Polytope of Type {12,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*1296h
if this polytope has a name.
Group : SmallGroup(1296,2061)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 108, 324, 54
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
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 : {6,6}*648e
   3-fold quotients : {12,6}*432a, {12,6}*432g
   6-fold quotients : {6,6}*216c, {6,6}*216d
   9-fold quotients : {12,6}*144a, {12,6}*144b, {12,6}*144c
   12-fold quotients : {3,6}*108
   18-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
   27-fold quotients : {12,2}*48, {4,6}*48a
   36-fold quotients : {3,6}*36, {6,3}*36
   54-fold quotients : {2,6}*24, {6,2}*24
   81-fold quotients : {4,2}*16
   108-fold quotients : {2,3}*12, {3,2}*12
   162-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*s2*s1*s0*s2*s1*s2*s1*s2> of order 3.
      18 facets:
         18 of {12}*24
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0> of order 3.
      18 facets:
         18 of {12}*24
      48 vertex figures:
         30 of {6}*12
         18 of {2}*4
   P/N, where N=<s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 3.
      24 facets:
         15 of {12}*24
         9 of {4}*8
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 3.
      18 facets:
         18 of {12}*24
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      18 facets:
         18 of {12}*24
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s1*s2*s1*s2, s0*s1*s2*s1*s0*s2> of order 9.
      6 facets:
         6 of {12}*24
      20 vertex figures:
         12 of {2}*4
         8 of {6}*12

Permutation Representation (GAP) : 
s0 := (  1,325)(  2,326)(  3,327)(  4,331)(  5,332)(  6,333)(  7,328)(  8,329)(  9,330)( 10,334)( 11,335)( 12,336)( 13,340)( 14,341)( 15,342)( 16,337)( 17,338)( 18,339)( 19,343)( 20,344)( 21,345)( 22,349)( 23,350)( 24,351)( 25,346)( 26,347)( 27,348)( 28,379)( 29,380)( 30,381)( 31,385)( 32,386)( 33,387)( 34,382)( 35,383)( 36,384)( 37,388)( 38,389)( 39,390)( 40,394)( 41,395)( 42,396)( 43,391)( 44,392)( 45,393)( 46,397)( 47,398)( 48,399)( 49,403)( 50,404)( 51,405)( 52,400)( 53,401)( 54,402)( 55,352)( 56,353)( 57,354)( 58,358)( 59,359)( 60,360)( 61,355)( 62,356)( 63,357)( 64,361)( 65,362)( 66,363)( 67,367)( 68,368)( 69,369)( 70,364)( 71,365)( 72,366)( 73,370)( 74,371)( 75,372)( 76,376)( 77,377)( 78,378)( 79,373)( 80,374)( 81,375)( 82,406)( 83,407)( 84,408)( 85,412)( 86,413)( 87,414)( 88,409)( 89,410)( 90,411)( 91,415)( 92,416)( 93,417)( 94,421)( 95,422)( 96,423)( 97,418)( 98,419)( 