Polytope of Type {12,8}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,8}*1728f
if this polytope has a name.
Group : SmallGroup(1728,31294)
Rank : 3
Schlafli Type : {12,8}
Number of vertices, edges, etc : 108, 432, 72
Order of s0s1s2 : 24
Order of s0s1s2s1 : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,4}*864c
   3-fold quotients : {4,8}*576b
   4-fold quotients : {12,4}*432b
   6-fold quotients : {4,4}*288
   9-fold quotients : {12,8}*192b
   12-fold quotients : {4,4}*144
   18-fold quotients : {12,4}*96a
   24-fold quotients : {4,4}*72
   27-fold quotients : {4,8}*64b
   36-fold quotients : {12,2}*48, {6,4}*48a
   54-fold quotients : {4,4}*32
   72-fold quotients : {6,2}*24
   108-fold quotients : {2,4}*16, {4,2}*16
   144-fold quotients : {3,2}*12
   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*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 2.
      36 facets:
         36 of {12}*24
      54 vertex figures:
         54 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 3.
      24 facets:
         24 of {12}*24
      36 vertex figures:
         36 of {8}*16
   P/N, where N=<s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 3.
      24 facets:
         24 of {12}*24
      36 vertex figures:
         36 of {8}*16
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 3.
      24 facets:
         24 of {12}*24
      36 vertex figures:
         36 of {8}*16
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 3.
      24 facets:
         24 of {12}*24
      36 vertex figures:
         36 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 6.
      12 facets:
         12 of {12}*24
      18 vertex figures:
         18 of {8}*16
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 6.
      12 facets:
         12 of {12}*24
      18 vertex figures:
         18 of {8}*16
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 9.
      8 facets:
         8 of {12}*24
      12 vertex figures:
         12 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1> of order 9.
      8 facets:
         8 of {12}*24
      12 vertex figures:
         12 of {8}*16

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

Twisty Puzzle