Polytope of Type {12,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,24}*1728r
if this polytope has a name.
Group : SmallGroup(1728,31294)
Rank : 3
Schlafli Type : {12,24}
Number of vertices, edges, etc : 36, 432, 72
Order of s₀s₁s₂ : 8
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
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 : {4,24}*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 : {4,8}*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)(109,352)(110,359)(111,357)(112,358)
(113,356)(114,354)(115,355)(116,353)(117,360)(118,370)(119,377)(120,375)
(121,376)(122,374)(123,372)(124,373)(125,371)(126,378)(127,361)(128,368)
(129,366)(130,367)(131,365)(132,363)(133,364)(134,362)(135,369)(136,325)
(137,332)(138,330)(139,331)(140,329)(141,327)(142,328)(143,326)(144,333)
(145,343)(146,350)(147,348)(148,349)(149,347)(150,345)(151,346)(152,344)
(153,351)(154,334)(155,341)(156,339)(157,340)(158,338)(159,336)(160,337)
(161,335)(162,342)(163,379)(164,386)(165,384)(166,385)(167,383)(168,381)
(169,382)(170,380)(171,387)(172,397)(173,404)(174,402)(175,403)(176,401)
(177,399)(178,400)(179,398)(180,405)(181,388)(182,395)(183,393)(184,394)
(185,392)(186,390)(187,391)(188,389)(189,396)(190,406)(191,413)(192,411)
(193,412)(194,410)(195,408)(196,409)(197,407)(198,414)(199,424)(200,431)
(201,429)(202,430)(203,428)(204,426)(205,427)(206,425)(207,432)(208,415)
(209,422)(210,420)(211,421)(212,419)(213,417)(214,418)(215,416)(216,423)
(433,757)(434,764)(435,762)(436,763)(437,761)(438,759)(439,760)(440,758)
(441,765)(442,775)(443,782)(444,780)(445,781)(446,779)(447,777)(448,778)
(449,776)(450,783)(451,766)(452,773)(453,771)(454,772)(455,770)(456,768)
(457,769)(458,767)(459,774)(460,784)(461,791)(462,789)(463,790)(464,788)
(465,786)(466,787)(467,785)(468,792)(469,802)(470,809)(471,807)(472,808)
(473,806)(474,804)(475,805)(476,803)(477,810)(478,793)(479,800)(480,798)
(481,799)(482,797)(483,795)(484,796)(485,794)(486,801)(487,838)(488,845)
(489,843)(490,844)(491,842)(492,840)(493,841)(494,839)(495,846)(496,856)
(497,863)(498,861)(499,862)(500,860)(501,858)(502,859)(503,857)(504,864)
(505,847)(506,854)(507,852)(508,853)(509,851)(510,849)(511,850)(512,848)
(513,855)(514,811)(515,818)(516,816)(517,817)(518,815)(519,813)(520,814)
(521,812)(522,819)(523,829)(524,836)(525,834)(526,835)(527,833)(528,831)
(529,832)(530,830)(531,837)(532,820)(533,827)(534,825)(535,826)(536,824)
(537,822)(538,823)(539,821)(540,828)(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)( 55,739)( 56,740)
( 57,741)( 58,747)( 59,745)( 60,746)( 61,743)( 62,744)( 63,742)( 64,730)
( 65,731)( 66,732)( 67,738)( 68,736)( 69,737)( 70,734)( 71,735)( 72,733)
( 73,748)( 74,749)( 75,750)( 76,756)( 77,754)( 78,755)( 79,752)( 80,753)
( 81,751)( 82,712)( 83,713)( 84,714)( 85,720)( 86,718)( 87,719)( 88,716)
( 89,717)( 90,715)( 91,703)( 92,704)( 93,705)( 94,711)( 95,709)( 96,710)
( 97,707)( 98,708)( 99,706)(100,721)(101,722)(102,723)(103,729)(104,727)
