Polytope of Type {12,3}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,3}*864
if this polytope has a name.
Group : SmallGroup(864,2225)
Rank : 3
Schlafli Type : {12,3}
Number of vertices, edges, etc : 144, 216, 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 :
   {12,3,2} of size 1728
Vertex Figure Of :
   {2,12,3} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,3}*432
   3-fold quotients : {12,3}*288
   6-fold quotients : {6,3}*144
   8-fold quotients : {6,3}*108
   9-fold quotients : {12,3}*96
   18-fold quotients : {6,3}*48
   24-fold quotients : {6,3}*36
   36-fold quotients : {3,3}*24
   72-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,6}*1728c
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
      14 facets:
         3 of {4}*8
         11 of {12}*24
      48 vertex figures:
         48 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2> of order 3.
      12 facets:
         12 of {12}*24
      48 vertex figures:
         48 of {3}*6

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

Twisty Puzzle