Polytope of Type {12,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*1728q
if this polytope has a name.
Group : SmallGroup(1728,31633)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 72, 432, 72
Order of s0s1s2 : 4
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 : {12,4}*576
   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 : {4,4}*64
   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*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      54 facets:
         36 of {6}*12
         18 of {12}*24
      36 vertex figures:
         36 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s2> of order 3.
      24 facets:
         24 of {12}*24
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s2> of order 3.
      24 facets:
         24 of {12}*24
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1> of order 4.
      30 facets:
         6 of {3}*6
         9 of {12}*24
         15 of {6}*12
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2> of order 6.
      18 facets:
         12 of {6}*12
         6 of {12}*24
      12 vertex figures:
         12 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s1*s0*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 6.
      18 facets:
         12 of {6}*12
         6 of {12}*24
      12 vertex figures:
         12 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)( 55,379)( 56,381)( 57,380)( 58,385)( 59,387)( 60,386)( 61,382)( 62,384)( 63,383)( 64,397)( 65,399)( 66,398)( 67,403)( 68,405)( 69,404)( 70,400)( 71,402)( 72,401)( 73,388)( 74,390)( 75,389)( 76,394)( 77,396)( 78,395)( 79,391)( 80,393)( 81,392)( 82,406)( 83,408)( 84,407)( 85,412)( 86,414)( 87,413)( 88,409)( 89,411)( 90,410)( 91,424)( 92,426)( 93,425)( 94,430)( 95,432)( 96,431)( 97,427)( 98,429)( 99,428)(100,415)(101,417)(102,416)(103,421)(104,423)(105,422)(106,418)(107,420)(108,419)(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)(163,298)(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)(487,811)(488,813)(489,812)(490,817)(491,819)(492,818)(493,814)(494,816)(495,815)(496,829)(497,831)(498,830)(499,835)(500,837)(501,836)(502,832)(503,834)(504,833)(505,820)(506,822)(507,821)(508,826)(509,828)(510,827)(511,823)(512,825)(513,824)(514,838)(515,840)(516,839)(517,844)(518,846)(519,845)(520,841)(521,843)(522,842)(523,856)(524,858)(525,857)(526,862)(527,864)(528,863)(529,859)(530,861)(531,860)(532,847)(533,849)(534,848)(535,853)(536,855)(537,854)(538,850)(539,852)(540,851)(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)(595,730)(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)( 55,821)( 56,820)( 57,822)( 58,825)( 59,824)( 60,823)( 61,826)( 62,828)( 63,827)( 64,812)( 65,811)( 66,813)( 67,816)( 68,815)( 69,814)( 70,817)( 71,819)( 72,818)( 73,830)( 74,829)( 75,831)( 76,834)( 77,833)( 78,832)( 79,835)( 80,837)( 81,836)( 82,848)( 83,847)( 84,849)( 85,852)( 86,851)( 87,850)( 88,853)( 89,855)( 90,854)( 91,839)( 92,838)( 93,840)( 94,843)( 95,842)( 96,841)( 97,844)( 98,846)( 