(105,728)(106,725)(107,726)(108,724)(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)(163,820)(164,821)(165,822)(166,828)(167,826)(168,827)
(169,824)(170,825)(171,823)(172,811)(173,812)(174,813)(175,819)(176,817)
(177,818)(178,815)(179,816)(180,814)(181,829)(182,830)(183,831)(184,837)
(185,835)(186,836)(187,833)(188,834)(189,832)(190,847)(191,848)(192,849)
(193,855)(194,853)(195,854)(196,851)(197,852)(198,850)(199,838)(200,839)
(201,840)(202,846)(203,844)(204,845)(205,842)(206,843)(207,841)(208,856)
(209,857)(210,858)(211,864)(212,862)(213,863)(214,860)(215,861)(216,859)
(217,523)(218,524)(219,525)(220,531)(221,529)(222,530)(223,527)(224,528)
(225,526)(226,514)(227,515)(228,516)(229,522)(230,520)(231,521)(232,518)
(233,519)(234,517)(235,532)(236,533)(237,534)(238,540)(239,538)(240,539)
(241,536)(242,537)(243,535)(244,496)(245,497)(246,498)(247,504)(248,502)
(249,503)(250,500)(251,501)(252,499)(253,487)(254,488)(255,489)(256,495)
(257,493)(258,494)(259,491)(260,492)(261,490)(262,505)(263,506)(264,507)
(265,513)(266,511)(267,512)(268,509)(269,510)(270,508)(271,469)(272,470)
(273,471)(274,477)(275,475)(276,476)(277,473)(278,474)(279,472)(280,460)
(281,461)(282,462)(283,468)(284,466)(285,467)(286,464)(287,465)(288,463)
(289,478)(290,479)(291,480)(292,486)(293,484)(294,485)(295,482)(296,483)
(297,481)(298,442)(299,443)(300,444)(301,450)(302,448)(303,449)(304,446)
(305,447)(306,445)(307,433)(308,434)(309,435)(310,441)(311,439)(312,440)
(313,437)(314,438)(315,436)(316,451)(317,452)(318,453)(319,459)(320,457)
(321,458)(322,455)(323,456)(324,454)(325,604)(326,605)(327,606)(328,612)
(329,610)(330,611)(331,608)(332,609)(333,607)(334,595)(335,596)(336,597)
(337,603)(338,601)(339,602)(340,599)(341,600)(342,598)(343,613)(344,614)
(345,615)(346,621)(347,619)(348,620)(349,617)(350,618)(351,616)(352,631)
(353,632)(354,633)(355,639)(356,637)(357,638)(358,635)(359,636)(360,634)
(361,622)(362,623)(363,624)(364,630)(365,628)(366,629)(367,626)(368,627)
(369,625)(370,640)(371,641)(372,642)(373,648)(374,646)(375,647)(376,644)
(377,645)(378,643)(379,550)(380,551)(381,552)(382,558)(383,556)(384,557)
(385,554)(386,555)(387,553)(388,541)(389,542)(390,543)(391,549)(392,547)
(393,548)(394,545)(395,546)(396,544)(397,559)(398,560)(399,561)(400,567)
(401,565)(402,566)(403,563)(404,564)(405,562)(406,577)(407,578)(408,579)
(409,585)(410,583)(411,584)(412,581)(413,582)(414,580)(415,568)(416,569)
(417,570)(418,576)(419,574)(420,575)(421,572)(422,573)(423,571)(424,586)
(425,587)(426,588)(427,594)(428,592)(429,593)(430,590)(431,591)(432,589);;
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)( 55,410)( 56,409)
( 57,411)( 58,407)( 59,406)( 60,408)( 61,413)( 62,412)( 63,414)( 64,428)
( 65,427)( 66,429)( 67,425)( 68,424)( 69,426)( 70,431)( 71,430)( 72,432)
( 73,419)( 74,418)( 75,420)( 76,416)( 77,415)( 78,417)( 79,422)( 80,421)
( 81,423)( 82,383)( 83,382)( 84,384)( 85,380)( 86,379)( 87,381)( 88,386)
( 89,385)( 90,387)( 91,401)( 92,400)( 93,402)( 94,398)( 95,397)( 96,399)
( 97,404)( 98,403)( 99,405)(100,392)(101,391)(102,393)(103,389)(104,388)
