Polytope of Type {24,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,12}*1728r
if this polytope has a name.
Group : SmallGroup(1728,31294)
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}*864j
   3-fold quotients : {24,4}*576b
   4-fold quotients : {6,12}*432h
   6-fold quotients : {12,4}*288
   12-fold quotients : {6,4}*144
   24-fold quotients : {6,4}*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.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 3.
      12 facets:
         12 of {24}*48
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2> of order 3.
      12 facets:
         12 of {24}*48
      24 vertex figures:
         24 of {12}*24

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

Twisty Puzzle