Polytope of Type {6,9}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,9}*1296f
if this polytope has a name.
Group : SmallGroup(1296,3492)
Rank : 3
Schlafli Type : {6,9}
Number of vertices, edges, etc : 72, 324, 108
Order of s0s1s2 : 12
Order of s0s1s2s1 : 18
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 : {3,9}*648
   27-fold quotients : {6,3}*48
   54-fold quotients : {3,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 3.
      36 facets:
         36 of {6}*12
      36 vertex figures:
         18 of {9}*18
         18 of {3}*6
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 3.
      36 facets:
         36 of {6}*12
      24 vertex figures:
         24 of {9}*18
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 3.
      36 facets:
         36 of {6}*12
      24 vertex figures:
         24 of {9}*18
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 9.
      12 facets:
         12 of {6}*12
      8 vertex figures:
         8 of {9}*18
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s0*s2*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 9.
      12 facets:
         12 of {6}*12
      16 vertex figures:
         4 of {9}*18
         12 of {3}*6
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s0*s2*s1, s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 9.
      12 facets:
         12 of {6}*12
      12 vertex figures:
         6 of {9}*18
         6 of {3}*6

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

Twisty Puzzle