Polytope of Type {6,123}

Play with this polytope as a twisty puzzle

s0 := ( 42, 83)( 43, 84)( 44, 85)( 45, 86)( 46, 87)( 47, 88)( 48, 89)( 49, 90)( 50, 91)( 51, 92)( 52, 93)( 53, 94)( 54, 95)( 55, 96)( 56, 97)( 57, 98)( 58, 99)( 59,100)( 60,101)( 61,102)( 62,103)( 63,104)( 64,105)( 65,106)( 66,107)( 67,108)( 68,109)( 69,110)( 70,111)( 71,112)( 72,113)( 73,114)( 74,115)( 75,116)( 76,117)( 77,118)( 78,119)( 79,120)( 80,121)( 81,122)( 82,123)(165,206)(166,207)(167,208)(168,209)(169,210)(170,211)(171,212)(172,213)(173,214)(174,215)(175,216)(176,217)(177,218)(178,219)(179,220)(180,221)(181,222)(182,223)(183,224)(184,225)(185,226)(186,227)(187,228)(188,229)(189,230)(190,231)(191,232)(192,233)(193,234)(194,235)(195,236)(196,237)(197,238)(198,239)(199,240)(200,241)(201,242)(202,243)(203,244)(204,245)(205,246)(288,329)(289,330)(290,331)(291,332)(292,333)(293,334)(294,335)(295,336)(296,337)(297,338)(298,339)(299,340)(300,341)(301,342)(302,343)(303,344)(304,345)(305,346)(306,347)(307,348)(308,349)(309,350)(310,351)(311,352)(312,353)(313,354)(314,355)(315,356)(316,357)(317,358)(318,359)(319,360)(320,361)(321,362)(322,363)(323,364)(324,365)(325,366)(326,367)(327,368)(328,369);;
s1 := (  1, 42)(  2, 82)(  3, 81)(  4, 80)(  5, 79)(  6, 78)(  7, 77)(  8, 76)(  9, 75)( 10, 74)( 11, 73)( 12, 72)( 13, 71)( 14, 70)( 15, 69)( 16, 68)( 17, 67)( 18, 66)( 19, 65)( 20, 64)( 21, 63)( 22, 62)( 23, 61)( 24, 60)( 25, 59)( 26, 58)( 27, 57)( 28, 56)( 29, 55)( 30, 54)( 31, 53)( 32, 52)( 33, 51)( 34, 50)( 35, 49)( 36, 48)( 37, 47)( 38, 46)( 39, 45)( 40, 44)( 41, 43)( 84,123)( 85,122)( 86,121)( 87,120)( 88,119)( 89,118)( 90,117)( 91,116)( 92,115)( 93,114)( 94,113)( 95,112)( 96,111)( 97,110)( 98,109)( 99,108)(100,107)(101,106)(102,105)(103,104)(124,288)(125,328)(126,327)(127,326)(128,325)(129,324)(130,323)(131,322)(132,321)(133,320)(134,319)(135,318)(136,317)(137,316)(138,315)(139,314)(140,313)(141,312)(142,311)(143,310)(144,309)(145,308)(146,307)(147,306)(148,305)(149,304)(150,303)(151,302)(152,301)(153,300)(154,299)(155,298)(156,297)(157,296)(158,295)(159,294)(160,293)(161,292)(162,291)(163,290)(164,289)(165,247)(166,287)(167,286)(168,285)(169,284)(170,283)(171,282)(172,281)(173,280)(174,279)(175,278)(176,277)(177,276)(178,275)(179,274)(180,273)(181,272)(182,271)(183,270)(184,269)(185,268)(186,267)(187,266)(188,265)(189,264)(190,263)(191,262)(192,261)(193,260)(194,259)(195,258)(196,257)(197,256)(198,255)(199,254)(200,253)(201,252)(202,251)(203,250)(204,249)(205,248)(206,329)(207,369)(208,368)(209,367)(210,366)(211,365)(212,364)(213,363)(214,362)(215,361)(216,360)(217,359)(218,358)(219,357)(220,356)(221,355)(222,354)(223,353)(224,352)(225,351)(226,350)(227,349)(228,348)(229,347)(230,346)(231,345)(232,344)(233,343)(234,342)(235,341)(236,340)(237,339)(238,338)(239,337)(240,336)(241,335)(242,334)(243,333)(244,332)(245,331)(246,330);;
s2 := (  1,125)(  2,124)(  3,164)(  4,163)(  5,162)(  6,161)(  7,160)(  8,159)(  9,158)( 10,157)( 11,156)( 12,155)( 13,154)( 14,153)( 15,152)( 16,151)( 17,150)( 18,149)( 19,148)( 20,147)( 21,146)( 22,145)( 23,144)( 24,143)( 25,142)( 26,141)( 27,140)( 28,139)( 29,138)( 30,137)( 31,136)( 32,135)( 33,134)( 34,133)( 35,132)( 36,131)( 37,130)( 38,129)( 39,128)( 40,127)( 41,126)( 42,207)( 43,206)( 44,246)( 45,245)( 46,244)( 47,243)( 48,242)( 49,241)( 50,240)( 51,239)( 52,238)( 53,237)( 54,236)( 55,235)( 56,234)( 57,233)( 58,232)( 59,231)( 60,230)( 61,229)( 62,228)( 63,227)( 64,226)( 65,225)( 66,224)( 67,223)( 68,222)( 69,221)( 70,220)( 71,219)( 72,218)( 73,217)( 74,216)( 75,215)( 76,214)( 77,213)( 78,212)( 79,211)( 80,210)( 81,209)( 82,208)( 83,166)( 84,165)( 85,205)( 86,204)( 87,203)( 88,202)( 89,201)( 90,200)( 91,199)( 92,198)( 93,197)( 94,196)( 95,195)( 96,194)( 97,193)( 98,192)( 99,191)(100,190)(101,189)(102,188)(103,187)(104,186)(105,185)(106,184)(107,183)(108,182)(109,181)(110,180)(111,179)(112,178)(113,177)(114,176)(115,175)(116,174)(117,173)(118,172)(119,171)(120,170)(121,169)(122,168)(123,167)(247,248)(249,287)(250,286)(251,285)(252,284)(253,283)(254,282)(255,281)(256,280)(257,279)(258,278)(259,277)(260,276)(261,275)(262,274)(263,273)(264,272)(265,271)(266,270)(267,269)(288,330)(289,329)(290,369)(291,368)(292,367)(293,366)(294,365)(295,364)(296,363)(297,362)(298,361)(299,360)(300,359)(301,358)(302,357)(303,356)(304,355)(305,354)(306,353)(307,352)(308,351)(309,350)(310,349)(311,348)(312,347)(313,346)(314,345)(315,344)(316,343)(317,342)(318,341)(319,340)(320,339)(321,338)(322,337)(323,336)(324,335)(325,334)(326,333)(327,332)(328,331);;
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*s0*s1*s0*s1*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(369)!( 42, 83)( 43, 84)( 44, 85)( 45, 86)( 46, 87)( 47, 88)( 48, 89)( 49, 90)( 50, 91)( 51, 92)( 52, 93)( 53, 94)( 54, 95)( 55, 96)( 56, 97)( 57, 98)( 58, 99)( 59,100)( 60,101)( 61,102)( 62,103)( 63,104)( 64,105)( 65,106)( 66,107)( 67,108)( 68,109)( 69,110)( 70,111)( 71,112)( 72,113)( 73,114)( 74,115)( 75,116)( 76,117)( 77,118)( 78,119)( 79,120)( 80,121)( 81,122)( 82,123)(165,206)(166,207)(167,208)(168,209)(169,210)(170,211)(171,212)(172,213)(173,214)(174,215)(175,216)(176,217)(177,218)(178,219)(179,220)(180,221)(181,222)(182,223)(183,224)(184,225)(185,226)(186,227)(187,228)(188,229)(189,230)(190,231)(191,232)(192,233)(193,234)(194,235)(195,236)(196,237)(197,238)(198,239)(199,240)(200,241)(201,242)(202,243)(203,244)(204,245)(205,246)(288,329)(289,330)(290,331)(291,332)(292,333)(293,334)(294,335)(295,336)(296,337)(297,338)(298,339)(299,340)(300,341)(301,342)(302,343)(303,344)(304,345)(305,346)(306,347)(307,348)(308,349)(309,350)(310,351)(311,352)(312,353)(313,354)(314,355)(315,356)(316,357)(317,358)(318,359)(319,360)(320,361)(321,362)(322,363)(323,364)(324,365)(325,366)(326,367)(327,368)(328,369);
