Polytope of Type {4,25}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,25}*1600
if this polytope has a name.
Group : SmallGroup(1600,5601)
Rank : 3
Schlafli Type : {4,25}
Number of vertices, edges, etc : 32, 400, 200
Order of s0s1s2 : 50
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 : {4,25}*800
   5-fold quotients : {4,5}*320
   10-fold quotients : {4,5}*160
   16-fold quotients : {2,25}*100
   80-fold quotients : {2,5}*20
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*s2*s1*s0*s1*s2*s1*s2> of order 2.
      120 facets:
         80 of {4}*8
         40 of {2}*4
      16 vertex figures:
         16 of {25}*50
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 2.
      100 facets:
         100 of {4}*8
      16 vertex figures:
         16 of {25}*50
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 2.
      100 facets:
         100 of {4}*8
      16 vertex figures:
         16 of {25}*50
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
      100 facets:
         100 of {4}*8
      16 vertex figures:
         16 of {25}*50
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1> of order 2.
      100 facets:
         100 of {4}*8
      16 vertex figures:
         16 of {25}*50
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2, s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 4.
      60 facets:
         40 of {4}*8
         20 of {2}*4
      8 vertex figures:
         8 of {25}*50
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
      60 facets:
         40 of {4}*8
         20 of {2}*4
      8 vertex figures:
         8 of {25}*50
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
      50 facets:
         50 of {4}*8
      8 vertex figures:
         8 of {25}*50
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
      50 facets:
         50 of {4}*8
      8 vertex figures:
         8 of {25}*50
   P/N, where N=<s0*s1*s0*s1, s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 4.
      70 facets:
         40 of {2}*4
         30 of {4}*8
      8 vertex figures:
         8 of {25}*50
   P/N, where N=<s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
      70 facets:
         30 of {4}*8
         40 of {2}*4
      8 vertex figures:
         8 of {25}*50
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
      50 facets:
         50 of {4}*8
      8 vertex figures:
         8 of {25}*50
   P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
      60 facets:
         20 of {2}*4
         40 of {4}*8
      8 vertex figures:
         8 of {25}*50
   P/N, where N=<s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
      60 facets:
         40 of {4}*8
         20 of {2}*4
      8 vertex figures:
         8 of {25}*50
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 4.
      50 facets:
         50 of {4}*8
      8 vertex figures:
         8 of {25}*50
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 4.
      50 facets:
         50 of {4}*8
      8 vertex figures:
         8 of {25}*50
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
      50 facets:
         50 of {4}*8
      8 vertex figures:
         8 of {25}*50
   P/N, where N=<s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
      60 facets:
         40 of {4}*8
         20 of {2}*4
      8 vertex figures:
         8 of {25}*50
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1> of order 4.
      50 facets:
         50 of {4}*8
      8 vertex figures:
         8 of {25}*50
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 4.
      50 facets:
         50 of {4}*8
      8 vertex figures:
         8 of {25}*50
   P/N, where N=<s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2> of order 4.
      60 facets:
         40 of {4}*8
         20 of {2}*4
      8 vertex figures:
         8 of {25}*50
   P/N, where N=<s0*s1*s0*s1, s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 8.
      40 facets:
         30 of {2}*4
         10 of {4}*8
      4 vertex figures:
         4 of {25}*50
   P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 8.
      30 facets:
         10 of {2}*4
         20 of {4}*8
      4 vertex figures:
         4 of {25}*50
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1, s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 8.
      25 facets:
         25 of {4}*8
      4 vertex figures:
         4 of {25}*50
   P/N, where N=<s0*s1*s0*s1, s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 8.
      35 facets:
         20 of {2}*4
         15 of {4}*8
      4 vertex figures:
         4 of {25}*50
   P/N, where N=<s0*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 8.
      30 facets:
         20 of {4}*8
         10 of {2}*4
      4 vertex figures:
         4 of {25}*50
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1, s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 8.
      40 facets:
         10 of {4}*8
         30 of {2}*4
      4 vertex figures:
         4 of {25}*50
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 8.
      30 facets:
         10 of {2}*4
         20 of {4}*8
      4 vertex figures:
         4 of {25}*50
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 8.
      30 facets:
         10 of {2}*4
         20 of {4}*8
      4 vertex figures:
         4 of {25}*50

Permutation Representation (GAP) : 
s0 := (  1,409)(  2,410)(  3,411)(  4,412)(  5,413)(  6,414)(  7,415)(  8,416)(  9,401)( 10,402)( 11,403)( 12,404)( 13,405)( 14,406)( 15,407)( 16,408)( 17,425)( 18,426)( 19,427)( 20,428)( 21,429)( 22,430)( 23,431)( 24,432)( 25,417)( 26,418)( 27,419)( 28,420)( 29,421)( 30,422)( 31,423)( 32,424)( 33,441)( 34,442)( 35,443)( 36,444)( 37,445)( 38,446)( 39,447)( 40,448)( 41,433)( 42,434)( 43,435)( 44,436)( 45,437)( 46,438)( 47,439)( 48,440)( 49,457)( 50,458)( 51,459)( 52,460)( 53,461)( 54,462)( 55,463)( 56,464)( 57,449)( 58,450)( 59,451)( 60,452)( 61,453)( 62,454)( 63,455)( 64,456)( 65,473)( 66,474)( 67,475)( 68,476)( 69,477)( 70,478)( 71,479)( 72,480)( 73,465)( 74,466)( 75,467)( 76,468)( 77,469)( 78,470)( 79,471)( 80,472)( 81,489)( 82,490)( 83,491)( 84,492)( 85,493)( 86,494)( 87,495)( 88,496)( 89,481)( 90,482)( 91,483)( 92,484)( 93,485)( 94,486)( 95,487)( 96,488)( 97,505)( 98,506)( 99,507)(100,508)(101,509)(102,510)(103,511)(104,512)(105,497)(106,498)(107,499)(108,500)(109,501)(110,502)(111,503)(112,504)(113,521)(114,522)(115,523)(116,524)(117,525)(118,526)(119,527)(120,528)(121,513)(122,514)(123,515)(124,516)(125,517)(126,518)(127,519)(128,520)(129,537)(130,538)(131,539)(132,540)(133,541)(134,542)(135,543)(136,544)(137,529)(138,530)(139,531)(140,532)(141,533)(142,534)(143,535)(144,536)(145,553)(146,554)(147,555)(148,556)(149,557)(150,558)(151,559)(152,560)(153,545)(154,546)(155,547)(156,548)(157,549)(158,550)(159,551)(160,552)(161,569)(162,570)(163,571)(164,572)(165,573)(166,574)(167,575)(168,576)(169,561)(170,562)(171,563)(172,564)(173,565)(174,566)(175,567)(176,568)(177,585)(178,586)(179,587)(180,588)(181,589)(182,590)(183,591)(184,592)(185,577)(186,578)(187,579)(188,580)(189,581)(190,582)(191,583)(192,584)(193,601)(194,602)(195,603)(196,604)(197,605)(198,606)(199,607)(200,608)(201,593)(202,594)(203,595)(204,596)(205,597)(206,598)(207,599)(208,600)(209,617)(210,618)(211,619)(212,620)(213,621)(214,622)(215,623)(216,624)(217,609)(218,610)(219,611)(220,612)(221,613)(222,614)(223,615)(224,616)(225,633)(226,634)(227,635)(228,636)(229,637)(230,638)(231,639)(232,640)(233,625)(234,626)(235,627)(236,628)(237,629)(238,630)(239,631)(240,632)(241,649)(242,650)(243,651)(244,652)(245,653)(246,654)(247,655)(248,656)(249,641)(250,642)(251,643)(252,644)(253,645)(254,646)(255,647)(256,648)(257,665)(258,666)(259,667)(260,668)(261,669)(262,670)(263,671)(264,672)(265,657)(266,658)(267,659)(268,660)(269,661)(270,662)(271,663)(272,664)(273,681)(274,682)(275,683)(276,684)(277,685)(278,686)(279,687)(280,688)(281,673)(282,674)(283,675)(284,676)(285,677)(286,678)(287,679)(288,680)(289,697)(290,698)(291,699)(292,700)(293,701)(294,702)(295,703)(296,704)(297,689)(298,690)(299,691)(300,692)(301,693)(302,694)(303,695)(304,696)(305,713)(306,714)(307,715)(308,716)(309,717)(310,718)(311,719)(312,720)(313,705)(314,706)(315,707)(316,708)(317,709)(318,710)(319,711)(320,712)(321,729)(322,730)(323,731)(324,732)(325,733)(326,734)(327,735)(328,736)(329,721)(330,722)(331,723)(332,724)(333,725)(334,726)(335,727)(336,728)(337,745)(338,746)(339,747)(340,748)(341,749)(342,750)(343,751)(344,752)(345,737)(346,738)(347,739)(348,740)(349,741)(350,742)(351,743)(352,744)(353,761)(354,762)(355,763)(356,764)(357,765)(358,766)(359,767)(360,768)(361,753)(362,754)(363,755)(364,756)(365,757)(366,758)(367,759)(368,760)(369,777)(370,778)(371,779)(372,780)(373,781)(374,782)(375,783)(376,784)(377,769)(378,770)(379,771)(380,772)(381,773)(382,774)(383,775)(384,776)(385,793)(386,794)(387,795)(388,796)(389,797)(390,798)(391,799)(392,800)(393,785)(394,786)(395,787)(396,788)(397,789)(398,790)(399,791)(400,792);;
s1 := (  3,  4)(  5,  6)(  9, 16)( 10, 15)( 11, 13)( 12, 14)( 17, 65)( 18, 66)( 19, 68)( 20, 67)( 21, 70)( 22, 69)( 23, 71)( 24, 72)( 25, 80)( 26, 79)( 27, 77)( 28, 78)( 29, 75)( 30, 76)( 31, 74)( 32, 73)( 33, 49)( 34, 50)( 35, 52)( 36, 51)( 37, 54)( 38, 53)( 39, 55)( 40, 56)( 41, 64)( 42, 63)( 43, 61)( 44, 62)( 45, 59)( 46, 60)( 47, 58)( 48, 57)( 81,385)( 82,386)( 83,388)( 84,387)( 85,390)( 86,389)( 87,391)( 88,392)( 89,400)( 90,399)( 91,397)( 92,398)( 93,395)( 94,396)( 95,394)( 96,393)( 97,369)( 98,370)( 99,372)(100,371)(101,374)(102,373)(103,375)(104,376)(105,384)(106,383)(107,381)(108,382)(109,379)(110,380)(111,378)(112,377)(113,353)(114,354)(115,356)(116,355)(117,358)(118,357)(119,359)(120,360)(121,368)(122,367)(123,365)(124,366)(125,363)(126,364)(127,362)(128,361)(129,337)(130,338)(131,340)(132,339)(133,342)(134,341)(135,343)(136,344)(137,352)(138,351)(139,349)(140,350)(141,347)(142,348)(143,346)(144,345)(145,321)(146,322)(147,324)(148,323)(149,326)(150,325)(151,327)(152,328)(153,336)(154,335)(155,333)(156,334)(157,331)(158,332)(159,330)(160,329)(161,305)(162,306)(163,308)(164,307)(165,310)(166,309)(167,311)(168,312)(169,320)(170,319)(171,317)(172,318)(173,315)(174,316)(175,314)(176,313)(177,289)(178,290)(179,292)(180,291)(181,294)(182,293)(183,295)(184,296)(185,304)(186,303)(187,301)(188,302)(189,299)(190,300)(191,298)(192,297)(193,273)(194,274)(195,276)(196,275)(197,278)(198,277)(199,279)(200,280)(201,288)(202,287)(203,285)(204,286)(205,283)(206,284)(207,282)(208,281)(209,257)(210,258)(211,260)(212,259)(213,262)(214,261)(215,263)(216,264)(217,272)(218,271)(219,269)(220,270)(221,267)(222,268)(223,266)(224,265)(225,241)(226,242)(227,244)(228,243)(229,246)(230,245)(231,247)(232,248)(233,256)(234,255)(235,253)(236,254)(237,251)(238,252)(239,250)(240,249)(403,404)(405,406)(409,416)(410,415)(411,413)(412,414)(417,465)(418,466)(419,468)(420,467)(421,470)(422,469)(423,471)(424,472)(425,480)(426,479)(427,477)(428,478)(429,475)(430,476)(431,474)(432,473)(433,449)(434,450)(435,452)(436,451)(437,454)(438,453)(439,455)(440,456)(441,464)(442,463)(443,461)(444,462)(445,459)(446,460)(447,458)(448,457)(481,785)(482,786)(483,788)(484,787)(485,790)(486,789)(487,791)(488,792)(489,800)(490,799)(491,797)(492,798)(493,795)(494,796)(495,794)(496,793)(497,769)(498,770)(499,772)(500,771)(501,774)(502,773)(503,775)(504,776)(505,784)(506,783)(507,781)(508,782)(509,779)(510,780)(511,778)(512,777)(513,753)(514,754)(515,756)(516,755)(517,758)(518,757)(519,759)(520,760)(521,768)(522,767)(523,765)(524,766)(525,763)(526,764)(527,762)(528,761)(529,737