Polytope of Type {24,3}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,3}*864
if this polytope has a name.
Group : SmallGroup(864,2225)
Rank : 3
Schlafli Type : {24,3}
Number of vertices, edges, etc : 144, 216, 18
Order of s0s1s2 : 12
Order of s0s1s2s1 : 24
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {24,3,2} of size 1728
Vertex Figure Of :
   {2,24,3} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,3}*432
   3-fold quotients : {24,3}*288
   6-fold quotients : {12,3}*144
   8-fold quotients : {6,3}*108
   9-fold quotients : {8,3}*96
   18-fold quotients : {4,3}*48
   24-fold quotients : {6,3}*36
   36-fold quotients : {4,3}*24
   72-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {24,6}*1728b
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
      10 facets:
         6 of {8}*16
         4 of {24}*48
      48 vertex figures:
         48 of {3}*6

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

Twisty Puzzle