,619)(228,618)(229,765)(230,768)(231,767)(232,766)
(233,761)(234,764)(235,763)(236,762)(237,833)(238,836)(239,835)(240,834)
(241,829)(242,832)(243,831)(244,830)(245,825)(246,828)(247,827)(248,826)
(249,821)(250,824)(251,823)(252,822)(253,817)(254,820)(255,819)(256,818)
(257,813)(258,816)(259,815)(260,814)(261,809)(262,812)(263,811)(264,810)
(265,805)(266,808)(267,807)(268,806)(269,801)(270,804)(271,803)(272,802)
(273,797)(274,800)(275,799)(276,798)(277,793)(278,796)(279,795)(280,794)
(281,789)(282,792)(283,791)(284,790)(285,785)(286,788)(287,787)(288,786)
(289,781)(290,784)(291,783)(292,782)(293,777)(294,780)(295,779)(296,778)
(297,773)(298,776)(299,775)(300,774)(301,769)(302,772)(303,771)(304,770)
(305,689)(306,692)(307,691)(308,690)(309,685)(310,688)(311,687)(312,686)
(313,757)(314,760)(315,759)(316,758)(317,753)(318,756)(319,755)(320,754)
(321,749)(322,752)(323,751)(324,750)(325,745)(326,748)(327,747)(328,746)
(329,741)(330,744)(331,743)(332,742)(333,737)(334,740)(335,739)(336,738)
(337,733)(338,736)(339,735)(340,734)(341,729)(342,732)(343,731)(344,730)
(345,725)(346,728)(347,727)(348,726)(349,721)(350,724)(351,723)(352,722)
(353,717)(354,720)(355,719)(356,718)(357,713)(358,716)(359,715)(360,714)
(361,709)(362,712)(363,711)(364,710)(365,705)(366,708)(367,707)(368,706)
(369,701)(370,704)(371,703)(372,702)(373,697)(374,700)(375,699)(376,698)
(377,693)(378,696)(379,695)(380,694)(381,841)(382,844)(383,843)(384,842)
(385,837)(386,840)(387,839)(388,838)(389,909)(390,912)(391,911)(392,910)
(393,905)(394,908)(395,907)(396,906)(397,901)(398,904)(399,903)(400,902)
(401,897)(402,900)(403,899)(404,898)(405,893)(406,896)(407,895)(408,894)
(409,889)(410,892)(411,891)(412,890)(413,885)(414,888)(415,887)(416,886)
(417,881)(418,884)(419,883)(420,882)(421,877)(422,880)(423,879)(424,878)
(425,873)(426,876)(427,875)(428,874)(429,869)(430,872)(431,871)(432,870)
(433,865)(434,868)(435,867)(436,866)(437,861)(438,864)(439,863)(440,862)
(441,857)(442,860)(443,859)(444,858)(445,853)(446,856)(447,855)(448,854)
(449,849)(450,852)(451,851)(452,850)(453,845)(454,848)(455,847)(456,846);;
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)
(865,866)(867,868)(869,870)(871,872)(873,874)(875,876)(877,878)(879,880)
(881,882)(883,884)(885,886)(887,888)(889,890)(891,892)(893,894)(895,896)
(897,898)(899,900)(901,902)(903,904)(905,906)(907,908)(909,910)(911,912);;
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*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(912)!(  3,  4)(  5, 73)(  6, 74)(  7, 76)(  8, 75)(  9, 69)( 10, 70)
( 11, 72)( 12, 71)( 13, 65)( 14, 66)( 15, 68)( 16, 67)( 17, 61)( 18, 62)
( 19, 64)( 20, 63)( 21, 57)( 22, 58)( 23, 60)( 24, 59)( 25, 53)( 26, 54)
( 27, 56)( 28, 55)( 29, 49)( 30, 50)( 31, 52)( 32, 51)( 33, 45)( 34, 46)
( 35, 48)( 36, 47)( 37, 41)( 38, 42)( 39, 44)( 40, 43)( 77,153)( 78,154)