(105,390)(106,395)(107,394)(108,396)(109,248)(110,247)(111,249)(112,245)
(113,244)(114,246)(115,251)(116,250)(117,252)(118,266)(119,265)(120,267)
(121,263)(122,262)(123,264)(124,269)(125,268)(126,270)(127,257)(128,256)
(129,258)(130,254)(131,253)(132,255)(133,260)(134,259)(135,261)(136,221)
(137,220)(138,222)(139,218)(140,217)(141,219)(142,224)(143,223)(144,225)
(145,239)(146,238)(147,240)(148,236)(149,235)(150,237)(151,242)(152,241)
(153,243)(154,230)(155,229)(156,231)(157,227)(158,226)(159,228)(160,233)
(161,232)(162,234)(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)(487,842)(488,841)
(489,843)(490,839)(491,838)(492,840)(493,845)(494,844)(495,846)(496,860)
(497,859)(498,861)(499,857)(500,856)(501,858)(502,863)(503,862)(504,864)
(505,851)(506,850)(507,852)(508,848)(509,847)(510,849)(511,854)(512,853)
(513,855)(514,815)(515,814)(516,816)(517,812)(518,811)(519,813)(520,818)
(521,817)(522,819)(523,833)(524,832)(525,834)(526,830)(527,829)(528,831)
(529,836)(530,835)(531,837)(532,824)(533,823)(534,825)(535,821)(536,820)
(537,822)(538,827)(539,826)(540,828)(541,680)(542,679)(543,681)(544,677)
(545,676)(546,678)(547,683)(548,682)(549,684)(550,698)(551,697)(552,699)
(553,695)(554,694)(555,696)(556,701)(557,700)(558,702)(559,689)(560,688)
(561,690)(562,686)(563,685)(564,687)(565,692)(566,691)(567,693)(568,653)
(569,652)(570,654)(571,650)(572,649)(573,651)(574,656)(575,655)(576,657)
(577,671)(578,670)(579,672)(580,668)(581,667)(582,669)(583,674)(584,673)
(585,675)(586,662)(587,661)(588,663)(589,659)(590,658)(591,660)(592,665)
(593,664)(594,666)(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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*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)(109,352)(110,359)(111,357)
(112,358)(113,356)(114,354)(115,355)(116,353)(117,360)(118,370)(119,377)
(120,375)(121,376)(122,374)(123,372)(124,373)(125,371)(126,378)(127,361)
(128,368)(129,366)(130,367)(131,365)(132,363)(133,364)(134,362)(135,369)
(136,325)(137,332)(138,330)(139,331)(140,329)(141,327)(142,328)(143,326)
(144,333)(145,343)(146,350)(147,348)(148,349)(149,347)(150,345)(151,346)
(152,344)(153,351)(154,334)(155,341)(156,339)(157,340)(158,338)(159,336)
(160,337)(161,335)(162,342)(163,379)(164,386)(165,384)(166,385)(167,383)
(168,381)(169,382)(170,380)(171,387)(172,397)(173,404)(174,402)(175,403)
(176,401)(177,399)(178,400)(179,398)(180,405)(181,388)(182,395)(183,393)
(184,394)(185,392)(186,390)(187,391)(188,389)(189,396)(190,406)(191,413)
(192,411)(193,412)(194,410)(195,408)(196,409)(197,407)(198,414)(199,424)
(200,431)(201,429)(202,430)(203,428)(204,426)(205,427)(206,425)(207,432)
(208,415)(209,422)(210,420)(211,421)(212,419)(213,417)(214,418)(215,416)
(216,423)(433,757)(434,764)(435,762)(436,763)(437,761)(438,759)(439,760)
(440,758)(441,765)(442,775)(443,782)(444,780)(445,781)(446,779)(447,777)
(448,778)(449,776)(450,783)(451,766)(452,773)(453,771)(454,772)(455,770)
(456,768)(457,769)(458,767)(459,774)(460,784)(461,791)(462,789)(463,790)