99,845)(100,857)(101,856)(102,858)(103,861)(104,860)(105,859)(106,862)(107,864)(108,863)(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)(163,713)(164,712)(165,714)(166,717)(167,716)(168,715)(169,718)(170,720)(171,719)(172,704)(173,703)(174,705)(175,708)(176,707)(177,706)(178,709)(179,711)(180,710)(181,722)(182,721)(183,723)(184,726)(185,725)(186,724)(187,727)(188,729)(189,728)(190,740)(191,739)(192,741)(193,744)(194,743)(195,742)(196,745)(197,747)(198,746)(199,731)(200,730)(201,732)(202,735)(203,734)(204,733)(205,736)(206,738)(207,737)(208,749)(209,748)(210,750)(211,753)(212,752)(213,751)(214,754)(215,756)(216,755)(217,605)(218,604)(219,606)(220,609)(221,608)(222,607)(223,610)(224,612)(225,611)(226,596)(227,595)(228,597)(229,600)(230,599)(231,598)(232,601)(233,603)(234,602)(235,614)(236,613)(237,615)(238,618)(239,617)(240,616)(241,619)(242,621)(243,620)(244,632)(245,631)(246,633)(247,636)(248,635)(249,634)(250,637)(251,639)(252,638)(253,623)(254,622)(255,624)(256,627)(257,626)(258,625)(259,628)(260,630)(261,629)(262,641)(263,640)(264,642)(265,645)(266,644)(267,643)(268,646)(269,648)(270,647)(271,551)(272,550)(273,552)(274,555)(275,554)(276,553)(277,556)(278,558)(279,557)(280,542)(281,541)(282,543)(283,546)(284,545)(285,544)(286,547)(287,549)(288,548)(289,560)(290,559)(291,561)(292,564)(293,563)(294,562)(295,565)(296,567)(297,566)(298,578)(299,577)(300,579)(301,582)(302,581)(303,580)(304,583)(305,585)(306,584)(307,569)(308,568)(309,570)(310,573)(311,572)(312,571)(313,574)(314,576)(315,575)(316,587)(317,586)(318,588)(319,591)(320,590)(321,589)(322,592)(323,594)(324,593)(325,497)(326,496)(327,498)(328,501)(329,500)(330,499)(331,502)(332,504)(333,503)(334,488)(335,487)(336,489)(337,492)(338,491)(339,490)(340,493)(341,495)(342,494)(343,506)(344,505)(345,507)(346,510)(347,509)(348,508)(349,511)(350,513)(351,512)(352,524)(353,523)(354,525)(355,528)(356,527)(357,526)(358,529)(359,531)(360,530)(361,515)(362,514)(363,516)(364,519)(365,518)(366,517)(367,520)(368,522)(369,521)(370,533)(371,532)(372,534)(373,537)(374,536)(375,535)(376,538)(377,540)(378,539)(379,443)(380,442)(381,444)(382,447)(383,446)(384,445)(385,448)(386,450)(387,449)(388,434)(389,433)(390,435)(391,438)(392,437)(393,436)(394,439)(395,441)(396,440)(397,452)(398,451)(399,453)(400,456)(401,455)(402,454)(403,457)(404,459)(405,458)(406,470)(407,469)(408,471)(409,474)(410,473)(411,472)(412,475)(413,477)(414,476)(415,461)(416,460)(417,462)(418,465)(419,464)(420,463)(421,466)(422,468)(423,467)(424,479)(425,478)(426,480)(427,483)(428,482)(429,481)(430,484)(431,486)(432,485);;
