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}*1728e
if this polytope has a name.
Group : SmallGroup(1728,31258)
Rank : 3
Schlafli Type : {12,8}
Number of vertices, edges, etc : 108, 432, 72
Order of s0s1s2 : 24
Order of s0s1s2s1 : 6
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}*576a
   4-fold quotients : {12,4}*432b
   6-fold quotients : {4,4}*288
   9-fold quotients : {12,8}*192a
   12-fold quotients : {4,4}*144
   18-fold quotients : {12,4}*96a, {6,8}*96
   24-fold quotients : {4,4}*72
   27-fold quotients : {4,8}*64a
   36-fold quotients : {12,2}*48, {6,4}*48a
   54-fold quotients : {4,4}*32, {2,8}*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=<s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2> of order 2.
      36 facets:
         36 of {12}*24
      54 vertex figures:
         54 of {8}*16
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      40 facets:
         8 of {6}*12
         32 of {12}*24
      54 vertex figures:
         54 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      36 facets:
         36 of {12}*24
      54 vertex figures:
         54 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*s2*s1*s0*s1*s2*s1> 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*s2*s1*s0*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*s2*s1*s0*s1*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*s2*s1*s0*s1*s2*s1, s1*s0*s1*s0*s2*s1*s0*s2*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=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 6.
      16 facets:
         8 of {6}*12
         8 of {12}*24
      18 vertex figures:
         18 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 6.
      16 facets:
         8 of {6}*12
         8 of {12}*24
      18 vertex figures:
         18 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s2*s1*s0*s2*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=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 9.
      8 facets:
         8 of {12}*24
      12 vertex figures:
         12 of {8}*16
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s0*s2*s1*s0*s1*s0*s2*s1*s0*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)( 55,271)( 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);;
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)( 55,194)( 56,196)( 57,192)( 58,197)( 59,190)( 60,195)( 61,191)( 62,193)( 63,198)( 64,203)( 65,205)( 66,201)( 67,206)( 68,199)( 69,204)( 70,200)( 71,202)( 72,207)( 73,212)( 74,214)( 75,210)( 76,215)( 77,208)( 78,213)( 79,209)( 80,211)( 81,216)( 82,167)( 83,169)( 84,165)( 85,170)( 86,163)( 87,168)( 88,164)( 89,166)( 90,171)( 91,176)( 92,178)( 93,174)( 94,179)( 95,172)( 96,177)( 97,173)( 98,175)( 99,180)(100,185)(101,187)(102,183)(103,188)(104,181)(105,186)(106,182)(107,184)(108,189)(217,329)(218,331)(219,327)(220,332)(221,325)(222,330)(223,326)(224,328)(225,333)(226,338)(227,340)(228,336)(229,341)(230,334)(231,339)(232,335)(233,337)(234,342)(235,347)(236,349)(237,345)(238,350)(239,343)(240,348)(241,344)(242,346)(243,351)(244,356)(245,358)(246,354)(247,359)(248,352)(249,357)(250,353)(251,355)(252,360)(253,365)(254,367)(255,363)(256,368)(257,361)(258,366)(259,362)(260,364)(261,369)(262,374)(263,376)(264,372)(265,377)(266,370)(267,375)(268,371)(269,373)(270,378)(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)(487,491)(488,493)(490,494)(496,500)(497,502)(499,503)(505,509)(506,511)(508,512)(514,518)(515,520)(517,521)(523,527)(524,529)(526,530)(532,536)(533,538)(535,539)(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)(595,626)(596,628)(597,624)(598,629)(599,622)(600,627)(601,623)(602,625)(603,630)(604,635)(605,637)(606,633)(607,638)(608,631)(609,636)(610,632)(611,634)(612,639)(613,644)(614,646)(615,642)(616,647)(617,640)(618,645)(619,641)(620,643)(621,648)(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)(703,734)(704,736)(705,732)(706,737)(707,730)(708,735)(709,731)(710,733)(711,738)(712,743)(713,745)(714,741)(715,746)(716,739)(717,744)(718,740)(719,742)(720,747)(721,752)(722,754)(723,750)(724,755)(725,748)(726,753)(727,749)(728,751)(729,756)(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)(811,815)(812,817)(814,818)(820,824)(821,826)(823,827)(829,833)(830,835)(832,836)(838,842)(839,844)(841,845)(847,851)(848,853)(850,854)(856,860)(857,862)(859,863);;
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, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
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)( 55,271)( 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);
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)( 55,194)( 56,196)( 57,192)( 58,197)( 59,190)( 60,195)( 61,191)( 62,193)( 63,198)( 64,203)( 65,205)( 66,201)( 67,206)( 68,199)( 69,204)( 70,200)( 71,202)( 72,207)( 73,212)( 74,214)( 75,210)( 76,215)( 77,208)( 78,213)( 79,209)( 80,211)( 81,216)( 82,167)( 83,169)( 84,165)( 85,170)( 86,163)( 87,168)( 88,164)( 89,166)( 90,171)( 91,176)( 92,178)( 93,174)( 94,179)( 95,172)( 96,177)( 97,173)( 98,175)( 99,180)(100,185)(101,187)(102,183)(103,188)(104,181)(105,186)(106,182)(107,184)(108,189)(217,329)(218,331)(219,327)(220,332)(221,325)(222,330)(223,326)(224,328)(225,333)(226,338)(227,340)(228,336)(229,341)(230,334)(231,339)(232,335)(233,337)(234,342)(235,347)(236,349)(237,345)(238,350)(239,343)(240,348)(241,344)(242,346)(243,351)(244,356)(245,358)(246,354)(247,359)(248,352)(249,357)(250,353)(251,355)(252,360)(253,365)(254,367)(255,363)(256,368)(257,361)(258,366)(259,362)(260,364)(261,369)(262,374)(263,376)(264,372)(265,377)(266,370)(267,375)(268,371)(269,373)(270,378)(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)(487,491)(488,493)(490,494)(496,500)(497,502)(499,503)(505,509)(506,511)(508,512)(514,518)(515,520)(517,521)(523,527)(524,529)(526,530)(532,536)(533,538)(535,539)(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)(595,626)(596,628)(597,624)(598,629)(599,622)(600,627)(601,623)(602,625)(603,630)(604,635)(605,637)(606,633)(607,638)(608,631)(609,636)(610,632)(611,634)(612,639)(613,644)(614,646)(615,642)(616,647)(617,640)(618,645)(619,641)(620,643)(621,648)(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)(703,734)(704,736)(705,732)(706,737)(707,730)(708,735)(709,731)(710,733)(711,738)(712,743)(713,745)(714,741)(715,746)(716,739)(717,744)(718,740)(719,742)(720,747)(721,752)(722,754)(723,750)(724,755)(725,748)(726,753)(727,749)(728,751)(729,756)(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)(811,815)(812,817)(814,818)(820,824)(821,826)(823,827)(829,833)(830,835)(832,836)(838,842)(839,844)(841,845)(847,851)(848,853)(850,854)(856,860)(857,862)(859,863);
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, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;
References : None.
to this polytope

Twisty Puzzle