99,420)(100,424)(101,425)(102,426)(103,430)(104,431)(105,432)(106,427)(107,428)(108,429)(109,460)(110,461)(111,462)(112,466)(113,467)(114,468)(115,463)(116,464)(117,465)(118,469)(119,470)(120,471)(121,475)(122,476)(123,477)(124,472)(125,473)(126,474)(127,478)(128,479)(129,480)(130,484)(131,485)(132,486)(133,481)(134,482)(135,483)(136,433)(137,434)(138,435)(139,439)(140,440)(141,441)(142,436)(143,437)(144,438)(145,442)(146,443)(147,444)(148,448)(149,449)(150,450)(151,445)(152,446)(153,447)(154,451)(155,452)(156,453)(157,457)(158,458)(159,459)(160,454)(161,455)(162,456)(163,568)(164,569)(165,570)(166,574)(167,575)(168,576)(169,571)(170,572)(171,573)(172,577)(173,578)(174,579)(175,583)(176,584)(177,585)(178,580)(179,581)(180,582)(181,586)(182,587)(183,588)(184,592)(185,593)(186,594)(187,589)(188,590)(189,591)(190,622)(191,623)(192,624)(193,628)(194,629)(195,630)(196,625)(197,626)(198,627)(199,631)(200,632)(201,633)(202,637)(203,638)(204,639)(205,634)(206,635)(207,636)(208,640)(209,641)(210,642)(211,646)(212,647)(213,648)(214,643)(215,644)(216,645)(217,595)(218,596)(219,597)(220,601)(221,602)(222,603)(223,598)(224,599)(225,600)(226,604)(227,605)(228,606)(229,610)(230,611)(231,612)(232,607)(233,608)(234,609)(235,613)(236,614)(237,615)(238,619)(239,620)(240,621)(241,616)(242,617)(243,618)(244,487)(245,488)(246,489)(247,493)(248,494)(249,495)(250,490)(251,491)(252,492)(253,496)(254,497)(255,498)(256,502)(257,503)(258,504)(259,499)(260,500)(261,501)(262,505)(263,506)(264,507)(265,511)(266,512)(267,513)(268,508)(269,509)(270,510)(271,541)(272,542)(273,543)(274,547)(275,548)(276,549)(277,544)(278,545)(279,546)(280,550)(281,551)(282,552)(283,556)(284,557)(285,558)(286,553)(287,554)(288,555)(289,559)(290,560)(291,561)(292,565)(293,566)(294,567)(295,562)(296,563)(297,564)(298,514)(299,515)(300,516)(301,520)(302,521)(303,522)(304,517)(305,518)(306,519)(307,523)(308,524)(309,525)(310,529)(311,530)(312,531)(313,526)(314,527)(315,528)(316,532)(317,533)(318,534)(319,538)(320,539)(321,540)(322,535)(323,536)(324,537);;
s1 := (  1,545)(  2,546)(  3,544)(  4,543)(  5,541)(  6,542)(  7,547)(  8,548)(  9,549)( 10,563)( 11,564)( 12,562)( 13,561)( 14,559)( 15,560)( 16,565)( 17,566)( 18,567)( 19,554)( 20,555)( 21,553)( 22,552)( 23,550)( 24,551)( 25,556)( 26,557)( 27,558)( 28,518)( 29,519)( 30,517)( 31,516)( 32,514)( 33,515)( 34,520)( 35,521)( 36,522)( 37,536)( 38,537)( 39,535)( 40,534)( 41,532)( 42,533)( 43,538)( 44,539)( 45,540)( 46,527)( 47,528)( 48,526)( 49,525)( 50,523)( 51,524)( 52,529)( 53,530)( 54,531)( 55,491)( 56,492)( 57,490)( 58,489)( 59,487)( 60,488)( 61,493)( 62,494)( 63,495)( 64,509)( 65,510)( 66,508)( 67,507)( 68,505)( 69,506)( 70,511)( 71,512)( 72,513)( 73,500)( 74,501)( 75,499)( 76,498)( 77,496)( 78,497)( 79,502)( 80,503)( 81,504)( 82,626)( 83,627)( 84,625)( 85,624)( 86,622)( 87,623)( 88,628)( 89,629)( 90,630)( 91,644)( 92,645)( 93,643)( 94,642)( 95,640)( 96,641)( 97,646)( 98,647)( 