Polytope of Type {24,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,12}*1728h
if this polytope has a name.
Group : SmallGroup(1728,12653)
Rank : 3
Schlafli Type : {24,12}
Number of vertices, edges, etc : 72, 432, 36
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {24,6}*864d, {12,12}*864d
   3-fold quotients : {8,12}*576a
   4-fold quotients : {12,6}*432e
   6-fold quotients : {8,6}*288, {4,12}*288
   8-fold quotients : {12,6}*216b
   12-fold quotients : {4,6}*144
   24-fold quotients : {4,6}*72
   27-fold quotients : {8,4}*64a
   54-fold quotients : {4,4}*32, {8,2}*32
   108-fold quotients : {2,4}*16, {4,2}*16
   216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope