Polytope of Type {8,68}

Play with this polytope as a twisty puzzle

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

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*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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 ];;
poly := F / rels;;
 

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

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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 >; 
 

Twisty Puzzle