99,648)(100,635)(101,636)(102,634)(103,633)(104,631)(105,632)(106,637)(107,638)(108,639)(109,599)(110,600)(111,598)(112,597)(113,595)(114,596)(115,601)(116,602)(117,603)(118,617)(119,618)(120,616)(121,615)(122,613)(123,614)(124,619)(125,620)(126,621)(127,608)(128,609)(129,607)(130,606)(131,604)(132,605)(133,610)(134,611)(135,612)(136,572)(137,573)(138,571)(139,570)(140,568)(141,569)(142,574)(143,575)(144,576)(145,590)(146,591)(147,589)(148,588)(149,586)(150,587)(151,592)(152,593)(153,594)(154,581)(155,582)(156,580)(157,579)(158,577)(159,578)(160,583)(161,584)(162,585)(163,383)(164,384)(165,382)(166,381)(167,379)(168,380)(169,385)(170,386)(171,387)(172,401)(173,402)(174,400)(175,399)(176,397)(177,398)(178,403)(179,404)(180,405)(181,392)(182,393)(183,391)(184,390)(185,388)(186,389)(187,394)(188,395)(189,396)(190,356)(191,357)(192,355)(193,354)(194,352)(195,353)(196,358)(197,359)(198,360)(199,374)(200,375)(201,373)(202,372)(203,370)(204,371)(205,376)(206,377)(207,378)(208,365)(209,366)(210,364)(211,363)(212,361)(213,362)(214,367)(215,368)(216,369)(217,329)(218,330)(219,328)(220,327)(221,325)(222,326)(223,331)(224,332)(225,333)(226,347)(227,348)(228,346)(229,345)(230,343)(231,344)(232,349)(233,350)(234,351)(235,338)(236,339)(237,337)(238,336)(239,334)(240,335)(241,340)(242,341)(243,342)(244,464)(245,465)(246,463)(247,462)(248,460)(249,461)(250,466)(251,467)(252,468)(253,482)(254,483)(255,481)(256,480)(257,478)(258,479)(259,484)(260,485)(261,486)(262,473)(263,474)(264,472)(265,471)(266,469)(267,470)(268,475)(269,476)(270,477)(271,437)(272,438)(273,436)(274,435)(275,433)(276,434)(277,439)(278,440)(279,441)(280,455)(281,456)(282,454)(283,453)(284,451)(285,452)(286,457)(287,458)(288,459)(289,446)(290,447)(291,445)(292,444)(293,442)(294,443)(295,448)(296,449)(297,450)(298,410)(299,411)(300,409)(301,408)(302,406)(303,407)(304,412)(305,413)(306,414)(307,428)(308,429)(309,427)(310,426)(311,424)(312,425)(313,430)(314,431)(315,432)(316,419)(317,420)(318,418)(319,417)(320,415)(321,416)(322,421)(323,422)(324,423);;
s2 := (  1,496)(  2,498)(  3,497)(  4,499)(  5,501)(  6,500)(  7,502)(  8,504)(  9,503)( 10,487)( 11,489)( 12,488)( 13,490)( 14,492)( 15,491)( 16,493)( 17,495)( 18,494)( 19,505)( 20,507)( 21,506)( 22,508)( 23,510)( 24,509)( 25,511)( 26,513)( 27,512)( 28,550)( 29,552)( 30,551)( 31,553)( 32,555)( 33,554)( 34,556)( 35,558)( 36,557)( 37,541)( 38,543)( 39,542)( 40,544)( 41,546)( 42,545)( 43,547)( 44,549)( 45,548)( 46,559)( 47,561)( 48,560)( 49,562)( 50,564)( 51,563)( 52,565)( 53,567)( 54,566)( 55,523)( 56,525)( 57,524)( 58,526)( 59,528)( 60,527)( 61,529)( 62,531)( 63,530)( 64,514)( 65,516)( 66,515)( 67,517)( 68,519)( 69,518)( 70,520)( 71,522)( 72,521)( 73,532)( 74,534)( 75,533)( 76,535)( 77,537)( 78,536)( 79,538)( 80,540)( 81,539)( 82,577)( 83,579)( 84,578)( 85,580)( 86,582)( 87,581)( 88,583)( 89,585)( 90,584)( 91,568)( 92,570)( 93,569)( 94,571)( 95,573)( 96,572)( 97,574)( 98,576)( 