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}*1728r
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}*864i
   3-fold quotients : {4,12}*576
   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 : {4,4}*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*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2> of order 2.
      36 facets:
         36 of {12}*24
      54 vertex figures:
         18 of {12}*24
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      24 facets:
         24 of {12}*24
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*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*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
      18 facets:
         18 of {12}*24
      30 vertex figures:
         9 of {12}*24
         15 of {6}*12
         6 of {3}*6
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1> of order 6.
      12 facets:
         12 of {12}*24
      18 vertex figures:
         6 of {12}*24
         12 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1> of order 6.
      12 facets:
         12 of {12}*24
      18 vertex figures:
         6 of {12}*24
         12 of {6}*12

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

Twisty Puzzle