)(530,738)(531,740)(532,739)(533,742)(534,741)(535,743)(536,744)(537,752)(538,751)(539,749)(540,750)(541,747)(542,748)(543,746)(544,745)(545,721)(546,722)(547,724)(548,723)(549,726)(550,725)(551,727)(552,728)(553,736)(554,735)(555,733)(556,734)(557,731)(558,732)(559,730)(560,729)(561,705)(562,706)(563,708)(564,707)(565,710)(566,709)(567,711)(568,712)(569,720)(570,719)(571,717)(572,718)(573,715)(574,716)(575,714)(576,713)(577,689)(578,690)(579,692)(580,691)(581,694)(582,693)(583,695)(584,696)(585,704)(586,703)(587,701)(588,702)(589,699)(590,700)(591,698)(592,697)(593,673)(594,674)(595,676)(596,675)(597,678)(598,677)(599,679)(600,680)(601,688)(602,687)(603,685)(604,686)(605,683)(606,684)(607,682)(608,681)(609,657)(610,658)(611,660)(612,659)(613,662)(614,661)(615,663)(616,664)(617,672)(618,671)(619,669)(620,670)(621,667)(622,668)(623,666)(624,665)(625,641)(626,642)(627,644)(628,643)(629,646)(630,645)(631,647)(632,648)(633,656)(634,655)(635,653)(636,654)(637,651)(638,652)(639,650)(640,649);;
s2 := (  1, 81)(  2, 90)(  3, 91)(  4, 84)(  5, 96)(  6, 87)(  7, 86)(  8, 93)(  9, 89)( 10, 82)( 11, 83)( 12, 92)( 13, 88)( 14, 95)( 15, 94)( 16, 85)( 17,145)( 18,154)( 19,155)( 20,148)( 21,160)( 22,151)( 23,150)( 24,157)( 25,153)( 26,146)( 27,147)( 28,156)( 29,152)( 30,159)( 31,158)( 32,149)( 33,129)( 34,138)( 35,139)( 36,132)( 37,144)( 38,135)( 39,134)( 40,141)( 41,137)( 42,130)( 43,131)( 44,140)( 45,136)( 46,143)( 47,142)( 48,133)( 49,113)( 50,122)( 51,123)( 52,116)( 53,128)( 54,119)( 55,118)( 56,125)( 57,121)( 58,114)( 59,115)( 60,124)( 61,120)( 62,127)( 63,126)( 64,117)( 65, 97)( 66,106)( 67,107)( 68,100)( 69,112)( 70,103)( 71,102)( 72,109)( 73,105)( 74, 98)( 75, 99)( 76,108)( 77,104)( 78,111)( 79,110)( 80,101)(161,385)(162,394)(163,395)(164,388)(165,400)(166,391)(167,390)(168,397)(169,393)(170,386)(171,387)(172,396)(173,392)(174,399)(175,398)(176,389)(177,369)(178,378)(179,379)(180,372)(181,384)(182,375)(183,374)(184,381)(185,377)(186,370)(187,371)(188,380)(189,376)(190,383)(191,382)(192,373)(193,353)(194,362)(195,363)(196,356)(197,368)(198,359)(199,358)(200,365)(201,361)(202,354)(203,355)(204,364)(205,360)(206,367)(207,366)(208,357)(209,337)(210,346)(211,347)(212,340)(213,352)(214,343)(215,342)(216,349)(217,345)(218,338)(219,339)(220,348)(221,344)(222,351)(223,350)(224,341)(225,321)(226,330)(227,331)(228,324)(229,336)(230,327)(231,326)(232,333)(233,329)(234,322)(235,323)(236,332)(237,328)(238,335)(239,334)(240,325)(241,305)(242,314)(243,315)(244,308)(245,320)(246,311)(247,310)(248,317)(249,313)(250,306)(251,307)(252,316)(253,312)(254,319)(255,318)(256,309)(257,289)(258,298)(259,299)(260,292)(261,304)(262,295)(263,294)(264,301)(265,297)(266,290)(267,291)(268,300)(269,296)(270,303)(271,302)(272,293)(274,282)(275,283)(277,288)(278,279)(280,285)(286,287)(401,481)(402,490)(403,491)(404,484)(405,496)(406,487)(407,486)(408,493)(409,489)(410,482)(411,483)(412,492)(413,488)(414,495)(415,494)(416,485)(417,545)(418,554)(419,555)(420,548)(421,560)(422,551)(423,550)(424,557)(425,553)(426,546)(427,547)(428,556)(429,552)(430,559)(431,558)(432,549)(433,529)(434,538)(435,539)(436,532)(437,544)(438,535)(439,534)(440,541)(441,537)(442,530)(443,531)(444,540)(445,536)(446,543)(447,542)(448,533)(449,513)(450,522)(451,523)(452,516)(453,528)(454,519)(455,518)(456,525)(457,521)(458,514)(459,515)(460,524)(461,520)(462,527)(463,526)(464,517)(465,497)(466,506)(467,507)(468,500)(469,512)(470,503)(471,502)(472,509)(473,505)(474,498)(475,499)(476,508)(477,504)(478,511)(479,510)(480,501)(561,785)(562,794)(563,795)(564,788)(565,800)(566,791)(567,790)(568,797)(569,793)(570,786)(571,787)(572,796)(573,792)(574,799)(575,798)(576,789)(577,769)(578,778)(579,779)(580,772)(581,784)(582,775)(583,774)(584,781)(585,777)(586,770)(587,771)(588,780)(589,776)(590,783)(591,782)(592,773)(593,753)(594,762)(595,763)(596,756)(597,768)(598,759)(599,758)(600,765)(601,761)(602,754)(603,755)(604,764)(605,760)(606,767)(607,766)(608,757)(609,737)(610,746)(611,747)(612,740)(613,752)(614,743)(615,742)(616,749)(617,745)(618,738)(619,739)(620,748)(621,744)(622,751)(623,750)(624,741)(625,721)(626,730)(627,731)(628,724)(629,736)(630,727)(631,726)(632,733)(633,729)(634,722)(635,723)(636,732)(637,728)(638,735)(639,734)(640,725)(641,705)(642,714)(643,715)(644,708)(645,720)(646,711)(647,710)(648,717)(649,713)(650,706)(651,707)(652,716)(653,712)(654,719)(655,718)(656,709)(657,689)(658,698)(659,699)(660,692)(661,704)(662,695)(663,694)(664,701)(665,697)(666,690)(667,691)(668,700)(669,696)(670,703)(671,702)(672,693)(674,682)(675,683)(677,688)(678,679)(680,685)(686,687);;
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*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*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 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(800)!(  1,409)(  2,410)(  3,411)(  4,412)(  5,413)(  6,414)(  7,415)(  8,416)(  9,401)( 10,402)( 11,403)( 12,404)( 13,405)( 14,406)( 15,407)( 16,408)( 17,425)( 18,426)( 19,427)( 20,428)( 21,429)( 22,430)( 23,431)( 24,432)( 25,417)( 26,418)( 27,419)( 28,420)( 29,421)( 30,422)( 31,423)( 32,424)( 33,441)( 34,442)( 35,443)( 36,444)( 37,445)( 38,446)( 39,447)( 40,448)( 41,433)( 42,434)( 43,435)( 44,436)( 45,437)( 46,438)( 47,439)( 48,440)( 49,457)( 50,458)( 51,459)( 52,460)( 53,461)( 54,462)( 55,463)( 56,464)( 57,449)( 58,450)( 59,451)( 60,452)( 61,453)( 62,454)( 63,455)( 64,456)( 65,473)( 66,474)( 67,475)( 68,476)( 69,477)( 70,478)( 71,479)( 72,480)( 73,465)( 74,466)( 75,467)( 76,468)( 77,469)( 78,470)( 79,471)( 80,472)( 81,489)( 82,490)( 83,491)( 84,492)( 85,493)( 86,494)( 87,495)( 88,496)( 89,481)( 90,482)( 91,483)( 92,484)( 93,485)( 94,486)( 95,487)( 96,488)( 97,505)( 98,506)( 99,507)(100,508)(101,509)(102,510)(103,511)(104,512)(105,497)(106,498)(107,499)(108,500)(109,501)(110,502)(111,503)(112,504)(113,521)(114,522)(115,523)(116,524)(117,525)(118,526)(119,527)(120,528)(121,513)(122,514)(123,515)(124,516)(125,517)(126,518)(127,519)(128,520)(129,537)(130,538)(131,539)(132,540)(133,541)(134,542)(135,543)(136,544)(137,529)(138,530)(139,531)(140,532)(141,533)(142,534)(143,535)(144,536)(145,553)(146,554)(147,555)(148,556)(149,557)(150,558)(151,559)(152,560)(153,545)(154,546)(155,547)(156,548)(157,549)(158,550)(159,551)(160,552)(161,569)(162,570)(163,571)(164,572)(165,573)(166,574)(167,575)(168,576)(169,561)(170,562)(171,563)(172,564)(173,565)(174,566)(175,567)(176,568)(177,585)(178,586)(179,587)(180,588)(181,589)(182,590)(183,591)(184,592)(185,577)(186,578)(187,579)(188,580)(189,581)(190,582)(191,583)(192,584)(193,601)(194,602)(195,603)(196,604)(197,605)(198,606)(199,607)(200,608)(201,593)(202,594)(203,595)(204,596)(205,597)(206,598)(207,599)(208,600)(209,617)(210,618)(211,619)(212,620)(213,621)(214,622)(215,623)(216,624)(217,609)(218,610)(219,611)(220,612)(221,613)(222,614)(223,615)(224,616)(225,633)(226,634)(227,635)(228,636)(229,637)(230,638)(231,639)(232,640)(233,625)(234,626)(235,627)(236,628)(237,629)(238,630)(239,631)(240,632)(241,649)(242,650)(243,651)(244,652)(245,653)(246,654)(247,655)(248,656)(249,641)(250,642)(251,643)(252,644)(253,645)(254,646)(255,647)(256,648)(257,665)(258,666)(259,667)(260,668)(261,669)(262,670)(263,671)(264,672)(265,657)(266,658)(267,659)(268,660)(269,661)(270,662)(271,663)(272,664)(273,681)(274,682)(275,683)(276,684)(277,685)(278,686)(279,687)(280,688)(281,673)(282,674)(283,675)(284,676)(285,677)(286,678)(287,679)(288,680)(289,697)(290,698)(291,699)(292,700)(293,701)(294,702)(295,703)(296,704)(297,689)(298,690)(299,691)(300,692)(301,693)(302,694)(303,695)(304,696)(305,713)(306,714)(307,715)(308,716)(309,717)(310,718)(311,719)(312,720)(313,705)(314,706)(315,707)(316,708)(317,709)(318,710)(319,711)(320,712)(321,729)(322,730)(323,731)(324,732)(325,733)(326,734)(327,735)(328,736)(329,721)(330,722)(331,723)(332,724)(333,725)(334,726)(335,727)(336,728)(337,745)(338,746)(339,747)(340,748)(341,749)(342,750)(343,751)(344,752)(345,737)(346,738)(347,739)(348,740)(349,741)(350,742)(351,743)(352,744)(353,761)(354,762)(355,763)(356,764)(357,765)(358,766)(359,767)(360,768)(361,753)(362,754)(363,755)(364,756)(365,757)(366,758)(367,759)(368,760)(369,777)(370,778)(371,779)(372,780)(373,781)(374,782)(375,783)(376,784)(377,769)(378,770)(379,771)(380,772)(381,773)(382,774)(383,775)(384,776)(385,793)(386,794)(387,795)(388,796)(389,797)(390,798)(391,799)(392,800)(393,785)(394,786)(395,787)(396,788)(397,789)(398,790)(399,791)(400,792);
s1 := Sym(800)!(  3,  4)(  5,  6)(  9, 16)( 10, 15)( 11, 13)( 12, 14)( 17, 65)( 18, 66)( 19, 68)( 20, 67)( 21, 70)( 22, 69)( 23, 71)( 24, 72)( 25, 80)( 26, 79)( 27, 77)( 28, 78)( 29, 75)( 30, 76)( 31, 74)( 32, 73)( 33, 49)( 34, 50)( 35, 52)( 36, 51)( 37, 54)( 38, 53)( 39, 55)( 40, 56)( 41, 64)( 42, 63)( 43, 61)( 44, 62)( 45, 59)( 46, 60)( 47, 58)( 48, 57)( 81,385)( 82,386)( 83,388)( 84,387)( 85,390)( 86,389)( 87,391)( 88,392)( 89,400)( 90,399)( 91,397)( 92,398)( 93,395)( 94,396)( 95,394)( 96,393)( 97,369)( 98,370)( 99,372)(100,371)(101,374)(102,373)(103,375)(104,376)(105,384)(106,383)(107,381)(108,382)(109,379)(110,380)(111,378)(112,377)(113,353)(114,354)(115,356)(116,355)(117,358)(118,357)(119,359)(120,360)(121,368)(122,367)(123,365)(124,366)(125,363)(126,364)(127,362)(128,361)(129,337)(130,338)(131,340)(132,339)(133,342)(134,341)(135,343)(136,344)(137,352)(138,351)(139,349)(140,350)(141,347)(142,348)(143,346)(144,345)(145,321)(146,322)(147,324)(148,323)(149,326)(150,325)(151,327)(152,328)(153,336)(154,335)(155,333)(156,334)(157,331)(158,332)(159,330)(160,329)(161,305)(162,306)(163,308)(164,307)(165,310)(166,309)(167,311)(168,312)(169,320)(170,319)(171,317)(172,318)(173,315)(174,316)(175,314)(176,313)(177,289)(178,290)(179,292)(180,291)(181,294)(182,293)(183,295)(184,296)(185,304)(186,303)(187,301)(188,302)(189,299)(190,300)(191,298)(192,297)(193,273)(194,274)(195,276)(196,275)(197,278)(198,277)(199,279)(200,280)(201,288)(202,287)(203,285)(204,286)(205,283)(206,284)(207,282)(208,281)(209,257)(210,258)(211,260)(212,259)(213,262)(214,261)(215,263)(216,264)(217,272)(218,271)(219,269)(220,270)(221,267)(222,268)(223,266)(224,265)(225,241)(226,242)(227,244)(228,243)(229,246)(230,245)(231,247)(232,248)(233,256)(234,255)(235,253)(236,254)(237,251)(238,252)(239,250)(240,249)(403,404)(405,406)(409,416)(410,415)(411,413)(412,414)(417,465)(418,466)(419,468)(420,467)(421,470)(422,469)(423,471)(424,472)(425,480)(426,479)(427,477)(428,478)(429,475)(430,476)(431,474)(432,473)(433,449)(434,450)(435,452)(436,451)(437,454)(438,453)(439,455)(440,456)(441,464)(442,463)(443,461)(444,462)(445,459)(446,460)(447,458)(448,457)(481,785)(482,786)(483,788)(484,787)(485,790)(486,789)(487,791)(488,792)(489,800)(490,799)(491,797)(492,798)(493,795)(494,796)(495,794)(496,793)(497,769)(498,770)(499,772)(500,771)(501,774)(502,773)(503,775)(504,776)(505,784)(506,783)(507,781)(508,782)(509,779)(510,780)(511,778)(512,777)(513,753)(514,754)(515,756)(516,755)(517,758)(518,757)(519,759)(520,760)(521,768)(522,767)(523,765)(524,766)(525,763)(526,764)(527,762)(528,761)(529,737)(530,738)(531,740)(532,739)(533,742)(534,741)(535,743)(536,744)(537,752)(538,751)(539,749)(540,750)(541,747)(542,748)(543,746)(544,745)(545,721)(546,722)(547,724)(548,723)(549,726)(550,725)(551,727)(552,728)(553,736)(554,735)(555,733)(556,734)(557,731)(558,732)(559,730)(560,729)(561,705)(562,706)(563,708)(564,707)(565,710)(566,709)(567,711)(568,712)(569,720)(570,719)(571,717)(572,718)(573,715)(574,716)(575,714)(576,713)(577,689)(578,690)(579,692)(580,691)(581,694)(582,693)(583,695)(584,696)(585,704)(586,703)(587,701)(588,702)(589,699)(590,700)(591,698)(592,697)(593,673)(594,674)(595,676)(596,675)(597,678)(598,677)(599,679)(600,680)(601,688)(602,687)(603,685)(604,686)(605,683)(606,684)(607,682)(608,681)(609,657)(610,658)(611,660)(612,659)(613,662)(614,661)(615,663)(616,664)(617,672)(618,671)(619,669)(620,670)(621,667)(622,668)(623,666)(624,665)(625,641)(626,642)(627,644)(628,643)(629,646)(630,645)(631,647)(632,648)(633,656)(634,655)(635,653)(636,654)(637,651)(638,652)(639,650)(640,649);
s2 := Sym(800)!(  1, 81)(  2, 90)(  3, 91)(  4, 84)(  5, 96)(  6, 87)(  7, 86)(  8, 93)(  9, 89)( 10, 82)( 11, 83)( 12, 92)( 13, 88)( 14, 95)( 15, 94)( 16, 85)( 17,145)( 18,154)( 19,155)( 20,148)( 21,160)( 22,151)( 23,150)( 24,157)( 25,153)( 26,146)( 27,147)( 28,156)( 29,152)( 30,159)( 31,158)( 32,149)( 33,129)( 34,138)( 35,139)( 36,132)( 37,144)( 38,135)( 39,134)( 40,141)( 41,137)( 42,130)( 43,131)( 44,140)( 45,136)( 46,143)( 47,142)( 48,133)( 49,113)( 50,122)( 51,123)( 52,116)( 53,128)( 54,119)( 55,118)( 56,125)( 57,121)( 58,114)( 59,115)( 60,124)( 61,120)( 62,127)( 63,126)( 64,117)( 65, 97)( 66,106)( 67,107)( 68,100)( 69,112)( 70,103)( 71,102)( 