99,575)(100,586)(101,588)(102,587)(103,589)(104,591)(105,590)(106,592)(107,594)(108,593)(109,631)(110,633)(111,632)(112,634)(113,636)(114,635)(115,637)(116,639)(117,638)(118,622)(119,624)(120,623)(121,625)(122,627)(123,626)(124,628)(125,630)(126,629)(127,640)(128,642)(129,641)(130,643)(131,645)(132,644)(133,646)(134,648)(135,647)(136,604)(137,606)(138,605)(139,607)(140,609)(141,608)(142,610)(143,612)(144,611)(145,595)(146,597)(147,596)(148,598)(149,600)(150,599)(151,601)(152,603)(153,602)(154,613)(155,615)(156,614)(157,616)(158,618)(159,617)(160,619)(161,621)(162,620)(163,415)(164,417)(165,416)(166,418)(167,420)(168,419)(169,421)(170,423)(171,422)(172,406)(173,408)(174,407)(175,409)(176,411)(177,410)(178,412)(179,414)(180,413)(181,424)(182,426)(183,425)(184,427)(185,429)(186,428)(187,430)(188,432)(189,431)(190,469)(191,471)(192,470)(193,472)(194,474)(195,473)(196,475)(197,477)(198,476)(199,460)(200,462)(201,461)(202,463)(203,465)(204,464)(205,466)(206,468)(207,467)(208,478)(209,480)(210,479)(211,481)(212,483)(213,482)(214,484)(215,486)(216,485)(217,442)(218,444)(219,443)(220,445)(221,447)(222,446)(223,448)(224,450)(225,449)(226,433)(227,435)(228,434)(229,436)(230,438)(231,437)(232,439)(233,441)(234,440)(235,451)(236,453)(237,452)(238,454)(239,456)(240,455)(241,457)(242,459)(243,458)(244,334)(245,336)(246,335)(247,337)(248,339)(249,338)(250,340)(251,342)(252,341)(253,325)(254,327)(255,326)(256,328)(257,330)(258,329)(259,331)(260,333)(261,332)(262,343)(263,345)(264,344)(265,346)(266,348)(267,347)(268,349)(269,351)(270,350)(271,388)(272,390)(273,389)(274,391)(275,393)(276,392)(277,394)(278,396)(279,395)(280,379)(281,381)(282,380)(283,382)(284,384)(285,383)(286,385)(287,387)(288,386)(289,397)(290,399)(291,398)(292,400)(293,402)(294,401)(295,403)(296,405)(297,404)(298,361)(299,363)(300,362)(301,364)(302,366)(303,365)(304,367)(305,369)(306,368)(307,352)(308,354)(309,353)(310,355)(311,357)(312,356)(313,358)(314,360)(315,359)(316,370)(317,372)(318,371)(319,373)(320,375)(321,374)(322,376)(323,378)(324,377);;
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*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*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 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(648)!(  1,325)(  2,326)(  3,327)(  4,331)(  5,332)(  6,333)(  7,328)(  8,329)(  9,330)( 10,334)( 11,335)( 12,336)( 13,340)( 14,341)( 15,342)( 16,337)( 17,338)( 18,339)( 19,343)( 20,344)( 21,345)( 22,349)( 23,350)( 24,351)( 25,346)( 26,347)( 27,348)( 28,379)( 29,380)( 30,381)( 31,385)( 32,386)( 33,387)( 34,382)( 35,383)( 36,384)( 37,388)( 38,389)( 39,390)( 40,394)( 41,395)( 42,396)( 43,391)( 44,392)( 45,393)( 46,397)( 47,398)( 48,399)( 49,403)( 50,404)( 51,405)( 52,400)( 53,401)( 54,402)( 55,352)( 56,353)( 57,354)( 58,358)( 59,359)( 60,360)( 61,355)( 62,356)( 63,357)( 64,361)( 65,362)( 