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}*1728q
if this polytope has a name.
Group : SmallGroup(1728,31258)
Rank : 3
Schlafli Type : {24,12}
Number of vertices, edges, etc : 72, 432, 36
Order of s0s1s2 : 8
Order of s0s1s2s1 : 6
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}*576a
   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}*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.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2> of order 2.
      18 facets:
         18 of {24}*48
      36 vertex figures:
         36 of {12}*24
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 3.
      12 facets:
         12 of {24}*48
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> 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)(109,217)(110,219)(111,218)(112,223)(113,225)(114,224)(115,220)(116,222)(117,221)(118,235)(119,237)(120,236)(121,241)(122,243)(123,242)(124,238)(125,240)(126,239)(127,226)(128,228)(129,227)(130,232)(131,234)(132,233)(133,229)(134,231)(135,230)(136,244)(137,246)(138,245)(139,250)(140,252)(141,251)(142,247)(143,249)(144,248)(145,262)(146,264)(147,263)(148,268)(149,270)(150,269)(151,265)(152,267)(153,266)(154,253)(155,255)(156,254)(157,259)(158,261)(159,260)(160,256)(161,258)(162,257)(163,298)(164,300)(165,299)(166,304)(167,306)(168,305)(169,301)(170,303)(171,302)(172,316)(173,318)(174,317)(175,322)(176,324)(177,323)(178,319)(179,321)(180,320)(181,307)(182,309)(183,308)(184,313)(185,315)(186,314)(187,310)(188,312)(189,311)(190,271)(191,273)(192,272)(193,277)(194,279)(195,278)(196,274)(197,276)(198,275)(199,289)(200,291)(201,290)(202,295)(203,297)(204,296)(205,292)(206,294)(207,293)(208,280)(209,282)(210,281)(211,286)(212,288)(213,287)(214,283)(215,285)(216,284)(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)(541,649)(542,651)(543,650)(544,655)(545,657)(546,656)(547,652)(548,654)(549,653)(550,667)(551,669)(552,668)(553,673)(554,675)(555,674)(556,670)(557,672)(558,671)(559,658)(560,660)(561,659)(562,664)(563,666)(564,665)(565,661)(566,663)(567,662)(568,676)(569,678)(570,677)(571,682)(572,684)(573,683)(574,679)(575,681)(576,680)(577,694)(578,696)(579,695)(580,700)(581,702)(582,701)(583,697)(584,699)(585,698)(586,685)(587,687)(588,686)(589,691)(590,693)(591,692)(592,688)(593,690)(594,689)(595,730)(596,732)(597,731)(598,736)(599,738)(600,737)(601,733)(602,735)(603,734)(604,748)(605,750)(606,749)(607,754)(608,756)(609,755)(610,751)(611,753)(612,752)(613,739)(614,741)(615,740)(616,745)(617,747)(618,746)(619,742)(620,744)(621,743)(622,703)(623,705)(624,704)(625,709)(626,711)(627,710)(628,706)(629,708)(630,707)(631,721)(632,723)(633,722)(634,727)(635,729)(636,728)(637,724)(638,726)(639,725)(640,712)(641,714)(642,713)(643,718)(644,720)(645,719)(646,715)(647,717)(648,716);;
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)( 55,271)( 56,278)( 57,276)( 58,277)( 59,275)( 60,273)( 61,274)( 62,272)( 63,279)( 64,289)( 65,296)( 66,294)( 67,295)( 68,293)( 69,291)( 70,292)( 71,290)( 72,297)( 73,280)( 74,287)( 75,285)( 76,286)( 77,284)( 78,282)( 79,283)( 80,281)( 81,288)( 82,298)( 83,305)( 84,303)( 85,304)( 86,302)( 87,300)( 88,301)( 89,299)( 90,306)( 91,316)( 92,323)( 93,321)( 94,322)( 95,320)( 96,318)( 97,319)( 98,317)( 99,324)(100,307)(101,314)(102,312)(103,313)(104,311)(105,309)(106,310)(107,308)(108,315)(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)(163,379)(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)(433,784)(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)(487,838)(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)(595,703)(596,710)(597,708)(598,709)(599,707)(600,705)(601,706)(602,704)(603,711)(604,721)(605,728)(606,726)(607,727)(608,725)(609,723)(610,724)(611,722)(612,729)(613,712)(614,719)(615,717)(616,718)(617,716)(618,714)(619,715)(620,713)(621,720)(622,730)(623,737)(624,735)(625,736)(626,734)(627,732)(628,733)(629,731)(630,738)(631,748)(632,755)(633,753)(634,754)(635,752)(636,750)(637,751)(638,749)(639,756)(640,739)(641,746)(642,744)(643,745)(644,743)(645,741)(646,742)(647,740)(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, 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*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*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 ];;
