Polytope of Type {2,4,51}

s0 := (1,2);;
s1 := (  3,209)(  4,210)(  5,207)(  6,208)(  7,213)(  8,214)(  9,211)( 10,212)( 11,217)( 12,218)( 13,215)( 14,216)( 15,221)( 16,222)( 17,219)( 18,220)( 19,225)( 20,226)( 21,223)( 22,224)( 23,229)( 24,230)( 25,227)( 26,228)( 27,233)( 28,234)( 29,231)( 30,232)( 31,237)( 32,238)( 33,235)( 34,236)( 35,241)( 36,242)( 37,239)( 38,240)( 39,245)( 40,246)( 41,243)( 42,244)( 43,249)( 44,250)( 45,247)( 46,248)( 47,253)( 48,254)( 49,251)( 50,252)( 51,257)( 52,258)( 53,255)( 54,256)( 55,261)( 56,262)( 57,259)( 58,260)( 59,265)( 60,266)( 61,263)( 62,264)( 63,269)( 64,270)( 65,267)( 66,268)( 67,273)( 68,274)( 69,271)( 70,272)( 71,277)( 72,278)( 73,275)( 74,276)( 75,281)( 76,282)( 77,279)( 78,280)( 79,285)( 80,286)( 81,283)( 82,284)( 83,289)( 84,290)( 85,287)( 86,288)( 87,293)( 88,294)( 89,291)( 90,292)( 91,297)( 92,298)( 93,295)( 94,296)( 95,301)( 96,302)( 97,299)( 98,300)( 99,305)(100,306)(101,303)(102,304)(103,309)(104,310)(105,307)(106,308)(107,313)(108,314)(109,311)(110,312)(111,317)(112,318)(113,315)(114,316)(115,321)(116,322)(117,319)(118,320)(119,325)(120,326)(121,323)(122,324)(123,329)(124,330)(125,327)(126,328)(127,333)(128,334)(129,331)(130,332)(131,337)(132,338)(133,335)(134,336)(135,341)(136,342)(137,339)(138,340)(139,345)(140,346)(141,343)(142,344)(143,349)(144,350)(145,347)(146,348)(147,353)(148,354)(149,351)(150,352)(151,357)(152,358)(153,355)(154,356)(155,361)(156,362)(157,359)(158,360)(159,365)(160,366)(161,363)(162,364)(163,369)(164,370)(165,367)(166,368)(167,373)(168,374)(169,371)(170,372)(171,377)(172,378)(173,375)(174,376)(175,381)(176,382)(177,379)(178,380)(179,385)(180,386)(181,383)(182,384)(183,389)(184,390)(185,387)(186,388)(187,393)(188,394)(189,391)(190,392)(191,397)(192,398)(193,395)(194,396)(195,401)(196,402)(197,399)(198,400)(199,405)(200,406)(201,403)(202,404)(203,409)(204,410)(205,407)(206,408);;
s2 := (  4,  5)(  7, 67)(  8, 69)(  9, 68)( 10, 70)( 11, 63)( 12, 65)( 13, 64)( 14, 66)( 15, 59)( 16, 61)( 17, 60)( 18, 62)( 19, 55)( 20, 57)( 21, 56)( 22, 58)( 23, 51)( 24, 53)( 25, 52)( 26, 54)( 27, 47)( 28, 49)( 29, 48)( 30, 50)( 31, 43)( 32, 45)( 33, 44)( 34, 46)( 35, 39)( 36, 41)( 37, 40)( 38, 42)( 71,139)( 72,141)( 73,140)( 74,142)( 75,203)( 76,205)( 77,204)( 78,206)( 79,199)( 80,201)( 81,200)( 82,202)( 83,195)( 84,197)( 85,196)( 86,198)( 87,191)( 88,193)( 89,192)( 90,194)( 91,187)( 92,189)( 93,188)( 94,190)( 95,183)( 96,185)( 97,184)( 98,186)( 