s2 := (  1,325)(  2,330)(  3,332)(  4,328)(  5,333)(  6,326)(  7,331)(  8,327)(  9,329)( 10,343)( 11,348)( 12,350)( 13,346)( 14,351)( 15,344)( 16,349)( 17,345)( 18,347)( 19,334)( 20,339)( 21,341)( 22,337)( 23,342)( 24,335)( 25,340)( 26,336)( 27,338)( 28,352)( 29,357)( 30,359)( 31,355)( 32,360)( 33,353)( 34,358)( 35,354)( 36,356)( 37,370)( 38,375)( 39,377)( 40,373)( 41,378)( 42,371)( 43,376)( 44,372)( 45,374)( 46,361)( 47,366)( 48,368)( 49,364)( 50,369)( 51,362)( 52,367)( 53,363)( 54,365)( 55,406)( 56,411)( 57,413)( 58,409)( 59,414)( 60,407)( 61,412)( 62,408)( 63,410)( 64,424)( 65,429)( 66,431)( 67,427)( 68,432)( 69,425)( 70,430)( 71,426)( 72,428)( 73,415)( 74,420)( 75,422)( 76,418)( 77,423)( 78,416)( 79,421)( 80,417)( 81,419)( 82,379)( 83,384)( 84,386)( 85,382)( 86,387)( 87,380)( 88,385)( 89,381)( 90,383)( 91,397)( 92,402)( 93,404)( 94,400)( 95,405)( 96,398)( 97,403)( 98,399)( 99,401)(100,388)(101,393)(102,395)(103,391)(104,396)(105,389)(106,394)(107,390)(108,392)(109,244)(110,249)(111,251)(112,247)(113,252)(114,245)(115,250)(116,246)(117,248)(118,262)(119,267)(120,269)(121,265)(122,270)(123,263)(124,268)(125,264)(126,266)(127,253)(128,258)(129,260)(130,256)(131,261)(132,254)(133,259)(134,255)(135,257)(136,217)(137,222)(138,224)(139,220)(140,225)(141,218)(142,223)(143,219)(144,221)(145,235)(146,240)(147,242)(148,238)(149,243)(150,236)(151,241)(152,237)(153,239)(154,226)(155,231)(156,233)(157,229)(158,234)(159,227)(160,232)(161,228)(162,230)(163,271)(164,276)(165,278)(166,274)(167,279)(168,272)(169,277)(170,273)(171,275)(172,289)(173,294)(174,296)(175,292)(176,297)(177,290)(178,295)(179,291)(180,293)(181,280)(182,285)(183,287)(184,283)(185,288)(186,281)(187,286)(188,282)(189,284)(190,298)(191,303)(192,305)(193,301)(194,306)(195,299)(196,304)(197,300)(198,302)(199,316)(200,321)(201,323)(202,319)(203,324)(204,317)(205,322)(206,318)(207,320)(208,307)(209,312)(210,314)(211,310)(212,315)(213,308)(214,313)(215,309)(216,311)(433,676)(434,681)(435,683)(436,679)(437,684)(438,677)(439,682)(440,678)(441,680)(442,694)(443,699)(444,701)(445,697)(446,702)(447,695)(448,700)(449,696)(450,698)(451,685)(452,690)(453,692)(454,688)(455,693)(456,686)(457,691)(458,687)(459,689)(460,649)(461,654)(462,656)(463,652)(464,657)(465,650)(466,655)(467,651)(468,653)(469,667)(470,672)(471,674)(472,670)(473,675)(474,668)(475,673)(476,669)(477,671)(478,658)(479,663)(480,665)(481,661)(482,666)(483,659)(484,664)(485,660)(486,662)(487,703)(488,708)(489,710)(490,706)(491,711)(492,704)(493,709)(494,705)(495,707)(496,721)(497,726)(498,728)(499,724)(500,729)(501,722)(502,727)(503,723)(504,725)(505,712)(506,717)(507,719)(508,715)(509,720)(510,713)(511,718)(512,714)(513,716)(514,730)(515,735)(516,737)(517,733)(518,738)(519,731)(520,736)(521,732)(522,734)(523,748)(524,753)(525,755)(526,751)(527,756)(528,749)(529,754)(530,750)(531,752)(532,739)(533,744)(534,746)(535,742)(536,747)(537,740)(538,745)(539,741)(540,743)(541,757)(542,762)(543,764)(544,760)(545,765)(546,758)(547,763)(548,759)(549,761)(550,775)(551,780)(552,782)(553,778)(554,783)(555,776)(556,781)(557,777)(558,779)(559,766)(560,771)(561,773)(562,769)(563,774)(564,767)(565,772)(566,768)(567,770)(568,784)(569,789)(570,791)(571,787)(572,792)(573,785)(574,790)(575,786)(576,788)(577,802)(578,807)(579,809)(580,805)(581,810)(582,803)(583,808)(584,804)(585,806)(586,793)(587,798)(588,800)(589,796)(590,801)(591,794)(592,799)(593,795)(594,797)(595,838)(596,843)(597,845