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)(109,217)(110,219)(111,218)(112,223)(113,225)(114,224)(115,220)(116,222)(117,221)(118,235)(119,237)(120,236)(121,241)(122,243)(123,242)(124,238)(125,240)(126,239)(127,226)(128,228)(129,227)(130,232)(131,234)(132,233)(133,229)(134,231)(135,230)(136,244)(137,246)(138,245)(139,250)(140,252)(141,251)(142,247)(143,249)(144,248)(145,262)(146,264)(147,263)(148,268)(149,270)(150,269)(151,265)(152,267)(153,266)(154,253)(155,255)(156,254)(157,259)(158,261)(159,260)(160,256)(161,258)(162,257)(163,298)(164,300)(165,299)(166,304)(167,306)(168,305)(169,301)(170,303)(171,302)(172,316)(173,318)(174,317)(175,322)(176,324)(177,323)(178,319)(179,321)(180,320)(181,307)(182,309)(183,308)(184,313)(185,315)(186,314)(187,310)(188,312)(189,311)(190,271)(191,273)(192,272)(193,277)(194,279)(195,278)(196,274)(197,276)(198,275)(199,289)(200,291)(201,290)(202,295)(203,297)(204,296)(205,292)(206,294)(207,293)(208,280)(209,282)(210,281)(211,286)(212,288)(213,287)(214,283)(215,285)(216,284)(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)(541,649)(542,651)(543,650)(544,655)(545,657)(546,656)(547,652)(548,654)(549,653)(550,667)(551,669)(552,668)(553,673)(554,675)(555,674)(556,670)(557,672)(558,671)(559,658)(560,660)(561,659)(562,664)(563,666)(564,665)(565,661)(566,663)(567,662)(568,676)(569,678)(570,677)(571,682)(572,684)(573,683)(574,679)(575,681)(576,680)(577,694)(578,696)(579,695)(580,700)(581,702)(582,701)(583,697)(584,699)(585,698)(586,685)(587,687)(588,686)(589,691)(590,693)(591,692)(592,688)(593,690)(594,689)(595,730)(596,732)(597,731)(598,736)(599,738)(600,737)(601,733)(602,735)(603,734)(604,748)(605,750)(606,749)(607,754)(608,756)(609,755)(610,751)(611,753)(612,752)(613,739)(614,741)(615,740)(616,745)(617,747)(618,746)(619,742)(620,744)(621,743)(622,703)(623,705)(624,704)(625,709)(626,711)(627,710)(628,706)(629,708)(630,707)(631,721)(632,723)(633,722)(634,727)(635,729)(636,728)(637,724)(638,726)(639,725)(640,712)(641,714)(642,713)(643,718)(644,720)(645,719)(646,715)(647,717)(648,716);
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)( 55,271)( 56,278)( 57,276)( 58,277)( 59,275)( 60,273)( 61,274)( 62,272)( 63,279)( 64,289)( 65,296)( 66,294)( 67,295)( 68,293)( 69,291)( 70,292)( 71,290)( 72,297)( 73,280)( 74,287)( 75,285)( 76,286)( 77,284)( 78,282)( 79,283)( 80,281)( 81,288)( 82,298)( 83,305)( 84,303)( 85,304)( 86,302)( 87,300)( 88,301)( 89,299)( 90,306)( 91,316)( 92,323)( 93,321)( 94,322)( 95,320)( 96,318)( 97,319)( 98,317)( 99,324)(100,307)(101,314)(102,312)(103,313)(104,311)(105,309)(106,310)(107,308)(108,315)(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)(163,379)(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)(433,784)(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)(487,838)(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)(595,703)(596,710)(597,708)(598,709)(599,707)(600,705)(601,706)(602,704)(603,711)(604,721)(605,728)(606,726)(607,727)(608,725)(609,723)(610,724)(611,722)(612,729)(613,712)(614,719)(615,717)(616,718)(617,716)(618,714)(619,715)(620,713)(621,720)(622,730)(623,737)(624,735)(625,736)(626,734)(627,732)(628,733)(629,731)(630,738)(631,748)(632,755)(633,753)(634,754)(635,752)(636,750)(637,751)(638,749)(639,756)(640,739)(641,746)(642,744)(643,745)(644,743)(645,741)(646,742)(647,740)(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, 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*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*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 >;
References : None.
to this polytope

Twisty Puzzle