( 79,156)( 80,155)( 81,225)( 82,226)( 83,228)( 84,227)( 85,221)( 86,222)
( 87,224)( 88,223)( 89,217)( 90,218)( 91,220)( 92,219)( 93,213)( 94,214)
( 95,216)( 96,215)( 97,209)( 98,210)( 99,212)(100,211)(101,205)(102,206)
(103,208)(104,207)(105,201)(106,202)(107,204)(108,203)(109,197)(110,198)
(111,200)(112,199)(113,193)(114,194)(115,196)(116,195)(117,189)(118,190)
(119,192)(120,191)(121,185)(122,186)(123,188)(124,187)(125,181)(126,182)
(127,184)(128,183)(129,177)(130,178)(131,180)(132,179)(133,173)(134,174)
(135,176)(136,175)(137,169)(138,170)(139,172)(140,171)(141,165)(142,166)
(143,168)(144,167)(145,161)(146,162)(147,164)(148,163)(149,157)(150,158)
(151,160)(152,159)(231,232)(233,301)(234,302)(235,304)(236,303)(237,297)
(238,298)(239,300)(240,299)(241,293)(242,294)(243,296)(244,295)(245,289)
(246,290)(247,292)(248,291)(249,285)(250,286)(251,288)(252,287)(253,281)
(254,282)(255,284)(256,283)(257,277)(258,278)(259,280)(260,279)(261,273)
(262,274)(263,276)(264,275)(265,269)(266,270)(267,272)(268,271)(305,381)
(306,382)(307,384)(308,383)(309,453)(310,454)(311,456)(312,455)(313,449)
(314,450)(315,452)(316,451)(317,445)(318,446)(319,448)(320,447)(321,441)
(322,442)(323,444)(324,443)(325,437)(326,438)(327,440)(328,439)(329,433)
(330,434)(331,436)(332,435)(333,429)(334,430)(335,432)(336,431)(337,425)
(338,426)(339,428)(340,427)(341,421)(342,422)(343,424)(344,423)(345,417)
(346,418)(347,420)(348,419)(349,413)(350,414)(351,416)(352,415)(353,409)
(354,410)(355,412)(356,411)(357,405)(358,406)(359,408)(360,407)(361,401)
(362,402)(363,404)(364,403)(365,397)(366,398)(367,400)(368,399)(369,393)
(370,394)(371,396)(372,395)(373,389)(374,390)(375,392)(376,391)(377,385)
(378,386)(379,388)(380,387)(457,685)(458,686)(459,688)(460,687)(461,757)
(462,758)(463,760)(464,759)(465,753)(466,754)(467,756)(468,755)(469,749)
(470,750)(471,752)(472,751)(473,745)(474,746)(475,748)(476,747)(477,741)
(478,742)(479,744)(480,743)(481,737)(482,738)(483,740)(484,739)(485,733)
(486,734)(487,736)(488,735)(489,729)(490,730)(491,732)(492,731)(493,725)
(494,726)(495,728)(496,727)(497,721)(498,722)(499,724)(500,723)(501,717)
(502,718)(503,720)(504,719)(505,713)(506,714)(507,716)(508,715)(509,709)
(510,710)(511,712)(512,711)(513,705)(514,706)(515,708)(516,707)(517,701)
(518,702)(519,704)(520,703)(521,697)(522,698)(523,700)(524,699)(525,693)
(526,694)(527,696)(528,695)(529,689)(530,690)(531,692)(532,691)(533,837)
(534,838)(535,840)(536,839)(537,909)(538,910)(539,912)(540,911)(541,905)
(542,906)(543,908)(544,907)(545,901)(546,902)(547,904)(548,903)(549,897)
(550,898)(551,900)(552,899)(553,893)(554,894)(555,896)(556,895)(557,889)
(558,890)(559,892)(560,891)(561,885)(562,886)(563,888)(564,887)(565,881)
(566,882)(567,884)(568,883)(569,877)(570,878)(571,880)(572,879)(573,873)
