Polytope of Type {68,10}

Play with this polytope as a twisty puzzle

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

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

Twisty Puzzle