Polytope of Type {24,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,12}*1728t
if this polytope has a name.
Group : SmallGroup(1728,31623)
Rank : 3
Schlafli Type : {24,12}
Number of vertices, edges, etc : 72, 432, 36
Order of s₀s₁s₂ : 8
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {12,12}*864i
   3-fold quotients : {8,12}*576b
   4-fold quotients : {12,6}*432h
   6-fold quotients : {4,12}*288
   12-fold quotients : {4,6}*144
   24-fold quotients : {4,6}*72
   27-fold quotients : {8,4}*64b
   54-fold quotients : {4,4}*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 := (  1,217)(  2,224)(  3,222)(  4,223)(  5,221)(  6,219)(  7,220)(  8,218)
(  9,225)( 10,235)( 11,242)( 12,240)( 13,241)( 14,239)( 15,237)( 16,238)
( 17,236)( 18,243)( 19,226)( 20,233)( 21,231)( 22,232)( 23,230)( 24,228)
( 25,229)( 26,227)( 27,234)( 28,244)( 29,251)( 30,249)( 31,250)( 32,248)
( 33,246)( 34,247)( 35,245)( 36,252)( 37,262)( 38,269)( 39,267)( 40,268)
( 41,266)( 42,264)( 43,265)( 44,263)( 45,270)( 46,253)( 47,260)( 48,258)
( 49,259)( 50,257)( 51,255)( 52,256)( 53,254)( 54,261)( 55,298)( 56,305)
( 57,303)( 58,304)( 59,302)( 60,300)( 61,301)( 62,299)( 63,306)( 64,316)
( 65,323)( 66,321)( 67,322)( 68,320)( 69,318)( 70,319)( 71,317)( 72,324)
( 73,307)( 74,314)( 75,312)( 76,313)( 77,311)( 78,309)( 79,310)( 80,308)
( 81,315)( 82,271)( 83,278)( 84,276)( 85,277)( 86,275)( 87,273)( 88,274)
( 89,272)( 90,279)( 91,289)( 92,296)( 93,294)( 94,295)( 95,293)( 96,291)
( 97,292)( 98,290)( 99,297)(100,280)(101,287)(102,285)(103,286)(104,284)
(105,282)(106,283)(107,281)(108,288)(109,325)(110,332)(111,330)(112,331)
(113,329)(114,327)(115,328)(116,326)(117,333)(118,343)(119,350)(120,348)
(121,349)(122,347)(123,345)(124,346)(125,344)(126,351)(127,334)(128,341)
(129,339)(130,340)(131,338)(132,336)(133,337)(134,335)(135,342)(136,352)
(137,359)(138,357)(139,358)(140,356)(141,354)(142,355)(143,353)(144,360)
(145,370)(146,377)(147,375)(148,376)(149,374)(150,372)(151,373)(152,371)
(153,378)(154,361)(155,368)(156,366)(157,367)(158,365)(159,363)(160,364)
(161,362)(162,369)(163,406)(164,413)(165,411)(166,412)(167,410)(168,408)
(169,409)(170,407)(171,414)(172,424)(173,431)(174,429)(175,430)(176,428)
(177,426)(178,427)(179,425)(180,432)(181,415)(182,422)(183,420)(184,421)
(185,419)(186,417)(187,418)(188,416)(189,423)(190,379)(191,386)(192,384)
(193,385)(194,383)(195,381)(196,382)(197,380)(198,387)(199,397)(200,404)
(201,402)(202,403)(203,401)(204,399)(205,400)(206,398)(207,405)(208,388)
(209,395)(210,393)(211,394)(212,392)(213,390)(214,391)(215,389)(216,396)
(433,784)(434,791)(435,789)(436,790)(437,788)(438,786)(439,787)(440,785)
(441,792)(442,802)(443,809)(444,807)(445,808)(446,806)(447,804)(448,805)
(449,803)(450,810)(451,793)(452,800)(453,798)(454,799)(455,797)(456,795)
(457,796)(458,794)(459,801)(460,757)(461,764)(462,762)(463,763)(464,761)
(465,759)(466,760)(467,758)(468,765)(469,775)(470,782)(471,780)(472,781)
(473,779)(474,777)(475,778)(476,776)(477,783)(478,766)(479,773)(480,771)
(481,772)(482,770)(483,768)(484,769)(485,767)(486,774)(487,811)(488,818)
(489,816)(490,817)(491,815)(492,813)(493,814)(494,812)(495,819)(496,829)