72,109)( 73,105)( 74, 98)( 75, 99)( 76,108)( 77,104)( 78,111)( 79,110)( 80,101)(161,385)(162,394)(163,395)(164,388)(165,400)(166,391)(167,390)(168,397)(169,393)(170,386)(171,387)(172,396)(173,392)(174,399)(175,398)(176,389)(177,369)(178,378)(179,379)(180,372)(181,384)(182,375)(183,374)(184,381)(185,377)(186,370)(187,371)(188,380)(189,376)(190,383)(191,382)(192,373)(193,353)(194,362)(195,363)(196,356)(197,368)(198,359)(199,358)(200,365)(201,361)(202,354)(203,355)(204,364)(205,360)(206,367)(207,366)(208,357)(209,337)(210,346)(211,347)(212,340)(213,352)(214,343)(215,342)(216,349)(217,345)(218,338)(219,339)(220,348)(221,344)(222,351)(223,350)(224,341)(225,321)(226,330)(227,331)(228,324)(229,336)(230,327)(231,326)(232,333)(233,329)(234,322)(235,323)(236,332)(237,328)(238,335)(239,334)(240,325)(241,305)(242,314)(243,315)(244,308)(245,320)(246,311)(247,310)(248,317)(249,313)(250,306)(251,307)(252,316)(253,312)(254,319)(255,318)(256,309)(257,289)(258,298)(259,299)(260,292)(261,304)(262,295)(263,294)(264,301)(265,297)(266,290)(267,291)(268,300)(269,296)(270,303)(271,302)(272,293)(274,282)(275,283)(277,288)(278,279)(280,285)(286,287)(401,481)(402,490)(403,491)(404,484)(405,496)(406,487)(407,486)(408,493)(409,489)(410,482)(411,483)(412,492)(413,488)(414,495)(415,494)(416,485)(417,545)(418,554)(419,555)(420,548)(421,560)(422,551)(423,550)(424,557)(425,553)(426,546)(427,547)(428,556)(429,552)(430,559)(431,558)(432,549)(433,529)(434,538)(435,539)(436,532)(437,544)(438,535)(439,534)(440,541)(441,537)(442,530)(443,531)(444,540)(445,536)(446,543)(447,542)(448,533)(449,513)(450,522)(451,523)(452,516)(453,528)(454,519)(455,518)(456,525)(457,521)(458,514)(459,515)(460,524)(461,520)(462,527)(463,526)(464,517)(465,497)(466,506)(467,507)(468,500)(469,512)(470,503)(471,502)(472,509)(473,505)(474,498)(475,499)(476,508)(477,504)(478,511)(479,510)(480,501)(561,785)(562,794)(563,795)(564,788)(565,800)(566,791)(567,790)(568,797)(569,793)(570,786)(571,787)(572,796)(573,792)(574,799)(575,798)(576,789)(577,769)(578,778)(579,779)(580,772)(581,784)(582,775)(583,774)(584,781)(585,777)(586,770)(587,771)(588,780)(589,776)(590,783)(591,782)(592,773)(593,753)(594,762)(595,763)(596,756)(597,768)(598,759)(599,758)(600,765)(601,761)(602,754)(603,755)(604,764)(605,760)(606,767)(607,766)(608,757)(609,737)(610,746)(611,747)(612,740)(613,752)(614,743)(615,742)(616,749)(617,745)(618,738)(619,739)(620,748)(621,744)(622,751)(623,750)(624,741)(625,721)(626,730)(627,731)(628,724)(629,736)(630,727)(631,726)(632,733)(633,729)(634,722)(635,723)(636,732)(637,728)(638,735)(639,734)(640,725)(641,705)(642,714)(643,715)(644,708)(645,720)(646,711)(647,710)(648,717)(649,713)(650,706)(651,707)(652,716)(653,712)(654,719)(655,718)(656,709)(657,689)(658,698)(659,699)(660,692)(661,704)(662,695)(663,694)(664,701)(665,697)(666,690)(667,691)(668,700)(669,696)(670,703)(671,702)(672,693)(674,682)(675,683)(677,688)(678,679)(680,685)(686,687);
poly := sub<Sym(800)|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*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*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 >;
References : None.
to this polytope

Twisty Puzzle