)(598,841)(599,846)(600,839)(601,844)(602,840)(603,842)(604,856)(605,861)(606,863)(607,859)(608,864)(609,857)(610,862)(611,858)(612,860)(613,847)(614,852)(615,854)(616,850)(617,855)(618,848)(619,853)(620,849)(621,851)(622,811)(623,816)(624,818)(625,814)(626,819)(627,812)(628,817)(629,813)(630,815)(631,829)(632,834)(633,836)(634,832)(635,837)(636,830)(637,835)(638,831)(639,833)(640,820)(641,825)(642,827)(643,823)(644,828)(645,821)(646,826)(647,822)(648,824);;
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*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*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, 
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)( 55,379)( 56,381)( 57,380)( 58,385)( 59,387)( 60,386)( 61,382)( 62,384)( 63,383)( 64,397)( 65,399)( 66,398)( 67,403)( 68,405)( 69,404)( 70,400)( 71,402)( 72,401)( 73,388)( 74,390)( 75,389)( 76,394)( 77,396)( 78,395)( 79,391)( 80,393)( 81,392)( 82,406)( 83,408)( 84,407)( 85,412)( 86,414)( 87,413)( 88,409)( 89,411)( 90,410)( 91,424)( 92,426)( 93,425)( 94,430)( 95,432)( 96,431)( 97,427)( 98,429)( 99,428)(100,415)(101,417)(102,416)(103,421)(104,423)(105,422)(106,418)(107,420)(108,419)(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)(163,298)(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)(487,811)(488,813)(489,812)(490,817)(491,819)(492,818)(493,814)(494,816)(495,815)(496,829)(497,831)(498,830)(499,835)(500,837)(501,836)(502,832)(503,834)(504,833)(505,820)(506,822)(507,821)(508,826)(509,828)(510,827)(511,823)(512,825)(513,824)(514,838)(515,840)(516,839)(517,844)(518,846)(519,845)(520,841)(521,843)(522,842)(523,856)(524,858)(525,857)(526,862)(527,864)(528,863)(529,859)(530,861)(531,860)(532,847)(533,849)(534,848)(535,853)(536,855)(537,854)(538,850)(539,852)(540,851)(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)(595,730)(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)( 55,821)( 56,820)( 57,822)( 58,825)( 59,824)( 60,823)( 61,826)( 62,828)( 63,827)( 64,812)( 65,811)( 66,813)( 67,816)( 68,815)( 69,814)( 70,817)( 71,819)( 72,818)( 73,830)( 74,829)( 75,831)( 76,834)( 77,833)( 78,832)( 79,835)( 80,837)( 81,836)( 82,848)( 83,847)( 84,849)( 85,852)( 86,851)( 87,850)( 88,853)( 89,855)( 90,854)( 91,839)( 92,838)( 93,840)( 94,843)( 95,842)( 96,841)( 97,844)( 98,846)( 99,845)(100,857)(101,856)(102,858)(103,861)(104,860)(105,859)(106,862)(107,864)(108,863)(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)(163,713)(164,712)(165,714)(166,717)(167,716)(168,715)(169,718)(170,720)(171,719)(172,704)(173,703)(174,705)(175,708)(176,707)(177,706)(178,709)(179,711)(180,710)(181,722)(182,721)(183,723)(184,726)(185,725)(186,724)(187,727)(188,729)(189,728)(190,740)(191,739)(192,741)(193,744)(194,743)(195,742)(196,745)(197,747)(198,746)(199,731)(200,730)(201,732)(202,735)(203,734)(204,733)(205,736)(206,738)(207,737)(208,749)(209,748)(210,750)(211,753)(212,752)(213,751)(214,754)(215,756)(216,755)(217,605)(218,604)(219,606)(220,609)(221,608)(222,607)(223,610)(224,612)(225