(497,836)(498,834)(499,835)(500,833)(501,831)(502,832)(503,830)(504,837)
(505,820)(506,827)(507,825)(508,826)(509,824)(510,822)(511,823)(512,821)
(513,828)(514,838)(515,845)(516,843)(517,844)(518,842)(519,840)(520,841)
(521,839)(522,846)(523,856)(524,863)(525,861)(526,862)(527,860)(528,858)
(529,859)(530,857)(531,864)(532,847)(533,854)(534,852)(535,853)(536,851)
(537,849)(538,850)(539,848)(540,855)(541,649)(542,656)(543,654)(544,655)
(545,653)(546,651)(547,652)(548,650)(549,657)(550,667)(551,674)(552,672)
(553,673)(554,671)(555,669)(556,670)(557,668)(558,675)(559,658)(560,665)
(561,663)(562,664)(563,662)(564,660)(565,661)(566,659)(567,666)(568,676)
(569,683)(570,681)(571,682)(572,680)(573,678)(574,679)(575,677)(576,684)
(577,694)(578,701)(579,699)(580,700)(581,698)(582,696)(583,697)(584,695)
(585,702)(586,685)(587,692)(588,690)(589,691)(590,689)(591,687)(592,688)
(593,686)(594,693)(595,730)(596,737)(597,735)(598,736)(599,734)(600,732)
(601,733)(602,731)(603,738)(604,748)(605,755)(606,753)(607,754)(608,752)
(609,750)(610,751)(611,749)(612,756)(613,739)(614,746)(615,744)(616,745)
(617,743)(618,741)(619,742)(620,740)(621,747)(622,703)(623,710)(624,708)
(625,709)(626,707)(627,705)(628,706)(629,704)(630,711)(631,721)(632,728)
(633,726)(634,727)(635,725)(636,723)(637,724)(638,722)(639,729)(640,712)
(641,719)(642,717)(643,718)(644,716)(645,714)(646,715)(647,713)(648,720);;
s1 := (  1,658)(  2,659)(  3,660)(  4,666)(  5,664)(  6,665)(  7,662)(  8,663)
(  9,661)( 10,649)( 11,650)( 12,651)( 13,657)( 14,655)( 15,656)( 16,653)
( 17,654)( 18,652)( 19,667)( 20,668)( 21,669)( 22,675)( 23,673)( 24,674)
( 25,671)( 26,672)( 27,670)( 28,685)( 29,686)( 30,687)( 31,693)( 32,691)
( 33,692)( 34,689)( 35,690)( 36,688)( 37,676)( 38,677)( 39,678)( 40,684)
( 41,682)( 42,683)( 43,680)( 44,681)( 45,679)( 46,694)( 47,695)( 48,696)
( 49,702)( 50,700)( 51,701)( 52,698)( 53,699)( 54,697)( 55,712)( 56,713)
( 57,714)( 58,720)( 59,718)( 60,719)( 61,716)( 62,717)( 63,715)( 64,703)
( 65,704)( 66,705)( 67,711)( 68,709)( 69,710)( 70,707)( 71,708)( 72,706)
( 73,721)( 74,722)( 75,723)( 76,729)( 77,727)( 78,728)( 79,725)( 80,726)
( 81,724)( 82,739)( 83,740)( 84,741)( 85,747)( 86,745)( 87,746)( 88,743)
( 89,744)( 90,742)( 91,730)( 92,731)( 93,732)( 94,738)( 95,736)( 96,737)
( 97,734)( 98,735)( 99,733)(100,748)(101,749)(102,750)(103,756)(104,754)
(105,755)(106,752)(107,753)(108,751)(109,793)(110,794)(111,795)(112,801)
(113,799)(114,800)(115,797)(116,798)(117,796)(118,784)(119,785)(120,786)
(121,792)(122,790)(123,791)(124,788)(125,789)(126,787)(127,802)(128,803)
(129,804)(130,810)(131,808)(132,809)(133,806)(134,807)(135,805)(136,766)
(137,767)(138,768)(139,774)(140,772)(141,773)(142,770)(143,771)(144,769)
(145,757)(146,758)(147,759)(148,765)(149,763)(150,764)(151,761)(152,762)
(153,760)(154,775)(155,776)(156,777)(157,783)(158,781)(159,782)(160,779)
(161,780)(162,778)(163,847)(164,848)(165,849)(166,855)(167,853)(168,854)
(169,851)(170,852)(171,850)(172,838)(173,839)(174,840)(175,846)(176,844)
(177,845)(178,842)(179,843)(180,841)(181,856)(182,857)(183,858)(184,864)
(185,862)(186,863)(187,860)(188,861)(189,859)(190,820)(191,821)(192,822)