(464,788)(465,786)(466,787)(467,785)(468,792)(469,802)(470,809)(471,807)
(472,808)(473,806)(474,804)(475,805)(476,803)(477,810)(478,793)(479,800)
(480,798)(481,799)(482,797)(483,795)(484,796)(485,794)(486,801)(487,838)
(488,845)(489,843)(490,844)(491,842)(492,840)(493,841)(494,839)(495,846)
(496,856)(497,863)(498,861)(499,862)(500,860)(501,858)(502,859)(503,857)
(504,864)(505,847)(506,854)(507,852)(508,853)(509,851)(510,849)(511,850)
(512,848)(513,855)(514,811)(515,818)(516,816)(517,817)(518,815)(519,813)
(520,814)(521,812)(522,819)(523,829)(524,836)(525,834)(526,835)(527,833)
(528,831)(529,832)(530,830)(531,837)(532,820)(533,827)(534,825)(535,826)
(536,824)(537,822)(538,823)(539,821)(540,828)(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)( 55,739)
( 56,740)( 57,741)( 58,747)( 59,745)( 60,746)( 61,743)( 62,744)( 63,742)
( 64,730)( 65,731)( 66,732)( 67,738)( 68,736)( 69,737)( 70,734)( 71,735)
( 72,733)( 73,748)( 74,749)( 75,750)( 76,756)( 77,754)( 78,755)( 79,752)
( 80,753)( 81,751)( 82,712)( 83,713)( 84,714)( 85,720)( 86,718)( 87,719)
( 88,716)( 89,717)( 90,715)( 91,703)( 92,704)( 93,705)( 94,711)( 95,709)
( 96,710)( 97,707)( 98,708)( 99,706)(100,721)(101,722)(102,723)(103,729)
(104,727)(105,728)(106,725)(107,726)(108,724)(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)(163,820)(164,821)(165,822)(166,828)(167,826)
(168,827)(169,824)(170,825)(171,823)(172,811)(173,812)(174,813)(175,819)
(176,817)(177,818)(178,815)(179,816)(180,814)(181,829)(182,830)(183,831)
(184,837)(185,835)(186,836)(187,833)(188,834)(189,832)(190,847)(191,848)
(192,849)(193,855)(194,853)(195,854)(196,851)(197,852)(198,850)(199,838)
(200,839)(201,840)(202,846)(203,844)(204,845)(205,842)(206,843)(207,841)
(208,856)(209,857)(210,858)(211,864)(212,862)(213,863)(214,860)(215,861)
(216,859)(217,523)(218,524)(219,525)(220,531)(221,529)(222,530)(223,527)
(224,528)(225,526)(226,514)(227,515)(228,516)(229,522)(230,520)(231,521)
(232,518)(233,519)(234,517)(235,532)(236,533)(237,534)(238,540)(239,538)
(240,539)(241,536)(242,537)(243,535)(244,496)(245,497)(246,498)(247,504)
(248,502)(249,503)(250,500)(251,501)(252,499)(253,487)(254,488)(255,489)
(256,495)(257,493)(258,494)(259,491)(260,492)(261,490)(262,505)(263,506)
(264,507)(265,513)(266,511)(267,512)(268,509)(269,510)(270,508)(271,469)
(272,470)(273,471)(274,477)(275,475)(276,476)(277,473)(278,474)(279,472)
(280,460)(281,461)(282,462)(283,468)(284,466)(285,467)(286,464)(287,465)
(288,463)(289,478)(290,479)(291,480)(292,486)(293,484)(294,485)(295,482)
(296,483)(297,481)(298,442)(299,443)(300,444)(301,450)(302,448)(303,449)
(304,446)(305,447)(306,445)(307,433)(308,434)(309,435)(310,441)(311,439)
(312,440)(313,437)(314,438)(315,436)(316,451)(317,452)(318,453)(319,459)
(320,457)(321,458)(322,455)(323,456)(324,454)(325,604)(326,605)(327,606)
(328,612)(329,610)(330,611)(331,608)(332,609)(333,607)(334,595)(335,596)
(336,597)(337,603)(338,601)(339,602)(340,599)(341,600)(342,598)(343,613)
(344,614)(345,615)(346,621)(347,619)(348,620)(349,617)(350,618)(351,616)