s1 := Sym(369)!(  1, 42)(  2, 82)(  3, 81)(  4, 80)(  5, 79)(  6, 78)(  7, 77)(  8, 76)(  9, 75)( 10, 74)( 11, 73)( 12, 72)( 13, 71)( 14, 70)( 15, 69)( 16, 68)( 17, 67)( 18, 66)( 19, 65)( 20, 64)( 21, 63)( 22, 62)( 23, 61)( 24, 60)( 25, 59)( 26, 58)( 27, 57)( 28, 56)( 29, 55)( 30, 54)( 31, 53)( 32, 52)( 33, 51)( 34, 50)( 35, 49)( 36, 48)( 37, 47)( 38, 46)( 39, 45)( 40, 44)( 41, 43)( 84,123)( 85,122)( 86,121)( 87,120)( 88,119)( 89,118)( 90,117)( 91,116)( 92,115)( 93,114)( 94,113)( 95,112)( 96,111)( 97,110)( 98,109)( 99,108)(100,107)(101,106)(102,105)(103,104)(124,288)(125,328)(126,327)(127,326)(128,325)(129,324)(130,323)(131,322)(132,321)(133,320)(134,319)(135,318)(136,317)(137,316)(138,315)(139,314)(140,313)(141,312)(142,311)(143,310)(144,309)(145,308)(146,307)(147,306)(148,305)(149,304)(150,303)(151,302)(152,301)(153,300)(154,299)(155,298)(156,297)(157,296)(158,295)(159,294)(160,293)(161,292)(162,291)(163,290)(164,289)(165,247)(166,287)(167,286)(168,285)(169,284)(170,283)(171,282)(172,281)(173,280)(174,279)(175,278)(176,277)(177,276)(178,275)(179,274)(180,273)(181,272)(182,271)(183,270)(184,269)(185,268)(186,267)(187,266)(188,265)(189,264)(190,263)(191,262)(192,261)(193,260)(194,259)(195,258)(196,257)(197,256)(198,255)(199,254)(200,253)(201,252)(202,251)(203,250)(204,249)(205,248)(206,329)(207,369)(208,368)(209,367)(210,366)(211,365)(212,364)(213,363)(214,362)(215,361)(216,360)(217,359)(218,358)(219,357)(220,356)(221,355)(222,354)(223,353)(224,352)(225,351)(226,350)(227,349)(228,348)(229,347)(230,346)(231,345)(232,344)(233,343)(234,342)(235,341)(236,340)(237,339)(238,338)(239,337)(240,336)(241,335)(242,334)(243,333)(244,332)(245,331)(246,330);
s2 := Sym(369)!(  1,125)(  2,124)(  3,164)(  4,163)(  5,162)(  6,161)(  7,160)(  8,159)(  9,158)( 10,157)( 11,156)( 12,155)( 13,154)( 14,153)( 15,152)( 16,151)( 17,150)( 18,149)( 19,148)( 20,147)( 21,146)( 22,145)( 23,144)( 24,143)( 25,142)( 26,141)( 27,140)( 28,139)( 29,138)( 30,137)( 31,136)( 32,135)( 33,134)( 34,133)( 35,132)( 36,131)( 37,130)( 38,129)( 39,128)( 40,127)( 41,126)( 42,207)( 43,206)( 44,246)( 45,245)( 46,244)( 47,243)( 48,242)( 49,241)( 50,240)( 51,239)( 52,238)( 53,237)( 54,236)( 55,235)( 56,234)( 57,233)( 58,232)( 59,231)( 60,230)( 61,229)( 62,228)( 63,227)( 64,226)( 65,225)( 66,224)( 67,223)( 68,222)( 69,221)( 70,220)( 71,219)( 72,218)( 73,217)( 74,216)( 75,215)( 76,214)( 77,213)( 78,212)( 79,211)( 80,210)( 81,209)( 82,208)( 83,166)( 84,165)( 85,205)( 86,204)( 87,203)( 88,202)( 89,201)( 90,200)( 91,199)( 92,198)( 93,197)( 94,196)( 95,195)( 96,194)( 97,193)( 98,192)( 99,191)(100,190)(101,189)(102,188)(103,187)(104,186)(105,185)(106,184)(107,183)(108,182)(109,181)(110,180)(111,179)(112,178)(113,177)(114,176)(115,175)(116,174)(117,173)(118,172)(119,171)(120,170)(121,169)(122,168)(123,167)(247,248)(249,287)(250,286)(251,285)(252,284)(253,283)(254,282)(255,281)(256,280)(257,279)(258,278)(259,277)(260,276)(261,275)(262,274)(263,273)(264,272)(265,271)(266,270)(267,269)(288,330)(289,329)(290,369)(291,368)(292,367)(293,366)(294,365)(295,364)(296,363)(297,362)(298,361)(299,360)(300,359)(301,358)(302,357)(303,356)(304,355)(305,354)(306,353)(307,352)(308,351)(309,350)(310,349)(311,348)(312,347)(313,346)(314,345)(315,344)(316,343)(317,342)(318,341)(319,340)(320,339)(321,338)(322,337)(323,336)(324,335)(325,334)(326,333)(327,332)(328,331);
poly := sub<Sym(369)|s0,s1,s2>;
 

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