(193,828)(194,826)(195,827)(196,824)(197,825)(198,823)(199,811)(200,812)
(201,813)(202,819)(203,817)(204,818)(205,815)(206,816)(207,814)(208,829)
(209,830)(210,831)(211,837)(212,835)(213,836)(214,833)(215,834)(216,832)
(217,496)(218,497)(219,498)(220,504)(221,502)(222,503)(223,500)(224,501)
(225,499)(226,487)(227,488)(228,489)(229,495)(230,493)(231,494)(232,491)
(233,492)(234,490)(235,505)(236,506)(237,507)(238,513)(239,511)(240,512)
(241,509)(242,510)(243,508)(244,523)(245,524)(246,525)(247,531)(248,529)
(249,530)(250,527)(251,528)(252,526)(253,514)(254,515)(255,516)(256,522)
(257,520)(258,521)(259,518)(260,519)(261,517)(262,532)(263,533)(264,534)
(265,540)(266,538)(267,539)(268,536)(269,537)(270,535)(271,442)(272,443)
(273,444)(274,450)(275,448)(276,449)(277,446)(278,447)(279,445)(280,433)
(281,434)(282,435)(283,441)(284,439)(285,440)(286,437)(287,438)(288,436)
(289,451)(290,452)(291,453)(292,459)(293,457)(294,458)(295,455)(296,456)
(297,454)(298,469)(299,470)(300,471)(301,477)(302,475)(303,476)(304,473)
(305,474)(306,472)(307,460)(308,461)(309,462)(310,468)(311,466)(312,467)
(313,464)(314,465)(315,463)(316,478)(317,479)(318,480)(319,486)(320,484)
(321,485)(322,482)(323,483)(324,481)(325,631)(326,632)(327,633)(328,639)
(329,637)(330,638)(331,635)(332,636)(333,634)(334,622)(335,623)(336,624)
(337,630)(338,628)(339,629)(340,626)(341,627)(342,625)(343,640)(344,641)
(345,642)(346,648)(347,646)(348,647)(349,644)(350,645)(351,643)(352,604)
(353,605)(354,606)(355,612)(356,610)(357,611)(358,608)(359,609)(360,607)
(361,595)(362,596)(363,597)(364,603)(365,601)(366,602)(367,599)(368,600)
(369,598)(370,613)(371,614)(372,615)(373,621)(374,619)(375,620)(376,617)
(377,618)(378,616)(379,577)(380,578)(381,579)(382,585)(383,583)(384,584)
(385,581)(386,582)(387,580)(388,568)(389,569)(390,570)(391,576)(392,574)
(393,575)(394,572)(395,573)(396,571)(397,586)(398,587)(399,588)(400,594)
(401,592)(402,593)(403,590)(404,591)(405,589)(406,550)(407,551)(408,552)
(409,558)(410,556)(411,557)(412,554)(413,555)(414,553)(415,541)(416,542)
(417,543)(418,549)(419,547)(420,548)(421,545)(422,546)(423,544)(424,559)
(425,560)(426,561)(427,567)(428,565)(429,566)(430,563)(431,564)(432,562);;
s2 := (  1,329)(  2,328)(  3,330)(  4,326)(  5,325)(  6,327)(  7,332)(  8,331)
(  9,333)( 10,347)( 11,346)( 12,348)( 13,344)( 14,343)( 15,345)( 16,350)
( 17,349)( 18,351)( 19,338)( 20,337)( 21,339)( 22,335)( 23,334)( 24,336)
( 25,341)( 26,340)( 27,342)( 28,356)( 29,355)( 30,357)( 31,353)( 32,352)
( 33,354)( 34,359)( 35,358)( 36,360)( 37,374)( 38,373)( 39,375)( 40,371)
( 41,370)( 42,372)( 43,377)( 44,376)( 45,378)( 46,365)( 47,364)( 48,366)
( 49,362)( 50,361)( 51,363)( 52,368)( 53,367)( 54,369)( 55,383)( 56,382)
( 57,384)( 58,380)( 59,379)( 60,381)( 61,386)( 62,385)( 63,387)( 64,401)
( 65,400)( 66,402)( 67,398)( 68,397)( 69,399)( 70,404)( 71,403)( 72,405)
( 73,392)( 74,391)( 75,393)( 76,389)( 77,388)( 78,390)( 79,395)( 80,394)
( 81,396)( 82,410)( 83,409)( 84,411)( 85,407)( 86,406)( 87,408)( 88,413)
( 89,412)( 90,414)( 91,428)( 92,427)( 93,429)( 94,425)( 95,424)( 96,426)
( 97,431)( 98,430)( 99,432)(100,419)(101,418)(102,420)(103,416)(104,415)