(352,631)(353,632)(354,633)(355,639)(356,637)(357,638)(358,635)(359,636)
(360,634)(361,622)(362,623)(363,624)(364,630)(365,628)(366,629)(367,626)
(368,627)(369,625)(370,640)(371,641)(372,642)(373,648)(374,646)(375,647)
(376,644)(377,645)(378,643)(379,550)(380,551)(381,552)(382,558)(383,556)
(384,557)(385,554)(386,555)(387,553)(388,541)(389,542)(390,543)(391,549)
(392,547)(393,548)(394,545)(395,546)(396,544)(397,559)(398,560)(399,561)
(400,567)(401,565)(402,566)(403,563)(404,564)(405,562)(406,577)(407,578)
(408,579)(409,585)(410,583)(411,584)(412,581)(413,582)(414,580)(415,568)
(416,569)(417,570)(418,576)(419,574)(420,575)(421,572)(422,573)(423,571)
(424,586)(425,587)(426,588)(427,594)(428,592)(429,593)(430,590)(431,591)
(432,589);
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)( 55,410)
( 56,409)( 57,411)( 58,407)( 59,406)( 60,408)( 61,413)( 62,412)( 63,414)
( 64,428)( 65,427)( 66,429)( 67,425)( 68,424)( 69,426)( 70,431)( 71,430)
( 72,432)( 73,419)( 74,418)( 75,420)( 76,416)( 77,415)( 78,417)( 79,422)
( 80,421)( 81,423)( 82,383)( 83,382)( 84,384)( 85,380)( 86,379)( 87,381)
( 88,386)( 89,385)( 90,387)( 91,401)( 92,400)( 93,402)( 94,398)( 95,397)
( 96,399)( 97,404)( 98,403)( 99,405)(100,392)(101,391)(102,393)(103,389)
(104,388)(105,390)(106,395)(107,394)(108,396)(109,248)(110,247)(111,249)
(112,245)(113,244)(114,246)(115,251)(116,250)(117,252)(118,266)(119,265)
(120,267)(121,263)(122,262)(123,264)(124,269)(125,268)(126,270)(127,257)
(128,256)(129,258)(130,254)(131,253)(132,255)(133,260)(134,259)(135,261)
(136,221)(137,220)(138,222)(139,218)(140,217)(141,219)(142,224)(143,223)
(144,225)(145,239)(146,238)(147,240)(148,236)(149,235)(150,237)(151,242)
(152,241)(153,243)(154,230)(155,229)(156,231)(157,227)(158,226)(159,228)
(160,233)(161,232)(162,234)(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)(487,842)
(488,841)(489,843)(490,839)(491,838)(492,840)(493,845)(494,844)(495,846)
(496,860)(497,859)(498,861)(499,857)(500,856)(501,858)(502,863)(503,862)
(504,864)(505,851)(506,850)(507,852)(508,848)(509,847)(510,849)(511,854)
(512,853)(513,855)(514,815)(515,814)(516,816)(517,812)(518,811)(519,813)
(520,818)(521,817)(522,819)(523,833)(524,832)(525,834)(526,830)(527,829)
(528,831)(529,836)(530,835)(531,837)(532,824)(533,823)(534,825)(535,821)
(536,820)(537,822)(538,827)(539,826)(540,828)(541,680)(542,679)(543,681)
(544,677)(545,676)(546,678)(547,683)(548,682)(549,684)(550,698)(551,697)
(552,699)(553,695)(554,694)(555,696)(556,701)(557,700)(558,702)(559,689)
(560,688)(561,690)(562,686)(563,685)(564,687)(565,692)(566,691)(567,693)
(568,653)(569,652)(570,654)(571,650)(572,649)(573,651)(574,656)(575,655)
(576,657)(577,671)(578,670)(579,672)(580,668)(581,667)(582,669)(583,674)
(584,673)(585,675)(586,662)(587,661)(588,663)(589,659)(590,658)(591,660)
(592,665)(593,664)(594,666)(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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1 >;

References : None.
to this polytope