,611)(226,596)(227,595)(228,597)(229,600)(230,599)(231,598)(232,601)(233,603)(234,602)(235,614)(236,613)(237,615)(238,618)(239,617)(240,616)(241,619)(242,621)(243,620)(244,632)(245,631)(246,633)(247,636)(248,635)(249,634)(250,637)(251,639)(252,638)(253,623)(254,622)(255,624)(256,627)(257,626)(258,625)(259,628)(260,630)(261,629)(262,641)(263,640)(264,642)(265,645)(266,644)(267,643)(268,646)(269,648)(270,647)(271,551)(272,550)(273,552)(274,555)(275,554)(276,553)(277,556)(278,558)(279,557)(280,542)(281,541)(282,543)(283,546)(284,545)(285,544)(286,547)(287,549)(288,548)(289,560)(290,559)(291,561)(292,564)(293,563)(294,562)(295,565)(296,567)(297,566)(298,578)(299,577)(300,579)(301,582)(302,581)(303,580)(304,583)(305,585)(306,584)(307,569)(308,568)(309,570)(310,573)(311,572)(312,571)(313,574)(314,576)(315,575)(316,587)(317,586)(318,588)(319,591)(320,590)(321,589)(322,592)(323,594)(324,593)(325,497)(326,496)(327,498)(328,501)(329,500)(330,499)(331,502)(332,504)(333,503)(334,488)(335,487)(336,489)(337,492)(338,491)(339,490)(340,493)(341,495)(342,494)(343,506)(344,505)(345,507)(346,510)(347,509)(348,508)(349,511)(350,513)(351,512)(352,524)(353,523)(354,525)(355,528)(356,527)(357,526)(358,529)(359,531)(360,530)(361,515)(362,514)(363,516)(364,519)(365,518)(366,517)(367,520)(368,522)(369,521)(370,533)(371,532)(372,534)(373,537)(374,536)(375,535)(376,538)(377,540)(378,539)(379,443)(380,442)(381,444)(382,447)(383,446)(384,445)(385,448)(386,450)(387,449)(388,434)(389,433)(390,435)(391,438)(392,437)(393,436)(394,439)(395,441)(396,440)(397,452)(398,451)(399,453)(400,456)(401,455)(402,454)(403,457)(404,459)(405,458)(406,470)(407,469)(408,471)(409,474)(410,473)(411,472)(412,475)(413,477)(414,476)(415,461)(416,460)(417,462)(418,465)(419,464)(420,463)(421,466)(422,468)(423,467)(424,479)(425,478)(426,480)(427,483)(428,482)(429,481)(430,484)(431,486)(432,485);
s2 := Sym(864)!(  1,325)(  2,330)(  3,332)(  4,328)(  5,333)(  6,326)(  7,331)(  8,327)(  9,329)( 10,343)( 11,348)( 12,350)( 13,346)( 14,351)( 15,344)( 16,349)( 17,345)( 18,347)( 19,334)( 20,339)( 21,341)( 22,337)( 23,342)( 24,335)( 25,340)( 26,336)( 27,338)( 28,352)( 29,357)( 30,359)( 31,355)( 32,360)( 33,353)( 34,358)( 35,354)( 36,356)( 37,370)( 38,375)( 39,377)( 40,373)( 41,378)( 42,371)( 43,376)( 44,372)( 45,374)( 46,361)( 47,366)( 48,368)( 49,364)( 50,369)( 51,362)( 52,367)( 53,363)( 54,365)( 55,406)( 56,411)( 57,413)( 58,409)( 59,414)( 60,407)( 61,412)( 62,408)( 63,410)( 64,424)( 65,429)( 66,431)( 67,427)( 68,432)( 69,425)( 70,430)( 71,426)( 72,428)( 73,415)( 74,420)( 75,422)( 76,418)( 77,423)( 78,416)( 79,421)( 80,417)( 81,419)( 82,379)( 83,384)( 84,386)( 85,382)( 86,387)( 87,380)( 88,385)( 89,381)( 90,383)( 91,397)( 92,402)( 93,404)( 94,400)( 95,405)( 96,398)( 97,403)( 98,399)( 