(574,874)(575,876)(576,875)(577,869)(578,870)(579,872)(580,871)(581,865)
(582,866)(583,868)(584,867)(585,861)(586,862)(587,864)(588,863)(589,857)
(590,858)(591,860)(592,859)(593,853)(594,854)(595,856)(596,855)(597,849)
(598,850)(599,852)(600,851)(601,845)(602,846)(603,848)(604,847)(605,841)
(606,842)(607,844)(608,843)(609,761)(610,762)(611,764)(612,763)(613,833)
(614,834)(615,836)(616,835)(617,829)(618,830)(619,832)(620,831)(621,825)
(622,826)(623,828)(624,827)(625,821)(626,822)(627,824)(628,823)(629,817)
(630,818)(631,820)(632,819)(633,813)(634,814)(635,816)(636,815)(637,809)
(638,810)(639,812)(640,811)(641,805)(642,806)(643,808)(644,807)(645,801)
(646,802)(647,804)(648,803)(649,797)(650,798)(651,800)(652,799)(653,793)
(654,794)(655,796)(656,795)(657,789)(658,790)(659,792)(660,791)(661,785)
(662,786)(663,788)(664,787)(665,781)(666,782)(667,784)(668,783)(669,777)
(670,778)(671,780)(672,779)(673,773)(674,774)(675,776)(676,775)(677,769)
(678,770)(679,772)(680,771)(681,765)(682,766)(683,768)(684,767);
s1 := Sym(912)!(  1,537)(  2,540)(  3,539)(  4,538)(  5,533)(  6,536)(  7,535)
(  8,534)(  9,605)( 10,608)( 11,607)( 12,606)( 13,601)( 14,604)( 15,603)
( 16,602)( 17,597)( 18,600)( 19,599)( 20,598)( 21,593)( 22,596)( 23,595)
( 24,594)( 25,589)( 26,592)( 27,591)( 28,590)( 29,585)( 30,588)( 31,587)
( 32,586)( 33,581)( 34,584)( 35,583)( 36,582)( 37,577)( 38,580)( 39,579)
( 40,578)( 41,573)( 42,576)( 43,575)( 44,574)( 45,569)( 46,572)( 47,571)
( 48,570)( 49,565)( 50,568)( 51,567)( 52,566)( 53,561)( 54,564)( 55,563)
( 56,562)( 57,557)( 58,560)( 59,559)( 60,558)( 61,553)( 62,556)( 63,555)
( 64,554)( 65,549)( 66,552)( 67,551)( 68,550)( 69,545)( 70,548)( 71,547)
( 72,546)( 73,541)( 74,544)( 75,543)( 76,542)( 77,461)( 78,464)( 79,463)
( 80,462)( 81,457)( 82,460)( 83,459)( 84,458)( 85,529)( 86,532)( 87,531)
( 88,530)( 89,525)( 90,528)( 91,527)( 92,526)( 93,521)( 94,524)( 95,523)
( 96,522)( 97,517)( 98,520)( 99,519)(100,518)(101,513)(102,516)(103,515)
(104,514)(105,509)(106,512)(107,511)(108,510)(109,505)(110,508)(111,507)
(112,506)(113,501)(114,504)(115,503)(116,502)(117,497)(118,500)(119,499)
(120,498)(121,493)(122,496)(123,495)(124,494)(125,489)(126,492)(127,491)
(128,490)(129,485)(130,488)(131,487)(132,486)(133,481)(134,484)(135,483)
(136,482)(137,477)(138,480)(139,479)(140,478)(141,473)(142,476)(143,475)
(144,474)(145,469)(146,472)(147,471)(148,470)(149,465)(150,468)(151,467)
(152,466)(153,613)(154,616)(155,615)(156,614)(157,609)(158,612)(159,611)
(160,610)(161,681)(162,684)(163,683)(164,682)(165,677)(166,680)(167,679)
(168,678)(169,673)(170,676)(171,675)(172,674)(173,669)(174,672)(175,671)
(176,670)(177,665)(178,668)(179,667)(180,666)(181,661)(182,664)(183,663)
(184,662)(185,657)(186,660)(187,659)(188,658)(189,653)(190,656)(191,655)
