Polytope of Type {171,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {171,4}*1368
if this polytope has a name.
Group : SmallGroup(1368,55)
Rank : 3
Schlafli Type : {171,4}
Number of vertices, edges, etc : 171, 342, 4
Order of s0s1s2 : 171
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {57,4}*456
   19-fold quotients : {9,4}*72
   57-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   None.

Permutation Representation (GAP) : 
s0 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13,217)( 14,218)( 15,220)( 16,219)( 17,225)( 18,226)( 19,228)( 20,227)( 21,221)( 22,222)( 23,224)( 24,223)( 25,205)( 26,206)( 27,208)( 28,207)( 29,213)( 30,214)( 31,216)( 32,215)( 33,209)( 34,210)( 35,212)( 36,211)( 37,193)( 38,194)( 39,196)( 40,195)( 41,201)( 42,202)( 43,204)( 44,203)( 45,197)( 46,198)( 47,200)( 48,199)( 49,181)( 50,182)( 51,184)( 52,183)( 53,189)( 54,190)( 55,192)( 56,191)( 57,185)( 58,186)( 59,188)( 60,187)( 61,169)( 62,170)( 63,172)( 64,171)( 65,177)( 66,178)( 67,180)( 68,179)( 69,173)( 70,174)( 71,176)( 72,175)( 73,157)( 74,158)( 75,160)( 76,159)( 77,165)( 78,166)( 79,168)( 80,167)( 81,161)( 82,162)( 83,164)( 84,163)( 85,145)( 86,146)( 87,148)( 88,147)( 89,153)( 90,154)( 91,156)( 92,155)( 93,149)( 94,150)( 95,152)( 96,151)( 97,133)( 98,134)( 99,136)(100,135)(101,141)(102,142)(103,144)(104,143)(105,137)(106,138)(107,140)(108,139)(109,121)(110,122)(111,124)(112,123)(113,129)(114,130)(115,132)(116,131)(117,125)(118,126)(119,128)(120,127)(229,465)(230,466)(231,468)(232,467)(233,461)(234,462)(235,464)(236,463)(237,457)(238,458)(239,460)(240,459)(241,681)(242,682)(243,684)(244,683)(245,677)(246,678)(247,680)(248,679)(249,673)(250,674)(251,676)(252,675)(253,669)(254,670)(255,672)(256,671)(257,665)(258,666)(259,668)(260,667)(261,661)(262,662)(263,664)(264,663)(265,657)(266,658)(267,660)(268,659)(269,653)(270,654)(271,656)(272,655)(273,649)(274,650)(275,652)(276,651)(277,645)(278,646)(279,648)(280,647)(281,641)(282,642)(283,644)(284,643)(285,637)(286,638)(287,640)(288,639)(289,633)(290,634)(291,636)(292,635)(293,629)(294,630)(295,632)(296,631)(297,625)(298,626)(299,628)(300,627)(301,621)(302,622)(303,624)(304,623)(305,617)(306,618)(307,620)(308,619)(309,613)(310,614)(311,616)(312,615)(313,609)(314,610)(315,612)(316,611)(317,605)(318,606)(319,608)(320,607)(321,601)(322,602)(323,604)(324,603)(325,597)(326,598)(327,600)(328,599)(329,593)(330,594)(331,596)(332,595)(333,589)(334,590)(335,592)(336,591)(337,585)(338,586)(339,588)(340,587)(341,581)(342,582)(343,584)(344,583)(345,577)(346,578)(347,580)(348,579)(349,573)(350,574)(351,576)(352,575)(353,569)(354,570)(355,572)(356,571)(357,565)(358,566)(359,568)(360,567)(361,561)(362,562)(363,564)(364,563)(365,557)(366,558)(367,560)(368,559)(369,553)(370,554)(371,556)(372,555)(373,549)(374,550)(375,552)(376,551)(377,545)(378,546)(379,548)(380,547)(381,541)(382,542)(383,544)(384,543)(385,537)(386,538)(387,540)(388,539)(389,533)(390,534)(391,536)(392,535)(393,529)(394,530)(395,532)(396,531)(397,525)(398,526)(399,528)(400,527)(401,521)(402,522)(403,524)(404,523)(405,517)(406,518)(407,520)(408,519)(409,513)(410,514)(411,516)(412,515)(413,509)(414,510)(415,512)(416,511)(417,505)(418,506)(419,508)(420,507)(421,501)(422,502)(423,504)(424,503)(425,497)(426,498)(427,500)(428,499)(429,493)(430,494)(431,496)(432,495)(433,489)(434,490)(435,492)(436,491)(437,485)(438,486)(439,488)(440,487)(441,481)(442,482)(443,484)(444,483)(445,477)(446,478)(447,480)(448,479)(449,473)(450,474)(451,476)(452,475)(453,469)(454,470)(455,472)(456,471);;
s1 := (  1,241)(  2,244)(  3,243)(  4,242)(  5,249)(  6,252)(  7,251)(  8,250)(  9,245)( 10,248)( 11,247)( 12,246)( 13,229)( 14,232)( 15,231)( 16,230)( 17,237)( 18,240)( 19,239)( 20,238)( 21,233)( 22,236)( 23,235)( 24,234)( 25,445)( 26,448)( 27,447)( 28,446)( 29,453)( 30,456)( 31,455)( 32,454)( 33,449)( 34,452)( 35,451)( 36,450)( 37,433)( 38,436)( 39,435)( 40,434)( 41,441)( 42,444)( 43,443)( 44,442)( 45,437)( 46,440)( 47,439)( 48,438)( 49,421)( 50,424)( 51,423)( 52,422)( 53,429)( 54,432)( 55,431)( 56,430)( 57,425)( 58,428)( 59,427)( 60,426)( 61,409)( 62,412)( 63,411)( 64,410)( 65,417)( 66,420)( 67,419)( 68,418)( 69,413)( 70,416)( 71,415)( 72,414)( 73,397)( 74,400)( 75,399)( 76,398)( 77,405)( 78,408)( 79,407)( 80,406)( 81,401)( 82,404)( 83,403)( 84,402)( 85,385)( 86,388)( 87,387)( 88,386)( 89,393)( 90,396)( 91,395)( 92,394)( 93,389)( 94,392)( 95,391)( 96,390)( 97,373)( 98,376)( 99,375)(100,374)(101,381)(102,384)(103,383)(104,382)(105,377)(106,380)(107,379)(108,378)(109,361)(110,364)(111,363)(112,362)(113,369)(114,372)(115,371)(116,370)(117,365)(118,368)(119,367)(120,366)(121,349)(122,352)(123,351)(124,350)(125,357)(126,360)(127,359)(128,358)(129,353)(130,356)(131,355)(132,354)(133,337)(134,340)(135,339)(136,338)(137,345)(138,348)(139,347)(140,346)(141,341)(142,344)(143,343)(144,342)(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)(152,334)(153,329)(154,332)(155,331)(156,330)(157,313)(158,316)(159,315)(160,314)(161,321)(162,324)(163,323)(164,322)(165,317)(166,320)(167,319)(168,318)(169,301)(170,304)(171,303)(172,302)(173,309)(174,312)(175,311)(176,310)(177,305)(178,308)(179,307)(180,306)(181,289)(182,292)(183,291)(184,290)(185,297)(186,300)(187,299)(188,298)(189,293)(190,296)(191,295)(192,294)(193,277)(194,280)(195,279)(196,278)(197,285)(198,288)(199,287)(200,286)(201,281)(202,284)(203,283)(204,282)(205,265)(206,268)(207,267)(208,266)(209,273)(210,276)(211,275)(212,274)(213,269)(214,272)(215,271)(216,270)(217,253)(218,256)(219,255)(220,254)(221,261)(222,264)(223,263)(224,262)(225,257)(226,260)(227,259)(228,258)(457,477)(458,480)(459,479)(460,478)(461,473)(462,476)(463,475)(464,474)(465,469)(466,472)(467,471)(468,470)(481,681)(482,684)(483,683)(484,682)(485,677)(486,680)(487,679)(488,678)(489,673)(490,676)(491,675)(492,674)(493,669)(494,672)(495,671)(496,670)(497,665)(498,668)(499,667)(500,666)(501,661)(502,664)(503,663)(504,662)(505,657)(506,660)(507,659)(508,658)(509,653)(510,656)(511,655)(512,654)(513,649)(514,652)(515,651)(516,650)(517,645)(518,648)(519,647)(520,646)(521,641)(522,644)(523,643)(524,642)(525,637)(526,640)(527,639)(528,638)(529,633)(530,636)(531,635)(532,634)(533,629)(534,632)(535,631)(536,630)(537,625)(538,628)(539,627)(540,626)(541,621)(542,624)(543,623)(544,622)(545,617)(546,620)(547,619)(548,618)(549,613)(550,616)(551,615)(552,614)(553,609)(554,612)(555,611)(556,610)(557,605)(558,608)(559,607)(560,606)(561,601)(562,604)(563,603)(564,602)(565,597)(566,600)(567,599)(568,598)(569,593)(570,596)(571,595)(572,594)(573,589)(574,592)(575,591)(576,590)(577,585)(578,588)(579,587)(580,586)(582,584);;
s2 := (  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)(271,272)(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)(287,288)(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)(303,304)(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)(319,320)(321,322)(323,324)(325,326)(327,328)(329,330)(331,332)(333,334)(335,336)(337,338)(339,340)(341,342)(343,344)(345,346)(347,348)(349,350)(351,352)(353,354)(355,356)(357,358)(359,360)(361,362)(363,364)(365,366)(367,368)(369,370)(371,372)(373,374)(375,376)(377,378)(379,380)(381,382)(383,384)(385,386)(387,388)(389,390)(391,392)(393,394)(395,396)(397,398)(399,400)(401,402)(403,404)(405,406)(407,408)(409,410)(411,412)(413,414)(415,416)(417,418)(419,420)(421,422)(423,424)(425,426)(427,428)(429,430)(431,432)(433,434)(435,436)(437,438)(439,440)(441,442)(443,444)(445,446)(447,448)(449,450)(451,452)(453,454)(455,456)(457,458)(459,460)(461,462)(463,464)(465,466)(467,468)(469,470)(471,472)(473,474)(475,476)(477,478)(479,480)(481,482)(483,484)(485,486)(487,488)(489,490)(491,492)(493,494)(495,496)(497,498)(499,500)(501,502)(503,504)(505,506)(507,508)(509,510)(511,512)(513,514)(515,516)(517,518)(519,520)(521,522)(523,524)(525,526)(527,528)(529,530)(531,532)(533,534)(535,536)(537,538)(539,540)(541,542)(543,544)(545,546)(547,548)(549,550)(551,552)(553,554)(555,556)(557,558)(559,560)(561,562)(563,564)(565,566)(567,568)(569,570)(571,572)(573,574)(575,576)(577,578)(579,580)(581,582)(583,584)(585,586)(587,588)(589,590)(591,592)(593,594)(595,596)(597,598)(599,600)(601,602)(603,604)(605,606)(607,608)(609,610)(611,612)(613,614)(615,616)(617,618)(619,620)(621,622)(623,624)(625,626)(627,628)(629,630)(631,632)(633,634)(635,636)(637,638)(639,640)(641,642)(643,644)(645,646)(647,648)(649,650)(651,652)(653,654)(655,656)(657,658)(659,660)(661,662)(663,664)(665,666)(667,668)(669,670)(671,672)(673,674)(675,676)(677,678)(679,680)(681,682)(683,684);;
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, 
s2*s1*s0*s2*s1*s2*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*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*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*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*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*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*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*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*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*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*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*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*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*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(684)!(  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13,217)( 14,218)( 15,220)( 16,219)( 17,225)( 18,226)( 19,228)( 20,227)( 21,221)( 22,222)( 23,224)( 24,223)( 25,205)( 26,206)( 27,208)( 28,207)( 29,213)( 30,214)( 31,216)( 32,215)( 33,209)( 34,210)( 35,212)( 36,211)( 37,193)( 38,194)( 39,196)( 40,195)( 41,201)( 42,202)( 43,204)( 44,203)( 45,197)( 46,198)( 47,200)( 48,199)( 49,181)( 50,182)( 51,184)( 52,183)( 53,189)( 54,190)( 55,192)( 56,191)( 57,185)( 58,186)( 59,188)( 60,187)( 61,169)( 62,170)( 63,172)( 64,171)( 65,177)( 66,178)( 67,180)( 68,179)( 69,173)( 70,174)( 71,176)( 72,175)( 73,157)( 74,158)( 75,160)( 76,159)( 77,165)( 78,166)( 79,168)( 80,167)( 81,161)( 82,162)( 83,164)( 84,163)( 85,145)( 86,146)( 87,148)( 88,147)( 89,153)( 90,154)( 91,156)( 92,155)( 93,149)( 94,150)( 95,152)( 96,151)( 97,133)( 98,134)( 99,136)(100,135)(101,141)(102,142)(103,144)(104,143)(105,137)(106,138)(107,140)(108,139)(109,121)(110,122)(111,124)(112,123)(113,129)(114,130)(115,132)(116,131)(117,125)(118,126)(119,128)(120,127)(229,465)(230,466)(231,468)(232,467)(233,461)(234,462)(235,464)(236,463)(237,457)(238,458)(239,460)(240,459)(241,681)(242,682)(243,684)(244,683)(245,677)(246,678)(247,680)(248,679)(249,673)(250,674)(251,676)(252,675)(253,669)(254,670)(255,672)(256,671)(257,665)(258,666)(259,668)(260,667)(261,661)(262,662)(263,664)(264,663)(265,657)(266,658)(267,660)(268,659)(269,653)(270,654)(271,656)(272,655)(273,649)(274,650)(275,652)(276,651)(277,645)(278,646)(279,648)(280,647)(281,641)(282,642)(283,644)(284,643)(285,637)(286,638)(287,640)(288,639)(289,633)(290,634)(291,636)(292,635)(293,629)(294,630)(295,632)(296,631)(297,625)(298,626)(299,628)(300,627)(301,621)(302,622)(303,624)(304,623)(305,617)(306,618)(307,620)(308,619)(309,613)(310,614)(311,616)(312,615)(313,609)(314,610)(315,612)(316,611)(317,605)(318,606)(319,608)(320,607)(321,601)(322,602)(323,604)(324,603)(325,597)(326,598)(327,600)(328,599)(329,593)(330,594)(331,596)(332,595)(333,589)(334,590)(335,592)(336,591)(337,585)(338,586)(339,588)(340,587)(341,581)(342,582)(343,584)(344,583)(345,577)(346,578)(347,580)(348,579)(349,573)(350,574)(351,576)(352,575)(353,569)(354,570)(355,572)(356,571)(357,565)(358,566)(359,568)(360,567)(361,561)(362,562)(363,564)(364,563)(365,557)(366,558)(367,560)(368,559)(369,553)(370,554)(371,556)(372,555)(373,549)(374,550)(375,552)(376,551)(377,545)(378,546)(379,548)(380,547)(381,541)(382,542)(383,544)(384,543)(385,537)(386,538)(387,540)(388,539)(389,533)(390,534)(391,536)(392,535)(393,529)(394,530)(395,532)(396,531)(397,525)(398,526)(399,528)(400,527)(401,521)(402,522)(403,524)(404,523)(405,517)(406,518)(407,520)(408,519)(409,513)(410,514)(411,516)(412,515)(413,509)(414,510)(415,512)(416,511)(417,505)(418,506)(419,508)(420,507)(421,501)(422,502)(423,504)(424,503)(425,497)(426,498)(427,500)(428,499)(429,493)(430,494)(431,496)(432,495)(433,489)(434,490)(435,492)(436,491)(437,485)(438,486)(439,488)(440,487)(441,481)(442,482)(443,484)(444,483)(445,477)(446,478)(447,480)(448,479)(449,473)(450,474)(451,476)(452,475)(453,469)(454,470)(455,472)(456,471);
s1 := Sym(684)!(  1,241)(  2,244)(  3,243)(  4,242)(  5,249)(  6,252)(  7,251)(  8,250)(  9,245)( 10,248)( 11,247)( 12,246)( 13,229)( 14,232)( 15,231)( 16,230)( 17,237)( 18,240)( 19,239)( 20,238)( 21,233)( 22,236)( 23,235)( 24,234)( 25,445)( 26,448)( 27,447)( 28,446)( 29,453)( 30,456)( 31,455)( 32,454)( 33,449)( 34,452)( 35,451)( 36,450)( 37,433)( 38,436)( 39,435)( 40,434)( 41,441)( 42,444)( 43,443)( 44,442)( 45,437)( 46,440)( 47,439)( 48,438)( 49,421)( 50,424)( 51,423)( 52,422)( 53,429)( 54,432)( 55,431)( 56,430)( 57,425)( 58,428)( 59,427)( 60,426)( 61,409)( 62,412)( 63,411)( 64,410)( 65,417)( 66,420)( 67,419)( 68,418)( 69,413)( 70,416)( 71,415)( 72,414)( 73,397)( 74,400)( 75,399)( 76,398)( 77,405)( 78,408)( 79,407)( 80,406)( 81,401)( 82,404)( 83,403)( 84,402)( 85,385)( 86,388)( 87,387)( 88,386)( 89,393)( 90,396)( 91,395)( 92,394)( 93,389)( 94,392)( 95,391)( 96,390)( 97,373)( 98,376)( 99,375)(100,374)(101,381)(102,384)(103,383)(104,382)(105,377)(106,380)(107,379)(108,378)(109,361)(110,364)(111,363)(112,362)(113,369)(114,372)(115,371)(116,370)(117,365)(118,368)(119,367)(120,366)(121,349)(122,352)(123,351)(124,350)(125,357)(126,360)(127,359)(128,358)(129,353)(130,356)(131,355)(132,354)(133,337)(134,340)(135,339)(136,338)(137,345)(138,348)(139,347)(140,346)(141,341)(142,344)(143,343)(144,342)(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)(152,334)(153,329)(154,332)(155,331)(156,330)(157,313)(158,316)(159,315)(160,314)(161,321)(162,324)(163,323)(164,322)(165,317)(166,320)(167,319)(168,318)(169,301)(170,304)(171,303)(172,302)(173,309)(174,312)(175,311)(176,310)(177,305)(178,308)(179,307)(180,306)(181,289)(182,292)(183,291)(184,290)(185,297)(186,300)(187,299)(188,298)(189,293)(190,296)(191,295)(192,294)(193,277)(194,280)(195,279)(196,278)(197,285)(198,288)(199,287)(200,286)(201,281)(202,284)(203,283)(204,282)(205,265)(206,268)(207,267)(208,266)(209,273)(210,276)(211,275)(212,274)(213,269)(214,272)(215,271)(216,270)(217,253)(218,256)(219,255)(220,254)(221,261)(222,264)(223,263)(224,262)(225,257)(226,260)(227,259)(228,258)(457,477)(458,480)(459,479)(460,478)(461,473)(462,476)(463,475)(464,474)(465,469)(466,472)(467,471)(468,470)(481,681)(482,684)(483,683)(484,682)(485,677)(486,680)(487,679)(488,678)(489,673)(490,676)(491,675)(492,674)(493,669)(494,672)(495,671)(496,670)(497,665)(498,668)(499,667)(500,666)(501,661)(502,664)(503,663)(504,662)(505,657)(506,660)(507,659)(508,658)(509,653)(510,656)(511,655)(512,654)(513,649)(514,652)(515,651)(516,650)(517,645)(518,648)(519,647)(520,646)(521,641)(522,644)(523,643)(524,642)(525,637)(526,640)(527,639)(528,638)(529,633)(530,636)(531,635)(532,634)(533,629)(534,632)(535,631)(536,630)(537,625)(538,628)(539,627)(540,626)(541,621)(542,624)(543,623)(544,622)(545,617)(546,620)(547,619)(548,618)(549,613)(550,616)(551,615)(552,614)(553,609)(554,612)(555,611)(556,610)(557,605)(558,608)(559,607)(560,606)(561,601)(562,604)(563,603)(564,602)(565,597)(566,600)(567,599)(568,598)(569,593)(570,596)(571,595)(572,594)(573,589)(574,592)(575,591)(576,590)(577,585)(578,588)(579,587)(580,586)(582,584);
s2 := Sym(684)!(  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)(271,272)(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)(287,288)(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)(303,304)(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)(319,320)(321,322)(323,324)(325,326)(327,328)(329,330)(331,332)(333,334)(335,336)(337,338)(339,340)(341,342)(343,344)(345,346)(347,348)(349,350)(351,352)(353,354)(355,356)(357,358)(359,360)(361,362)(363,364)(365,366)(367,368)(369,370)(371,372)(373,374)(375,376)(377,378)(379,380)(381,382)(383,384)(385,386)(387,388)(389,390)(391,392)(393,394)(395,396)(397,398)(399,400)(401,402)(403,404)(405,406)(407,408)(409,410)(411,412)(413,414)(415,416)(417,418)(419,420)(421,422)(423,424)(425,426)(427,428)(429,430)(431,432)(433,434)(435,436)(437,438)(439,440)(441,442)(443,444)(445,446)(447,448)(449,450)(451,452)(453,454)(455,456)(457,458)(459,460)(461,462)(463,464)(465,466)(467,468)(469,470)(471,472)(473,474)(475,476)(477,478)(479,480)(481,482)(483,484)(485,486)(487,488)(489,490)(491,492)(493,494)(495,496)(497,498)(499,500)(501,502)(503,504)(505,506)(507,508)(509,510)(511,512)(513,514)(515,516)(517,518)(519,520)(521,522)(523,524)(525,526)(527,528)(529,530)(531,532)(533,534)(535,536)(537,538)(539,540)(541,542)(543,544)(545,546)(547,548)(549,550)(551,552)(553,554)(555,556)(557,558)(559,560)(561,562)(563,564)(565,566)(567,568)(569,570)(571,572)(573,574)(575,576)(577,578)(579,580)(581,582)(583,584)(585,586)(587,588)(589,590)(591,592)(593,594)(595,596)(597,598)(599,600)(601,602)(603,604)(605,606)(607,608)(609,610)(611,612)(613,614)(615,616)(617,618)(619,620)(621,622)(623,624)(625,626)(627,628)(629,630)(631,632)(633,634)(635,636)(637,638)(639,640)(641,642)(643,644)(645,646)(647,648)(649,650)(651,652)(653,654)(655,656)(657,658)(659,660)(661,662)(663,664)(665,666)(667,668)(669,670)(671,672)(673,674)(675,676)(677,678)(679,680)(681,682)(683,684);
poly := sub<Sym(684)|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, 
s2*s1*s0*s2*s1*s2*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*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*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*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*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*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*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*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*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*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*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*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*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*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope

Twisty Puzzle