(105,417)(106,422)(107,421)(108,423)(109,221)(110,220)(111,222)(112,218)
(113,217)(114,219)(115,224)(116,223)(117,225)(118,239)(119,238)(120,240)
(121,236)(122,235)(123,237)(124,242)(125,241)(126,243)(127,230)(128,229)
(129,231)(130,227)(131,226)(132,228)(133,233)(134,232)(135,234)(136,248)
(137,247)(138,249)(139,245)(140,244)(141,246)(142,251)(143,250)(144,252)
(145,266)(146,265)(147,267)(148,263)(149,262)(150,264)(151,269)(152,268)
(153,270)(154,257)(155,256)(156,258)(157,254)(158,253)(159,255)(160,260)
(161,259)(162,261)(163,275)(164,274)(165,276)(166,272)(167,271)(168,273)
(169,278)(170,277)(171,279)(172,293)(173,292)(174,294)(175,290)(176,289)
(177,291)(178,296)(179,295)(180,297)(181,284)(182,283)(183,285)(184,281)
(185,280)(186,282)(187,287)(188,286)(189,288)(190,302)(191,301)(192,303)
(193,299)(194,298)(195,300)(196,305)(197,304)(198,306)(199,320)(200,319)
(201,321)(202,317)(203,316)(204,318)(205,323)(206,322)(207,324)(208,311)
(209,310)(210,312)(211,308)(212,307)(213,309)(214,314)(215,313)(216,315)
(433,761)(434,760)(435,762)(436,758)(437,757)(438,759)(439,764)(440,763)
(441,765)(442,779)(443,778)(444,780)(445,776)(446,775)(447,777)(448,782)
(449,781)(450,783)(451,770)(452,769)(453,771)(454,767)(455,766)(456,768)
(457,773)(458,772)(459,774)(460,788)(461,787)(462,789)(463,785)(464,784)
(465,786)(466,791)(467,790)(468,792)(469,806)(470,805)(471,807)(472,803)
(473,802)(474,804)(475,809)(476,808)(477,810)(478,797)(479,796)(480,798)
(481,794)(482,793)(483,795)(484,800)(485,799)(486,801)(487,815)(488,814)
(489,816)(490,812)(491,811)(492,813)(493,818)(494,817)(495,819)(496,833)
(497,832)(498,834)(499,830)(500,829)(501,831)(502,836)(503,835)(504,837)
(505,824)(506,823)(507,825)(508,821)(509,820)(510,822)(511,827)(512,826)
(513,828)(514,842)(515,841)(516,843)(517,839)(518,838)(519,840)(520,845)
(521,844)(522,846)(523,860)(524,859)(525,861)(526,857)(527,856)(528,858)
(529,863)(530,862)(531,864)(532,851)(533,850)(534,852)(535,848)(536,847)
(537,849)(538,854)(539,853)(540,855)(541,653)(542,652)(543,654)(544,650)
(545,649)(546,651)(547,656)(548,655)(549,657)(550,671)(551,670)(552,672)
(553,668)(554,667)(555,669)(556,674)(557,673)(558,675)(559,662)(560,661)
(561,663)(562,659)(563,658)(564,660)(565,665)(566,664)(567,666)(568,680)
(569,679)(570,681)(571,677)(572,676)(573,678)(574,683)(575,682)(576,684)
(577,698)(578,697)(579,699)(580,695)(581,694)(582,696)(583,701)(584,700)
(585,702)(586,689)(587,688)(588,690)(589,686)(590,685)(591,687)(592,692)
(593,691)(594,693)(595,707)(596,706)(597,708)(598,704)(599,703)(600,705)
(601,710)(602,709)(603,711)(604,725)(605,724)(606,726)(607,722)(608,721)
(609,723)(610,728)(611,727)(612,729)(613,716)(614,715)(615,717)(616,713)
(617,712)(618,714)(619,719)(620,718)(621,720)(622,734)(623,733)(624,735)
(625,731)(626,730)(627,732)(628,737)(629,736)(630,738)(631,752)(632,751)
(633,753)(634,749)(635,748)(636,750)(637,755)(638,754)(639,756)(640,743)
(641,742)(642,744)(643,740)(644,739)(645,741)(646,746)(647,745)(648,747);;
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, s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope