Polytope of Type {4,81}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,81}*1296
if this polytope has a name.
Group : SmallGroup(1296,630)
Rank : 3
Schlafli Type : {4,81}
Number of vertices, edges, etc : 8, 324, 162
Order of s0s1s2 : 162
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,81}*648
   3-fold quotients : {4,27}*432
   4-fold quotients : {2,81}*324
   6-fold quotients : {4,27}*216
   9-fold quotients : {4,9}*144
   12-fold quotients : {2,27}*108
   18-fold quotients : {4,9}*72
   27-fold quotients : {4,3}*48
   36-fold quotients : {2,9}*36
   54-fold quotients : {4,3}*24
   108-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1> of order 2.
      108 facets:
         54 of {2}*4
         54 of {4}*8
      4 vertex figures:
         4 of {81}*162

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