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}*1296i
if this polytope has a name.
Group : SmallGroup(1296,2077)
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
   Self-Petrie
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}*648f
   3-fold quotients : {12,6}*432c, {12,6}*432g
   6-fold quotients : {6,6}*216a, {6,6}*216d
   9-fold quotients : {12,6}*144a, {12,6}*144b, {12,6}*144c
   12-fold quotients : {6,3}*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> 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*s2*s1*s0*s2*s1*s2> of order 3.
      18 facets:
         18 of {12}*24
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s2*s1*s2*s1*s0*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*s0*s1*s0*s1> of order 3.
      24 facets:
         9 of {4}*8
         15 of {12}*24
      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*s0*s2*s1*s0*s1*s0*s2*s1, s0*s2*s1*s0*s1*s0*s1*s0*s1*s2> of order 9.
      10 facets:
         4 of {12}*24
         6 of {4}*8
      12 vertex figures:
         12 of {6}*12

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