66,363)( 67,367)( 68,368)( 69,369)( 70,364)( 71,365)( 72,366)( 73,370)( 74,371)( 75,372)( 76,376)( 77,377)( 78,378)( 79,373)( 80,374)( 81,375)( 82,406)( 83,407)( 84,408)( 85,412)( 86,413)( 87,414)( 88,409)( 89,410)( 90,411)( 91,415)( 92,416)( 93,417)( 94,421)( 95,422)( 96,423)( 97,418)( 98,419)( 99,420)(100,424)(101,425)(102,426)(103,430)(104,431)(105,432)(106,427)(107,428)(108,429)(109,460)(110,461)(111,462)(112,466)(113,467)(114,468)(115,463)(116,464)(117,465)(118,469)(119,470)(120,471)(121,475)(122,476)(123,477)(124,472)(125,473)(126,474)(127,478)(128,479)(129,480)(130,484)(131,485)(132,486)(133,481)(134,482)(135,483)(136,433)(137,434)(138,435)(139,439)(140,440)(141,441)(142,436)(143,437)(144,438)(145,442)(146,443)(147,444)(148,448)(149,449)(150,450)(151,445)(152,446)(153,447)(154,451)(155,452)(156,453)(157,457)(158,458)(159,459)(160,454)(161,455)(162,456)(163,568)(164,569)(165,570)(166,574)(167,575)(168,576)(169,571)(170,572)(171,573)(172,577)(173,578)(174,579)(175,583)(176,584)(177,585)(178,580)(179,581)(180,582)(181,586)(182,587)(183,588)(184,592)(185,593)(186,594)(187,589)(188,590)(189,591)(190,622)(191,623)(192,624)(193,628)(194,629)(195,630)(196,625)(197,626)(198,627)(199,631)(200,632)(201,633)(202,637)(203,638)(204,639)(205,634)(206,635)(207,636)(208,640)(209,641)(210,642)(211,646)(212,647)(213,648)(214,643)(215,644)(216,645)(217,595)(218,596)(219,597)(220,601)(221,602)(222,603)(223,598)(224,599)(225,600)(226,604)(227,605)(228,606)(229,610)(230,611)(231,612)(232,607)(233,608)(234,609)(235,613)(236,614)(237,615)(238,619)(239,620)(240,621)(241,616)(242,617)(243,618)(244,487)(245,488)(246,489)(247,493)(248,494)(249,495)(250,490)(251,491)(252,492)(253,496)(254,497)(255,498)(256,502)(257,503)(258,504)(259,499)(260,500)(261,501)(262,505)(263,506)(264,507)(265,511)(266,512)(267,513)(268,508)(269,509)(270,510)(271,541)(272,542)(273,543)(274,547)(275,548)(276,549)(277,544)(278,545)(279,546)(280,550)(281,551)(282,552)(283,556)(284,557)(285,558)(286,553)(287,554)(288,555)(289,559)(290,560)(291,561)(292,565)(293,566)(294,567)(295,562)(296,563)(297,564)(298,514)(299,515)(300,516)(301,520)(302,521)(303,522)(304,517)(305,518)(306,519)(307,523)(308,524)(309,525)(310,529)(311,530)(312,531)(313,526)(314,527)(315,528)(316,532)(317,533)(318,534)(319,538)(320,539)(321,540)(322,535)(323,536)(324,537);
s1 := Sym(648)!(  1,545)(  2,546)(  3,544)(  4,543)(  5,541)(  6,542)(  7,547)(  8,548)(  9,549)( 10,563)( 11,564)( 12,562)( 13,561)( 14,559)( 15,560)( 16,565)( 17,566)( 18,567)( 19,554)( 20,555)( 21,553)( 22,552)( 23,550)( 24,551)( 25,556)( 26,557)( 27,558)( 28,518)( 29,519)( 30,517)( 31,516)( 32,514)( 33,515)( 34,520)( 35,521)( 36,522)( 37,536)( 38,537)( 39,535)( 40,534)( 41,532)( 42,533)( 43,538)( 44,539)( 45,540)( 46,527)( 47,528)( 48,526)( 49,525)( 50,523)( 51,524)( 52,529)( 53,530)( 54,531)( 55,491)( 56,492)( 57,490)( 58,489)( 59,487)( 60,488)( 61,493)( 62,494)( 63,495)( 64,509)( 65,510)( 66,508)( 67,507)( 68,505)( 69,506)( 70,511)( 71,512)( 72,513)( 73,500)( 74,501)( 75,499)( 76,498)( 77,496)( 78,497)( 79,502)( 80,503)( 81,504)( 82,626)( 83,627)( 84,625)( 85,624)( 86,622)( 87,623)( 88,628)( 89,629)( 90,630)( 91,644)( 92,645)( 93,643)( 94,642)( 95,640)( 96,641)( 97,646)( 98,647)( 99,648)(100,635)(101,636)(102,634)(103,633)(104,631)(105,632)(106,637)(107,638)(108,639)(109,599)(110,600)(111,598)(112,597)(113,595)(114,596)(115,601)(116,602)(117,603)(118,617)(119,618)(120,616)(121,615)(122,613)(123,614)(124,619)(125,620)(126,621)(127,608)(128,609)(129,607)(130,606)(131,604)(132,605)(133,610)(134,611)(135,612)(136,572)(137,573)(138,571)(139,570)(140,568)(141,569)(142,574)(143,575)(144,576)(145,590)(146,591)(147,589)(148,588)(149,586)(150,587)(151,592)(152,593)(153,594)(154,581)(155,582)(156,580)(157,579)(158,577)(159,578)(160,583)(161,584)(162,585)(163,383)(164,384)(165,382)(166,381)(167,379)(168,380)(169,385)(170,386)(171,387)(172,401)(173,402)(174,400)(175,399)(176,397)(177,398)(178,403)(179,404)(180,405)(181,392)(182,393)(183,391)(184,390)(185,388)(186,389)(187,394)(188,395)(189,396)(190,356)(191,357)(192,355)(193,354)(194,352)(195,353)(196,358)(197,359)(198,360)(199,374)(200,375)(201,373)(202,372)(203,370)(204,371)(205,376)(206,377)(207,378)(208,365)(209,366)(210,364)(211,363)(212,361)(213,362)(214,367)(215,368)(216,369)(217,329)(218,330)(219,328)(220,327)(221,325)(222,326)(223,331)(224,332)(225,333)(226,347)(227,348)(228,346)(229,345)(230,343)(231,344)(232,349)(233,350)(234,351)(235,338)(236,339)(237,337)(238,336)(239,334)(240,335)(241,340)(242,341)(243,342)(244,464)(245,465)(246,463)(247,462)(248,460)(249,461)(250,466)(251,467)(252,468)(253,482)(254,483)(255,481)(256,480)(257,478)(258,479)(259,484)(260,485)(261,486)(262,473)(263,474)(264,472)(265,471)(266,469)(267,470)(268,475)(269,476)(270,477)(271,437)(272,438)(273,436)(274,435)(275,433)(276,434)(277,439)(278,440)(279,441)(280,455)(281,456)(282,454)(283,453)(284,451)(285,452)(286,457)(287,458)(288,459)(289,446)(290,447)(291,445)(292,444)(293,442)(294,443)(295,448)(296,449)(297,450)(298,410)(299,411)(300,409)(301,408)(302,406)(303,407)(304,412)(305,413)(306,414)(307,428)(308,429)(309,427)(310,426)(311,424)(312,425)(313,430)(314,431)(315,432)(316,419)(317,420)(318,418)(319,417)(320,415)(321,416)(322,421)(323,422)(324,423);
