Polytope of Type {6,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12}*1728g
if this polytope has a name.
Group : SmallGroup(1728,46303)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 72, 432, 144
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 : {6,6}*864c
   3-fold quotients : {6,12}*576c, {6,12}*576d
   6-fold quotients : {3,12}*288, {6,6}*288a, {6,6}*288b
   8-fold quotients : {6,6}*216d
   9-fold quotients : {6,12}*192b
   12-fold quotients : {3,6}*144, {6,3}*144
   18-fold quotients : {3,12}*96, {6,6}*96
   24-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
   36-fold quotients : {3,6}*48, {6,3}*48
   48-fold quotients : {3,6}*36, {6,3}*36
   72-fold quotients : {3,3}*24, {2,6}*24, {6,2}*24
   144-fold quotients : {2,3}*12, {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> of order 2.
      96 facets:
         48 of {3}*6
         48 of {6}*12
      36 vertex figures:
         36 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 2.
      72 facets:
         72 of {6}*12
      36 vertex figures:
         36 of {12}*24
   P/N, where N=<s0*s1*s0*s1> of order 3.
      60 facets:
         18 of {2}*4
         42 of {6}*12
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 3.
      48 facets:
         48 of {6}*12
      30 vertex figures:
         21 of {12}*24
         9 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 3.
      48 facets:
         48 of {6}*12
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 3.
      48 facets:
         48 of {6}*12
      24 vertex figures:
         24 of {12}*24

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

Twisty Puzzle