Polytope of Type {4,24}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,24}*1728f
if this polytope has a name.
Group : SmallGroup(1728,31257)
Rank : 3
Schlafli Type : {4,24}
Number of vertices, edges, etc : 36, 432, 216
Order of s0s1s2 : 24
Order of s0s1s2s1 : 6
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 : {4,12}*864d
   3-fold quotients : {4,8}*576a
   4-fold quotients : {4,12}*432b
   6-fold quotients : {4,4}*288
   9-fold quotients : {4,24}*192a
   12-fold quotients : {4,4}*144
   18-fold quotients : {4,12}*96a, {2,24}*96
   24-fold quotients : {4,4}*72
   27-fold quotients : {4,8}*64a
   36-fold quotients : {2,12}*48, {4,6}*48a
   54-fold quotients : {4,4}*32, {2,8}*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*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 2.
      108 facets:
         108 of {4}*8
      18 vertex figures:
         18 of {24}*48
   P/N, where N=<s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 2.
      108 facets:
         108 of {4}*8
      18 vertex figures:
         18 of {24}*48
   P/N, where N=<s0*s1*s0*s1> of order 2.
      120 facets:
         24 of {2}*4
         96 of {4}*8
      18 vertex figures:
         18 of {24}*48
   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
      12 vertex figures:
         12 of {24}*48
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      72 facets:
         72 of {4}*8
      12 vertex figures:
         12 of {24}*48
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 3.
      72 facets:
         72 of {4}*8
      12 vertex figures:
         12 of {24}*48
   P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1> of order 3.
      72 facets:
         72 of {4}*8
      12 vertex figures:
         12 of {24}*48
   P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1> of order 6.
      36 facets:
         36 of {4}*8
      6 vertex figures:
         6 of {24}*48
   P/N, where N=<s0*s1*s0*s1, s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 6.
      48 facets:
         24 of {2}*4
         24 of {4}*8
      6 vertex figures:
         6 of {24}*48
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1> of order 6.
      48 facets:
         24 of {2}*4
         24 of {4}*8
      6 vertex figures:
         6 of {24}*48
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 6.
      36 facets:
         36 of {4}*8
      6 vertex figures:
         6 of {24}*48
   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*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 6.
      36 facets:
         36 of {4}*8
      6 vertex figures:
         6 of {24}*48
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 6.
      36 facets:
         36 of {4}*8
      6 vertex figures:
         6 of {24}*48
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1> of order 9.
      24 facets:
         24 of {4}*8
      4 vertex figures:
         4 of {24}*48
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 9.
      24 facets:
         24 of {4}*8
      4 vertex figures:
         4 of {24}*48

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

Twisty Puzzle