Polytope of Type {6,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12}*1728i
if this polytope has a name.
Group : SmallGroup(1728,46356)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 72, 432, 144
Order of s0s1s2 : 12
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 : {6,12}*864c
   3-fold quotients : {6,12}*576b, {6,12}*576f
   4-fold quotients : {6,12}*432g
   6-fold quotients : {6,12}*288a, {6,12}*288b
   8-fold quotients : {6,6}*216d
   9-fold quotients : {6,4}*192b
   12-fold quotients : {6,12}*144a, {6,12}*144b, {6,12}*144c, {3,12}*144, {6,12}*144d
   18-fold quotients : {6,4}*96
   24-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
   36-fold quotients : {2,12}*48, {6,4}*48a, {3,4}*48, {6,4}*48b, {6,4}*48c
   48-fold quotients : {3,6}*36, {6,3}*36
   72-fold quotients : {3,4}*24, {2,6}*24, {6,2}*24
   108-fold quotients : {2,4}*16
   144-fold quotients : {2,3}*12, {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=<s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 2.
      72 facets:
         72 of {6}*12
      36 vertex figures:
         36 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1> of order 2.
      72 facets:
         72 of {6}*12
      36 vertex figures:
         36 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1> of order 2.
      84 facets:
         24 of {3}*6
         60 of {6}*12
      36 vertex figures:
         36 of {12}*24
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 2.
      72 facets:
         72 of {6}*12
      48 vertex figures:
         24 of {12}*24
         24 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 3.
      48 facets:
         48 of {6}*12
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2> of order 3.
      48 facets:
         48 of {6}*12
      48 vertex figures:
         12 of {12}*24
         36 of {4}*8
   P/N, where N=<s0*s1*s0*s1> of order 3.
      60 facets:
         18 of {2}*4
         42 of {6}*12
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 4.
      42 facets:
         12 of {3}*6
         30 of {6}*12
      24 vertex figures:
         12 of {12}*24
         12 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 4.
      36 facets:
         36 of {6}*12
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2> of order 4.
      36 facets:
         36 of {6}*12
      30 vertex figures:
         24 of {6}*12
         6 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0> of order 4.
      36 facets:
         36 of {6}*12
      30 vertex figures:
         12 of {12}*24
         12 of {3}*6
         6 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 6.
      24 facets:
         24 of {6}*12
      32 vertex figures:
         4 of {12}*24
         12 of {4}*8
         12 of {2}*4
         4 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2, s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 6.
      24 facets:
         24 of {6}*12
      24 vertex figures:
         6 of {12}*24
         18 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1, s0*s2*s1*s2*s1*s0*s1*s2*s1> of order 6.
      24 facets:
         24 of {6}*12
      24 vertex figures:
         6 of {12}*24
         18 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 12.
      12 facets:
         12 of {6}*12
      20 vertex figures:
         4 of {6}*12
         12 of {2}*4
         3 of {4}*8
         1 of {12}*24

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

Twisty Puzzle