Polytope of Type {12,24}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,24}*1728u
if this polytope has a name.
Group : SmallGroup(1728,33554)
Rank : 3
Schlafli Type : {12,24}
Number of vertices, edges, etc : 36, 432, 72
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 : {6,24}*864h, {12,12}*864l
   3-fold quotients : {12,8}*576a
   4-fold quotients : {6,12}*432i
   6-fold quotients : {6,8}*288, {12,4}*288
   8-fold quotients : {6,12}*216c
   9-fold quotients : {4,24}*192a
   12-fold quotients : {6,4}*144
   18-fold quotients : {4,12}*96a, {2,24}*96
   24-fold quotients : {6,4}*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=<s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      24 facets:
         24 of {12}*24
      12 vertex figures:
         12 of {24}*48
   P/N, where N=<s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 3.
      24 facets:
         24 of {12}*24
      12 vertex figures:
         12 of {24}*48
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
      48 facets:
         36 of {4}*8
         12 of {12}*24
      12 vertex figures:
         12 of {24}*48

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