Polytope of Type {16,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {16,6}*1728b
if this polytope has a name.
Group : SmallGroup(1728,31412)
Rank : 3
Schlafli Type : {16,6}
Number of vertices, edges, etc : 144, 432, 54
Order of s0s1s2 : 48
Order of s0s1s2s1 : 6
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 : {8,6}*864b
   3-fold quotients : {16,6}*576
   4-fold quotients : {4,6}*432b
   6-fold quotients : {8,6}*288
   9-fold quotients : {16,6}*192
   12-fold quotients : {4,6}*144
   18-fold quotients : {8,6}*96
   24-fold quotients : {4,6}*72
   27-fold quotients : {16,2}*64
   36-fold quotients : {4,6}*48a
   54-fold quotients : {8,2}*32
   72-fold quotients : {2,6}*24
   108-fold quotients : {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*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 3.
      18 facets:
         18 of {16}*32
      48 vertex figures:
         48 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 3.
      18 facets:
         18 of {16}*32
      72 vertex figures:
         36 of {6}*12
         36 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 3.
      18 facets:
         18 of {16}*32
      48 vertex figures:
         48 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      18 facets:
         18 of {16}*32
      48 vertex figures:
         48 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s2*s1> of order 9.
      6 facets:
         6 of {16}*32
      32 vertex figures:
         8 of {6}*12
         24 of {2}*4
   P/N, where N=<s1*s2*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 9.
      6 facets:
         6 of {16}*32
      24 vertex figures:
         12 of {2}*4
         12 of {6}*12

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

Twisty Puzzle