s2 := Sym(648)!(  1,496)(  2,498)(  3,497)(  4,499)(  5,501)(  6,500)(  7,502)(  8,504)(  9,503)( 10,487)( 11,489)( 12,488)( 13,490)( 14,492)( 15,491)( 16,493)( 17,495)( 18,494)( 19,505)( 20,507)( 21,506)( 22,508)( 23,510)( 24,509)( 25,511)( 26,513)( 27,512)( 28,550)( 29,552)( 30,551)( 31,553)( 32,555)( 33,554)( 34,556)( 35,558)( 36,557)( 37,541)( 38,543)( 39,542)( 40,544)( 41,546)( 42,545)( 43,547)( 44,549)( 45,548)( 46,559)( 47,561)( 48,560)( 49,562)( 50,564)( 51,563)( 52,565)( 53,567)( 54,566)( 55,523)( 56,525)( 57,524)( 58,526)( 59,528)( 60,527)( 61,529)( 62,531)( 63,530)( 64,514)( 65,516)( 66,515)( 67,517)( 68,519)( 69,518)( 70,520)( 71,522)( 72,521)( 73,532)( 74,534)( 75,533)( 76,535)( 77,537)( 78,536)( 79,538)( 80,540)( 81,539)( 82,577)( 83,579)( 84,578)( 85,580)( 86,582)( 87,581)( 88,583)( 89,585)( 90,584)( 91,568)( 92,570)( 93,569)( 94,571)( 95,573)( 96,572)( 97,574)( 98,576)( 99,575)(100,586)(101,588)(102,587)(103,589)(104,591)(105,590)(106,592)(107,594)(108,593)(109,631)(110,633)(111,632)(112,634)(113,636)(114,635)(115,637)(116,639)(117,638)(118,622)(119,624)(120,623)(121,625)(122,627)(123,626)(124,628)(125,630)(126,629)(127,640)(128,642)(129,641)(130,643)(131,645)(132,644)(133,646)(134,648)(135,647)(136,604)(137,606)(138,605)(139,607)(140,609)(141,608)(142,610)(143,612)(144,611)(145,595)(146,597)(147,596)(148,598)(149,600)(150,599)(151,601)(152,603)(153,602)(154,613)(155,615)(156,614)(157,616)(158,618)(159,617)(160,619)(161,621)(162,620)(163,415)(164,417)(165,416)(166,418)(167,420)(168,419)(169,421)(170,423)(171,422)(172,406)(173,408)(174,407)(175,409)(176,411)(177,410)(178,412)(179,414)(180,413)(181,424)(182,426)(183,425)(184,427)(185,429)(186,428)(187,430)(188,432)(189,431)(190,469)(191,471)(192,470)(193,472)(194,474)(195,473)(196,475)(197,477)(198,476)(199,460)(200,462)(201,461)(202,463)(203,465)(204,464)(205,466)(206,468)(207,467)(208,478)(209,480)(210,479)(211,481)(212,483)(213,482)(214,484)(215,486)(216,485)(217,442)(218,444)(219,443)(220,445)(221,447)(222,446)(223,448)(224,450)(225,449)(226,433)(227,435)(228,434)(229,436)(230,438)(231,437)(232,439)(233,441)(234,440)(235,451)(236,453)(237,452)(238,454)(239,456)(240,455)(241,457)(242,459)(243,458)(244,334)(245,336)(246,335)(247,337)(248,339)(249,338)(250,340)(251,342)(252,341)(253,325)(254,327)(255,326)(256,328)(257,330)(258,329)(259,331)(260,333)(261,332)(262,343)(263,345)(264,344)(265,346)(266,348)(267,347)(268,349)(269,351)(270,350)(271,388)(272,390)(273,389)(274,391)(275,393)(276,392)(277,394)(278,396)(279,395)(280,379)(281,381)(282,380)(283,382)(284,384)(285,383)(286,385)(287,387)(288,386)(289,397)(290,399)(291,398)(292,400)(293,402)(294,401)(295,403)(296,405)(297,404)(298,361)(299,363)(300,362)(301,364)(302,366)(303,365)(304,367)(305,369)(306,368)(307,352)(308,354)(309,353)(310,355)(311,357)(312,356)(313,358)(314,360)(315,359)(316,370)(317,372)(318,371)(319,373)(320,375)(321,374)(322,376)(323,378)(324,377);
poly := sub<Sym(648)|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*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*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 >;
References : None.
to this polytope

Twisty Puzzle