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}*1728d
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}*864c
   3-fold quotients : {4,4}*576
   4-fold quotients : {12,4}*432b
   6-fold quotients : {4,4}*288
   9-fold quotients : {12,4}*192a
   12-fold quotients : {4,4}*144
   18-fold quotients : {12,4}*96a
   24-fold quotients : {4,4}*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*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      38 facets:
         34 of {12}*24
         4 of {6}*12
      108 vertex figures:
         108 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 2.
      36 facets:
         36 of {12}*24
      108 vertex figures:
         108 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      36 facets:
         36 of {12}*24
      108 vertex figures:
         108 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*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=<s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*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*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*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*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*s2*s1*s0*s2*s1*s0*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*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2> of order 4.
      19 facets:
         17 of {12}*24
         2 of {6}*12
      57 vertex figures:
         51 of {4}*8
         6 of {2}*4
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 4.
      18 facets:
         18 of {12}*24
      54 vertex figures:
         54 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 4.
      19 facets:
         17 of {12}*24
         2 of {6}*12
      54 vertex figures:
         54 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2> of order 4.
      18 facets:
         18 of {12}*24
      54 vertex figures:
         54 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
      18 facets:
         18 of {12}*24
      57 vertex figures:
         51 of {4}*8
         6 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*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*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2> of order 6.
      14 facets:
         10 of {12}*24
         4 of {6}*12
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 6.
      14 facets:
         10 of {12}*24
         4 of {6}*12
      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*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 6.
      12 facets:
         12 of {12}*24
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 6.
      12 facets:
         12 of {12}*24
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 6.
      12 facets:
         12 of {12}*24
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 6.
      12 facets:
         12 of {12}*24
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*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=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 6.
      12 facets:
         12 of {12}*24
      36 vertex figures:
         36 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 6.
      12 facets:
         12 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, s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2> of order 9.
      8 facets:
         8 of {12}*24
      24 vertex figures:
         24 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 9.
      8 facets:
         8 of {12}*24
      24 vertex figures:
         24 of {4}*8
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 12.
      7 facets:
         5 of {12}*24
         2 of {6}*12
      21 vertex figures:
         15 of {4}*8
         6 of {2}*4
   P/N, where N=<s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2> of order 12.
      6 facets:
         6 of {12}*24
      18 vertex figures:
         18 of {4}*8
   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, s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2> of order 12.
      7 facets:
         2 of {6}*12
         5 of {12}*24
      21 vertex figures:
         15 of {4}*8
         6 of {2}*4

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

Twisty Puzzle