Polytope of Type {4,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12}*1728d
if this polytope has a name.
Group : SmallGroup(1728,31633)
Rank : 3
Schlafli Type : {4,12}
Number of vertices, edges, etc : 72, 432, 216
Order of s0s1s2 : 12
Order of s0s1s2s1 : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,12}*864d
   3-fold quotients : {4,4}*576
   4-fold quotients : {4,12}*432b
   6-fold quotients : {4,4}*288
   9-fold quotients : {4,12}*192a
   12-fold quotients : {4,4}*144
   18-fold quotients : {4,12}*96a
   24-fold quotients : {4,4}*72
   27-fold quotients : {4,4}*64
   36-fold quotients : {2,12}*48, {4,6}*48a
   54-fold quotients : {4,4}*32
   72-fold quotients : {2,6}*24
   108-fold quotients : {2,4}*16, {4,2}*16
   144-fold quotients : {2,3}*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*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2> of order 2.
      108 facets:
         108 of {4}*8
      38 vertex figures:
         34 of {12}*24
         4 of {6}*12
   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.
      108 facets:
         108 of {4}*8
      36 vertex figures:
         36 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      108 facets:
         108 of {4}*8
      36 vertex figures:
         36 of {12}*24
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      114 facets:
         102 of {4}*8
         12 of {2}*4
      36 vertex figures:
         36 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      72 facets:
         72 of {4}*8
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 3.
      72 facets:
         72 of {4}*8
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1> of order 3.
      72 facets:
         72 of {4}*8
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 3.
      72 facets:
         72 of {4}*8
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2> of order 4.
      57 facets:
         51 of {4}*8
         6 of {2}*4
      19 vertex figures:
         17 of {12}*24
         2 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 4.
      54 facets:
         54 of {4}*8
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*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.
      54 facets:
         54 of {4}*8
      19 vertex figures:
         17 of {12}*24
         2 of {6}*12
   P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 4.
      54 facets:
         54 of {4}*8
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 4.
      57 facets:
         51 of {4}*8
         6 of {2}*4
      18 vertex figures:
         18 of {12}*24
   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*s1*s2*s1*s0*s2*s1*s0*s2> of order 6.
      42 facets:
         30 of {4}*8
         12 of {2}*4
      12 vertex figures:
         12 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2> of order 6.
      36 facets:
         36 of {4}*8
      14 vertex figures:
         10 of {12}*24
         4 of {6}*12
   P/N, where N=<s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 6.
      36 facets:
         36 of {4}*8
      14 vertex figures:
         10 of {12}*24
         4 of {6}*12
   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.
      36 facets:
         36 of {4}*8
      12 vertex figures:
         12 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1> of order 6.
      36 facets:
         36 of {4}*8
      12 vertex figures:
         12 of {12}*24
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1> of order 6.
      36 facets:
         36 of {4}*8
      12 vertex figures:
         12 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 6.
      36 facets:
         36 of {4}*8
      12 vertex figures:
         12 of {12}*24
   P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0> of order 6.
      42 facets:
         30 of {4}*8
         12 of {2}*4
      12 vertex figures:
         12 of {12}*24
   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*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 6.
      36 facets:
         36 of {4}*8
      12 vertex figures:
         12 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1> of order 6.
      36 facets:
         36 of {4}*8
      12 vertex figures:
         12 of {12}*24
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 9.
      24 facets:
         24 of {4}*8
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s1> of order 9.
      24 facets:
         24 of {4}*8
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0, s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 12.
      21 facets:
         15 of {4}*8
         6 of {2}*4
      7 vertex figures:
         5 of {12}*24
         2 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1, s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 12.
      18 facets:
         18 of {4}*8
      6 vertex figures:
         6 of {12}*24
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2> of order 12.
      21 facets:
         15 of {4}*8
         6 of {2}*4
      7 vertex figures:
         2 of {6}*12
         5 of {12}*24

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