99,179)(100,181)(101,180)(102,182)(103,175)(104,177)(105,176)(106,178)(107,171)(108,173)(109,172)(110,174)(111,167)(112,169)(113,168)(114,170)(115,163)(116,165)(117,164)(118,166)(119,159)(120,161)(121,160)(122,162)(123,155)(124,157)(125,156)(126,158)(127,151)(128,153)(129,152)(130,154)(131,147)(132,149)(133,148)(134,150)(135,143)(136,145)(137,144)(138,146)(208,209)(211,271)(212,273)(213,272)(214,274)(215,267)(216,269)(217,268)(218,270)(219,263)(220,265)(221,264)(222,266)(223,259)(224,261)(225,260)(226,262)(227,255)(228,257)(229,256)(230,258)(231,251)(232,253)(233,252)(234,254)(235,247)(236,249)(237,248)(238,250)(239,243)(240,245)(241,244)(242,246)(275,343)(276,345)(277,344)(278,346)(279,407)(280,409)(281,408)(282,410)(283,403)(284,405)(285,404)(286,406)(287,399)(288,401)(289,400)(290,402)(291,395)(292,397)(293,396)(294,398)(295,391)(296,393)(297,392)(298,394)(299,387)(300,389)(301,388)(302,390)(303,383)(304,385)(305,384)(306,386)(307,379)(308,381)(309,380)(310,382)(311,375)(312,377)(313,376)(314,378)(315,371)(316,373)(317,372)(318,374)(319,367)(320,369)(321,368)(322,370)(323,363)(324,365)(325,364)(326,366)(327,359)(328,361)(329,360)(330,362)(331,355)(332,357)(333,356)(334,358)(335,351)(336,353)(337,352)(338,354)(339,347)(340,349)(341,348)(342,350);;
s3 := (  3,143)(  4,146)(  5,145)(  6,144)(  7,139)(  8,142)(  9,141)( 10,140)( 11,203)( 12,206)( 13,205)( 14,204)( 15,199)( 16,202)( 17,201)( 18,200)( 19,195)( 20,198)( 21,197)( 22,196)( 23,191)( 24,194)( 25,193)( 26,192)( 27,187)( 28,190)( 29,189)( 30,188)( 31,183)( 32,186)( 33,185)( 34,184)( 35,179)( 36,182)( 37,181)( 38,180)( 39,175)( 40,178)( 41,177)( 42,176)( 43,171)( 44,174)( 45,173)( 46,172)( 47,167)( 48,170)( 49,169)( 50,168)( 51,163)( 52,166)( 53,165)( 54,164)( 55,159)( 56,162)( 57,161)( 58,160)( 59,155)( 60,158)( 61,157)( 62,156)( 63,151)( 64,154)( 65,153)( 66,152)( 67,147)( 68,150)( 69,149)( 70,148)( 71, 75)( 72, 78)( 73, 77)( 74, 76)( 79,135)( 80,138)( 81,137)( 82,136)( 83,131)( 84,134)( 85,133)( 86,132)( 87,127)( 88,130)( 89,129)( 90,128)( 91,123)( 92,126)( 93,125)( 94,124)( 95,119)( 96,122)( 97,121)( 98,120)( 99,115)(100,118)(101,117)(102,116)(103,111)(104,114)(105,113)(106,112)(108,110)(207,347)(208,350)(209,349)(210,348)(211,343)(212,346)(213,345)(214,344)(215,407)(216,410)(217,409)(218,408)(219,403)(220,406)(221,405)(222,404)(223,399)(224,402)(225,401)(226,400)(227,395)(228,398)(229,397)(230,396)(231,391)(232,394)(233,393)(234,392)(235,387)(236,390)(237,389)(238,388)(239,383)(240,386)(241,385)(242,384)(243,379)(244,382)(245,381)(246,380)(247,375)(248,378)(249,377)(250,376)(251,371)(252,374)(253,373)(254,372)(255,367)(256,370)(257,369)(258,368)(259,363)(260,366)(261,365)(262,364)(263,359)(264,362)(265,361)(266,360)(267,355)(268,358)(269,357)(270,356)(271,351)(272,354)(273,353)(274,352)(275,279)(276,282)(277,281)(278,280)(283,339)(284,342)(285,341)(286,340)(287,335)(288,338)(289,337)(290,336)(291,331)(292,334)(293,333)(294,332)(295,327)(296,330)(297,329)(298,328)(299,323)(300,326)(301,325)(302,324)(303,319)(304,322)(305,321)(306,320)(307,315)(308,318)(309,317)(310,316)(312,314);;
poly := Group([s0,s1,s2,s3]);;

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

s0 := Sym(410)!(1,2);
s1 := Sym(410)!(  3,209)(  4,210)(  5,207)(  6,208)(  7,213)(  8,214)(  9,211)( 10,212)( 11,217)( 12,218)( 13,215)( 14,216)( 15,221)( 16,222)( 17,219)( 18,220)( 19,225)( 20,226)( 21,223)( 22,224)( 23,229)( 24,230)( 25,227)( 26,228)( 27,233)( 28,234)( 29,231)( 30,232)( 31,237)( 32,238)( 33,235)( 34,236)( 35,241)( 36,242)( 37,239)( 38,240)( 39,245)( 40,246)( 41,243)( 42,244)( 43,249)( 44,250)( 45,247)( 46,248)( 47,253)( 48,254)( 49,251)( 50,252)( 51,257)( 52,258)( 53,255)( 54,256)( 55,261)( 56,262)( 57,259)( 58,260)( 59,265)( 60,266)( 61,263)( 62,264)( 63,269)( 64,270)( 65,267)( 66,268)( 67,273)( 68,274)( 69,271)( 70,272)( 71,277)( 72,278)( 73,275)( 74,276)( 75,281)( 76,282)( 77,279)( 78,280)( 79,285)( 80,286)( 81,283)( 82,284)( 83,289)( 84,290)( 85,287)( 86,288)( 87,293)( 88,294)( 89,291)( 90,292)( 91,297)( 92,298)( 93,295)( 94,296)( 95,301)( 96,302)( 97,299)( 98,300)( 99,305)(100,306)(101,303)(102,304)(103,309)(104,310)(105,307)(106,308)(107,313)(108,314)(109,311)(110,312)(111,317)(112,318)(113,315)(114,316)(115,321)(116,322)(117,319)(118,320)(119,325)(120,326)(121,323)(122,324)(123,329)(124,330)(125,327)(126,328)(127,333)(128,334)(129,331)(130,332)(131,337)(132,338)(133,335)(134,336)(135,341)(136,342)(137,339)(138,340)(139,345)(140,346)(141,343)(142,344)(143,349)(144,350)(145,347)(146,348)(147,353)(148,354)(149,351)(150,352)(151,357)(152,358)(153,355)(154,356)(155,361)(156,362)(157,359)(158,360)(159,365)(160,366)(161,363)(162,364)(163,369)(164,370)(165,367)(166,368)(167,373)(168,374)(169,371)(170,372)(171,377)(172,378)(173,375)(174,376)(175,381)(176,382)(177,379)(178,380)(179,385)(180,386)(181,383)(182,384)(183,389)(184,390)(185,387)(186,388)(187,393)(188,394)(189,391)(190,392)(191,397)(192,398)(193,395)(194,396)(195,401)(196,402)(197,399)(198,400)(199,405)(200,406)(201,403)(202,404)(203,409)(204,410)(205,407)(206,408);
s2 := Sym(410)!(  4,  5)(  7, 67)(  8, 69)(  9, 68)( 10, 70)( 11, 63)( 12, 65)( 13, 64)( 14, 66)( 15, 59)( 16, 61)( 17, 60)( 18, 62)( 19, 55)( 20, 57)( 21, 56)( 22, 58)( 23, 51)( 24, 53)( 25, 52)( 26, 54)( 27, 47)( 28, 49)( 29, 48)( 30, 50)( 31, 43)( 32, 45)( 33, 44)( 34, 46)( 35, 39)( 36, 41)( 37, 40)( 38, 42)( 71,139)( 72,141)( 73,140)( 74,142)( 75,203)( 76,205)( 77,204)( 78,206)( 79,199)( 80,201)( 81,200)( 82,202)( 83,195)( 84,197)( 85,196)( 86,198)( 87,191)( 88,193)( 89,192)( 90,194)( 91,187)( 92,189)( 93,188)( 94,190)( 95,183)( 96,185)( 97,184)( 98,186)( 99,179)(100,181)(101,180)(102,182)(103,175)(104,177)(105,176)(106,178)(107,171)(108,173)(109,172)(110,174)(111,167)(112,169)(113,168)(114,170)(115,163)(116,165)(117,164)(118,166)(119,159)(120,161)(121,160)(122,162)(123,155)(124,157)(125,156)(126,158)(127,151)(128,153)(129,152)(130,154)(131,147)(132,149)(133,148)(134,150)(135,143)(136,145)(137,144)(138,146)(208,209)(211,271)(212,273)(213,272)(214,274)(215,267)(216,269)(217,268)(218,270)(219,263)(220,265)(221,264)(222,266)(223,259)(224,261)(225,260)(226,262)(227,255)(228,257)(229,256)(230,258)(231,251)(232,253)(233,252)(234,254)(235,247)(236,249)(237,248)(238,250)(239,243)(240,245)(241,244)(242,246)(275,343)(276,345)(277,344)(278,346)(279,407)(280,409)(281,408)(282,410)(283,403)(284,405)(285,404)(286,406)(287,399)(288,401)(289,400)(290,402)(291,395)(292,397)(293,396)(294,398)(295,391)(296,393)(297,392)(298,394)(299,387)(300,389)(301,388)(302,390)(303,383)(304,385)(305,384)(306,386)(307,379)(308,381)(309,380)(310,382)(311,375)(312,377)(313,376)(314,378)(315,371)(316,373)(317,372)(318,374)(319,367)(320,369)(321,368)(322,370)(323,363)(324,365)(325,364)(326,366)(327,359)(328,361)(329,360)(330,362)(331,355)(332,357)(333,356)(334,358)(335,351)(336,353)(337,352)(338,354)(339,347)(340,349)(341,348)(342,350);
s3 := Sym(410)!(  3,143)(  4,146)(  5,145)(  6,144)(  7,139)(  8,142)(  9,141)( 10,140)( 11,203)( 12,206)( 13,205)( 14,204)( 15,199)( 16,202)( 17,201)( 18,200)( 19,195)( 20,198)( 21,197)( 22,196)( 23,191)( 24,194)( 25,193)( 26,192)( 27,187)( 28,190)( 29,189)( 30,188)( 31,183)( 32,186)( 33,185)( 34,184)( 35,179)( 36,182)( 37,181)( 38,180)( 39,175)( 40,178)( 41,177)( 42,176)( 43,171)( 44,174)( 45,173)( 46,172)( 47,167)( 48,170)( 49,169)( 50,168)( 51,163)( 52,166)( 53,165)( 54,164)( 55,159)( 56,162)( 57,161)( 58,160)( 59,155)( 60,158)( 61,157)( 62,156)( 63,151)( 64,154)( 65,153)( 66,152)( 67,147)( 68,150)( 69,149)( 70,148)( 71, 75)( 72, 78)( 73, 77)( 74, 76)( 79,135)( 80,138)( 81,137)( 82,136)( 83,131)( 84,134)( 85,133)( 86,132)( 87,127)( 88,130)( 89,129)( 90,128)( 91,123)( 92,126)( 93,125)( 94,124)( 95,119)( 96,122)( 97,121)( 98,120)( 99,115)(100,118)(101,117)(102,116)(103,111)(104,114)(105,113)(106,112)(108,110)(207,347)(208,350)(209,349)(210,348)(211,343)(212,346)(213,345)(214,344)(215,407)(216,410)(217,409)(218,408)(219,403)(220,406)(221,405)(222,404)(223,399)(224,402)(225,401)(226,400)(227,395)(228,398)(229,397)(230,396)(231,391)(232,394)(233,393)(234,392)(235,387)(236,390)(237,389)(238,388)(239,383)(240,386)(241,385)(242,384)(243,379)(244,382)(245,381)(246,380)(247,375)(248,378)(249,377)(250,376)(251,371)(252,374)(253,373)(254,372)(255,367)(256,370)(257,369)(258,368)(259,363)(260,366)(261,365)(262,364)(263,359)(264,362)(265,361)(266,360)(267,355)(268,358)(269,357)(270,356)(271,351)(272,354)(273,353)(274,352)(275,279)(276,282)(277,281)(278,280)(283,339)(284,342)(285,341)(286,340)(287,335)(288,338)(289,337)(290,336)(291,331)(292,334)(293,333)(294,332)(295,327)(296,330)(297,329)(298,328)(299,323)(300,326)(301,325)(302,324)(303,319)(304,322)(305,321)(306,320)(307,315)(308,318)(309,317)(310,316)(312,314);
poly := sub<Sym(410)|s0,s1,s2,s3>;

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;