99,401)(100,388)(101,393)(102,395)(103,391)(104,396)(105,389)(106,394)(107,390)(108,392)(109,244)(110,249)(111,251)(112,247)(113,252)(114,245)(115,250)(116,246)(117,248)(118,262)(119,267)(120,269)(121,265)(122,270)(123,263)(124,268)(125,264)(126,266)(127,253)(128,258)(129,260)(130,256)(131,261)(132,254)(133,259)(134,255)(135,257)(136,217)(137,222)(138,224)(139,220)(140,225)(141,218)(142,223)(143,219)(144,221)(145,235)(146,240)(147,242)(148,238)(149,243)(150,236)(151,241)(152,237)(153,239)(154,226)(155,231)(156,233)(157,229)(158,234)(159,227)(160,232)(161,228)(162,230)(163,271)(164,276)(165,278)(166,274)(167,279)(168,272)(169,277)(170,273)(171,275)(172,289)(173,294)(174,296)(175,292)(176,297)(177,290)(178,295)(179,291)(180,293)(181,280)(182,285)(183,287)(184,283)(185,288)(186,281)(187,286)(188,282)(189,284)(190,298)(191,303)(192,305)(193,301)(194,306)(195,299)(196,304)(197,300)(198,302)(199,316)(200,321)(201,323)(202,319)(203,324)(204,317)(205,322)(206,318)(207,320)(208,307)(209,312)(210,314)(211,310)(212,315)(213,308)(214,313)(215,309)(216,311)(433,676)(434,681)(435,683)(436,679)(437,684)(438,677)(439,682)(440,678)(441,680)(442,694)(443,699)(444,701)(445,697)(446,702)(447,695)(448,700)(449,696)(450,698)(451,685)(452,690)(453,692)(454,688)(455,693)(456,686)(457,691)(458,687)(459,689)(460,649)(461,654)(462,656)(463,652)(464,657)(465,650)(466,655)(467,651)(468,653)(469,667)(470,672)(471,674)(472,670)(473,675)(474,668)(475,673)(476,669)(477,671)(478,658)(479,663)(480,665)(481,661)(482,666)(483,659)(484,664)(485,660)(486,662)(487,703)(488,708)(489,710)(490,706)(491,711)(492,704)(493,709)(494,705)(495,707)(496,721)(497,726)(498,728)(499,724)(500,729)(501,722)(502,727)(503,723)(504,725)(505,712)(506,717)(507,719)(508,715)(509,720)(510,713)(511,718)(512,714)(513,716)(514,730)(515,735)(516,737)(517,733)(518,738)(519,731)(520,736)(521,732)(522,734)(523,748)(524,753)(525,755)(526,751)(527,756)(528,749)(529,754)(530,750)(531,752)(532,739)(533,744)(534,746)(535,742)(536,747)(537,740)(538,745)(539,741)(540,743)(541,757)(542,762)(543,764)(544,760)(545,765)(546,758)(547,763)(548,759)(549,761)(550,775)(551,780)(552,782)(553,778)(554,783)(555,776)(556,781)(557,777)(558,779)(559,766)(560,771)(561,773)(562,769)(563,774)(564,767)(565,772)(566,768)(567,770)(568,784)(569,789)(570,791)(571,787)(572,792)(573,785)(574,790)(575,786)(576,788)(577,802)(578,807)(579,809)(580,805)(581,810)(582,803)(583,808)(584,804)(585,806)(586,793)(587,798)(588,800)(589,796)(590,801)(591,794)(592,799)(593,795)(594,797)(595,838)(596,843)(597,845)(598,841)(599,846)(600,839)(601,844)(602,840)(603,842)(604,856)(605,861)(606,863)(607,859)(608,864)(609,857)(610,862)(611,858)(612,860)(613,847)(614,852)(615,854)(616,850)(617,855)(618,848)(619,853)(620,849)(621,851)(622,811)(623,816)(624,818)(625,814)(626,819)(627,812)(628,817)(629,813)(630,815)(631,829)(632,834)(633,836)(634,832)(635,837)(636,830)(637,835)(638,831)(639,833)(640,820)(641,825)(642,827)(643,823)(644,828)(645,821)(646,826)(647,822)(648,824);
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*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*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, 
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