(192,654)(193,649)(194,652)(195,651)(196,650)(197,645)(198,648)(199,647)
(200,646)(201,641)(202,644)(203,643)(204,642)(205,637)(206,640)(207,639)
(208,638)(209,633)(210,636)(211,635)(212,634)(213,629)(214,632)(215,631)
(216,630)(217,625)(218,628)(219,627)(220,626)(221,621)(222,624)(223,623)
(224,622)(225,617)(226,620)(227,619)(228,618)(229,765)(230,768)(231,767)
(232,766)(233,761)(234,764)(235,763)(236,762)(237,833)(238,836)(239,835)
(240,834)(241,829)(242,832)(243,831)(244,830)(245,825)(246,828)(247,827)
(248,826)(249,821)(250,824)(251,823)(252,822)(253,817)(254,820)(255,819)
(256,818)(257,813)(258,816)(259,815)(260,814)(261,809)(262,812)(263,811)
(264,810)(265,805)(266,808)(267,807)(268,806)(269,801)(270,804)(271,803)
(272,802)(273,797)(274,800)(275,799)(276,798)(277,793)(278,796)(279,795)
(280,794)(281,789)(282,792)(283,791)(284,790)(285,785)(286,788)(287,787)
(288,786)(289,781)(290,784)(291,783)(292,782)(293,777)(294,780)(295,779)
(296,778)(297,773)(298,776)(299,775)(300,774)(301,769)(302,772)(303,771)
(304,770)(305,689)(306,692)(307,691)(308,690)(309,685)(310,688)(311,687)
(312,686)(313,757)(314,760)(315,759)(316,758)(317,753)(318,756)(319,755)
(320,754)(321,749)(322,752)(323,751)(324,750)(325,745)(326,748)(327,747)
(328,746)(329,741)(330,744)(331,743)(332,742)(333,737)(334,740)(335,739)
(336,738)(337,733)(338,736)(339,735)(340,734)(341,729)(342,732)(343,731)
(344,730)(345,725)(346,728)(347,727)(348,726)(349,721)(350,724)(351,723)
(352,722)(353,717)(354,720)(355,719)(356,718)(357,713)(358,716)(359,715)
(360,714)(361,709)(362,712)(363,711)(364,710)(365,705)(366,708)(367,707)
(368,706)(369,701)(370,704)(371,703)(372,702)(373,697)(374,700)(375,699)
(376,698)(377,693)(378,696)(379,695)(380,694)(381,841)(382,844)(383,843)
(384,842)(385,837)(386,840)(387,839)(388,838)(389,909)(390,912)(391,911)
(392,910)(393,905)(394,908)(395,907)(396,906)(397,901)(398,904)(399,903)
(400,902)(401,897)(402,900)(403,899)(404,898)(405,893)(406,896)(407,895)
(408,894)(409,889)(410,892)(411,891)(412,890)(413,885)(414,888)(415,887)
(416,886)(417,881)(418,884)(419,883)(420,882)(421,877)(422,880)(423,879)
(424,878)(425,873)(426,876)(427,875)(428,874)(429,869)(430,872)(431,871)
(432,870)(433,865)(434,868)(435,867)(436,866)(437,861)(438,864)(439,863)
(440,862)(441,857)(442,860)(443,859)(444,858)(445,853)(446,856)(447,855)
(448,854)(449,849)(450,852)(451,851)(452,850)(453,845)(454,848)(455,847)
(456,846);
s2 := Sym(912)!(  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)(865,866)(867,868)(869,870)(871,872)(873,874)(875,876)(877,878)
(879,880)(881,882)(883,884)(885,886)(887,888)(889,890)(891,892)(893,894)
(895,896)(897,898)(899,900)(901,902)(903,904)(905,906)(907,908)(909,910)
(911,912);
poly := sub<Sym(912)|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*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