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}*1728d
if this polytope has a name.
Group : SmallGroup(1728,12762)
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}*864b
   3-fold quotients : {12,8}*576b
   4-fold quotients : {6,4}*432a
   6-fold quotients : {12,4}*288
   8-fold quotients : {6,4}*216
   12-fold quotients : {6,4}*144
   24-fold quotients : {6,4}*72
   27-fold quotients : {4,8}*64b
   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*s0*s1*s2*s1*s0*s1*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*s0*s1*s2*s1*s0*s2*s1> of order 3.
      32 facets:
         20 of {12}*24
         12 of {4}*8
      36 vertex figures:
         36 of {8}*16
   P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 3.
      24 facets:
         24 of {12}*24
      36 vertex figures:
         36 of {8}*16

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

Twisty Puzzle