Polytope of Type {24,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,4}*1728g
if this polytope has a name.
Group : SmallGroup(1728,31293)
Rank : 3
Schlafli Type : {24,4}
Number of vertices, edges, etc : 216, 432, 36
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 : {12,4}*864d
   3-fold quotients : {24,4}*576b
   4-fold quotients : {6,4}*432b
   6-fold quotients : {12,4}*288
   9-fold quotients : {24,4}*192b
   12-fold quotients : {6,4}*144
   18-fold quotients : {12,4}*96a
   24-fold quotients : {6,4}*72
   27-fold quotients : {8,4}*64b
   36-fold quotients : {12,2}*48, {6,4}*48a
   54-fold quotients : {4,4}*32
   72-fold quotients : {6,2}*24
   108-fold quotients : {2,4}*16, {4,2}*16
   144-fold quotients : {3,2}*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*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 3.
      12 facets:
         12 of {24}*48
      72 vertex figures:
         72 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
      18 facets:
         9 of {8}*16
         9 of {24}*48
      72 vertex figures:
         72 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1> of order 3.
      12 facets:
         12 of {24}*48
      72 vertex figures:
         72 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 3.
      12 facets:
         12 of {24}*48
      72 vertex figures:
         72 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 9.
      8 facets:
         6 of {8}*16
         2 of {24}*48
      24 vertex figures:
         24 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 9.
      6 facets:
         3 of {24}*48
         3 of {8}*16
      24 vertex figures:
         24 of {4}*8

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

Twisty Puzzle