Polytope of Type {12,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,4}*1728c
if this polytope has a name.
Group : SmallGroup(1728,31633)
Rank : 3
Schlafli Type : {12,4}
Number of vertices, edges, etc : 216, 432, 72
Order of s0s1s2 : 12
Order of s0s1s2s1 : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,4}*864d
   3-fold quotients : {12,4}*576
   4-fold quotients : {6,4}*432b
   6-fold quotients : {12,4}*288
   9-fold quotients : {12,4}*192a
   12-fold quotients : {6,4}*144
   18-fold quotients : {12,4}*96a
   24-fold quotients : {6,4}*72
   27-fold quotients : {4,4}*64
   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=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      54 facets:
         36 of {6}*12
         18 of {12}*24
      108 vertex figures:
         108 of {4}*8
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 2.
      36 facets:
         36 of {12}*24
      114 vertex figures:
         102 of {4}*8
         12 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1> of order 3.
      24 facets:
         24 of {12}*24
      72 vertex figures:
         72 of {4}*8
   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
      72 vertex figures:
         72 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 3.
      24 facets:
         24 of {12}*24
      72 vertex figures:
         72 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
      36 facets:
         18 of {4}*8
         18 of {12}*24
      72 vertex figures:
         72 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1> of order 4.
      30 facets:
         6 of {3}*6
         9 of {12}*24
         15 of {6}*12
      54 vertex figures:
         54 of {4}*8
   P/N, where N=<s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 6.
      12 facets:
         12 of {12}*24
      42 vertex figures:
         30 of {4}*8
         12 of {2}*4
   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.
      18 facets:
         12 of {6}*12
         6 of {12}*24
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 6.
      18 facets:
         6 of {12}*24
         12 of {6}*12
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s2> of order 6.
      18 facets:
         12 of {6}*12
         6 of {12}*24
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2> of order 6.
      18 facets:
         12 of {6}*12
         6 of {12}*24
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 6.
      24 facets:
         9 of {4}*8
         12 of {6}*12
         3 of {12}*24
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s0*s1> of order 6.
      30 facets:
         18 of {2}*4
         6 of {12}*24
         6 of {6}*12
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s0> of order 6.
      12 facets:
         12 of {12}*24
      42 vertex figures:
         30 of {4}*8
         12 of {2}*4
   P/N, where N=<s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 9.
      12 facets:
         6 of {12}*24
         6 of {4}*8
      24 vertex figures:
         24 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s0*s1*s2> of order 9.
      16 facets:
         12 of {4}*8
         4 of {12}*24
      24 vertex figures:
         24 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 12.
      12 facets:
         3 of {12}*24
         6 of {3}*6
         3 of {6}*12
      18 vertex figures:
         18 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 18.
      8 facets:
         4 of {6}*12
         1 of {12}*24
         3 of {4}*8
      12 vertex figures:
         12 of {4}*8
   P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1> of order 18.
      10 facets:
         2 of {12}*24
         2 of {6}*12
         6 of {2}*4
      12 vertex figures:
         12 of {4}*8
   P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s0*s1*s2> of order 18.
      12 facets:
         6 of {2}*4
         3 of {4}*8
         1 of {12}*24
         2 of {6}*12
      12 vertex figures:
         12 of {4}*8

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

Twisty Puzzle