Polytope of Type {8,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,12}*1728f
if this polytope has a name.
Group : SmallGroup(1728,31294)
Rank : 3
Schlafli Type : {8,12}
Number of vertices, edges, etc : 72, 432, 108
Order of s0s1s2 : 24
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 : {8,4}*576b
   4-fold quotients : {4,12}*432b
   6-fold quotients : {4,4}*288
   9-fold quotients : {8,12}*192b
   12-fold quotients : {4,4}*144
   18-fold quotients : {4,12}*96a
   24-fold quotients : {4,4}*72
   27-fold quotients : {8,4}*64b
   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*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1> of order 2.
      54 facets:
         54 of {8}*16
      36 vertex figures:
         36 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 3.
      36 facets:
         36 of {8}*16
      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.
      36 facets:
         36 of {8}*16
      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.
      36 facets:
         36 of {8}*16
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2> of order 3.
      36 facets:
         36 of {8}*16
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1> of order 6.
      18 facets:
         18 of {8}*16
      12 vertex figures:
         12 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2, s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1> of order 6.
      18 facets:
         18 of {8}*16
      12 vertex figures:
         12 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 9.
      12 facets:
         12 of {8}*16
      8 vertex figures:
         8 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 9.
      12 facets:
         12 of {8}*16
      8 vertex figures:
         8 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)( 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)(109,136)(110,143)(111,141)(112,142)(113,140)(114,138)(115,139)(116,137)(117,144)(118,145)(119,152)(120,150)(121,151)(122,149)(123,147)(124,148)(125,146)(126,153)(127,154)(128,161)(129,159)(130,160)(131,158)(132,156)(133,157)(134,155)(135,162)(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)(217,244)(218,251)(219,249)(220,250)(221,248)(222,246)(223,247)(224,245)(225,252)(226,253)(227,260)(228,258)(229,259)(230,257)(231,255)(232,256)(233,254)(234,261)(235,262)(236,269)(237,267)(238,268)(239,266)(240,264)(241,265)(242,263)(243,270)(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)(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)(649,784)(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)(703,838)(704,845)(705,843)(706,844)(707,842)(708,840)(709,841)(710,839)(711,846)(712,847)(713,854)(714,852)(715,853)(716,851)(717,849)(718,850)(719,848)(720,855)(721,856)(722,863)(723,861)(724,862)(725,860)(726,858)(727,859)(728,857)(729,864)(730,811)(731,818)(732,816)(733,817)(734,815)(735,813)(736,814)(737,812)(738,819)(739,820)(740,827)(741,825)(742,826)(743,824)(744,822)(745,823)(746,821)(747,828)(748,829)(749,836)(750,834)(751,835)(752,833)(753,831)(754,832)(755,830)(756,837);;
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)(109,784)(110,785)(111,786)(112,792)(113,790)(114,791)(115,788)(116,789)(117,787)(118,802)(119,803)(120,804)(121,810)(122,808)(123,809)(124,806)(125,807)(126,805)(127,793)(128,794)(129,795)(130,801)(131,799)(132,800)(133,797)(134,798)(135,796)(136,757)(137,758)(138,759)(139,765)(140,763)(141,764)(142,761)(143,762)(144,760)(145,775)(146,776)(147,777)(148,783)(149,781)(150,782)(151,779)(152,780)(153,778)(154,766)(155,767)(156,768)(157,774)(158,772)(159,773)(160,770)(161,771)(162,769)(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)(217,514)(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)(325,595)(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)(109,230)(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)(433,689)(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*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*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, 
s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*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)( 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)(109,136)(110,143)(111,141)(112,142)(113,140)(114,138)(115,139)(116,137)(117,144)(118,145)(119,152)(120,150)(121,151)(122,149)(123,147)(124,148)(125,146)(126,153)(127,154)(128,161)(129,159)(130,160)(131,158)(132,156)(133,157)(134,155)(135,162)(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)(217,244)(218,251)(219,249)(220,250)(221,248)(222,246)(223,247)(224,245)(225,252)(226,253)(227,260)(228,258)(229,259)(230,257)(231,255)(232,256)(233,254)(234,261)(235,262)(236,269)(237,267)(238,268)(239,266)(240,264)(241,265)(242,263)(243,270)(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)(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)(649,784)(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)(703,838)(704,845)(705,843)(706,844)(707,842)(708,840)(709,841)(710,839)(711,846)(712,847)(713,854)(714,852)(715,853)(716,851)(717,849)(718,850)(719,848)(720,855)(721,856)(722,863)(723,861)(724,862)(725,860)(726,858)(727,859)(728,857)(729,864)(730,811)(731,818)(732,816)(733,817)(734,815)(735,813)(736,814)(737,812)(738,819)(739,820)(740,827)(741,825)(742,826)(743,824)(744,822)(745,823)(746,821)(747,828)(748,829)(749,836)(750,834)(751,835)(752,833)(753,831)(754,832)(755,830)(756,837);
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)(109,784)(110,785)(111,786)(112,792)(113,790)(114,791)(115,788)(116,789)(117,787)(118,802)(119,803)(120,804)(121,810)(122,808)(123,809)(124,806)(125,807)(126,805)(127,793)(128,794)(129,795)(130,801)(131,799)(132,800)(133,797)(134,798)(135,796)(136,757)(137,758)(138,759)(139,765)(140,763)(141,764)(142,761)(143,762)(144,760)(145,775)(146,776)(147,777)(148,783)(149,781)(150,782)(151,779)(152,780)(153,778)(154,766)(155,767)(156,768)(157,774)(158,772)(159,773)(160,770)(161,771)(162,769)(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)(217,514)(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)(325,595)(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)(109,230)(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)(433,689)(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*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*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, 
s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1 >;
References : None.
to this polytope

Twisty Puzzle