//load "D:\\Sporadic\\ON\\onsupp.mag";
// presentation for group
//    hh = <x1,x2,t2 > = Centralizer(ON, 2A) ; kinldy provided by Lempken

/*

hh<x,y,t> := Group< x,y,t | x^2, y^2, t^2, (y * x)^5, (x * t)^6, (y * t * x * t)^4,
    (y * t)^8, (t * x * y * x * t * x)^3, y * t * y * t * x * t * y * t * y * 
    x * y * x * t * y * t * y * t * y * t * x * y * x,
    t * y * t * x * y * x * t * y * t * x * y * x * t * y * t * y * x * t * 
    y * t * x * y ,

    t * y * x * y * t * y * t * y * x * 
    t * y * x * y * t * y * t * y * x * 
    t * y * t * y ,

    x * y * x * t * x * t * y * x * t * 
    x * y * x * y * x * t * y * t * x * 
    y * t * x * t * y * x * y * x * t >  ;

  x2 := x^t;
  y2 := y^t; 
  y20 := y*x2*y2;
  y28 := y*x2*x*y2;
  y7 := y28^4;
  y5 := y20^4;                     
  l272 := sub<hh | y7, x^(y5^3*y7*y5^2), t>;    // = L_2(7):2 =Stab(hh, ??)
PH:=CosetEnumerationProcess(hh,l272: CosetLimit := 40000,Print :=true); 
StartEnumeration(~PH); 
pH,H:=CosetAction(PH);
Hb:=sub<H|x@pH,y@pH,t@pH>;
load "myccs.lib";
kcch:=LexLowestBaseConjugacyClassRepresentatives(Hb);
for p in PrimeDivisors(#Hb) do InstallPrimePowermap(~kcch,p);end for;
Translate2LaTeX(kcch,"H","O`N",["x","y","t"], home cat "kcch.tex");
cth:=CharacterTable(H);
load "chartab2gap.mag";
CharTab2GAP(H,"H");
S:=SylowSubgroup(H,2);
NS:=NormalSubgroups(S); 
NS:=[x`subgroup:x in NS]; 
EANS:=[x:x in NS|IsElementaryAbelian(x)];
EANS8:=[x:x in EANS|#x eq 8];
A:=EANS8[1];
zz:=EANS[1].1;
tt:=EANS[2].2;
uu:=EANS[3].1;
ss:=[x:x in S|Order(x) eq 16][1];
rr:=[d:d in D|Order(d) eq 3 and sub<D|d,ss> eq D][1];
cch:=ConjugacyClasses(H);
cch5:=[x[3]:x in cch|x[1] eq 5];
Ccc:=Class(H,cch5[1]);
for d in Ccc do 
  if sub<H|d,ss,rr> eq H then cc:=d;
    break;
  end if; 
end for;

K:=PSL(3,2);
C:=Centralizer(H,A);
D:=Normalizer(H,A);
//ext:=ExtensionsOfSolubleGroup(C,K);
load home cat "ExtZ4To3byPSL3_2.mag";
P:=pCore(H,2);
ll:=P.1;
Sy1:=Sym(112);
e:=ext[2];
pE,E:=CosetAction(e,sub<e|e.1*e.2>);//has degree 1536=#D
SubE:=Subgroups(E);
SubE:=[x`subgroup:x in SubE];
CoE:=[x:x in SubE|#Core(E,x) eq 1];
E1:=[x:x in CoE|Index(E,x) eq 112][1];
pE,E:=CosetAction(E,E1);//has degree 112
MaxE:=MaximalSubgroups(E); 
MaxE:=[x`subgroup:x in MaxE];
D1:=[x:x in MaxE|IsIsomorphic(D,x)][1];
bo,bj:=IsIsomorphic(D,D1);
rr1:=rr@bj;
ss1:=ss@bj;
cce:=ConjugacyClasses(E);
cce3:=[x[3]:x in cce|x[1] eq 3][1];
Ccg:=Class(E,cce3);
zz1:=zz@bj;
tt1:=tt@bj;
uu1:=uu@bj;
zz21:=zz2@bj;
tt21:=tt2@bj;
uu21:=uu2@bj;
ggcand:=[g:g in Ccg|zz1^g eq tt1 and tt1^g eq zz1*tt1 and uu1^g eq uu1];
ggOnC1:=[[(x^g)@@fC1:x in [zz21,tt21,uu21]]:g in ggcand]; 
//shows that only one action is possible for g on C1, namely [ z2^2 * u2^2 * t2^-1, t2 * u2^2 * z2^-1, z2^2 * t2^2 * u2 ] 
gg:=ggcand[1];

C1:=sub<E|zz21,tt21,uu21>;
C1p<z2,t2,u2>,fC1:=FPGroup(C1);

////////////////////////////////////////////
//Compatible Pairs
///////////////////////////////////////

cth:=CharacterTable(H); 
cte:=CharacterTable(E);
ctdh:=CharacterTable(D);
ctde:=CharacterTable(D1);
cch:=ConjugacyClasses(H);
cce:=ConjugacyClasses(E);
ccdh:=ConjugacyClasses(D);
ccde:=ConjugacyClasses(D1);
cchr:=[x[3]:x in cch];
ccer:=[x[3]:x in cce];
ccdhr:=[x[3]:x in ccdh];
ccder:=[x@bj:x in ccdhr];//ccder:=[x[3]:x in ccde];
load "scp.lib"; 
hs,es:=RestrictionsOnIsomorphicSubgroups(cth,ctdh,ccdhr,cte,ctde,ccder);
pos:=[x:x in [1..#cth]|IsFaithful(cth[x])][1];
scp:=SearchCompatiblePairs(cth,cte,ctdh,hs,es,true,true,true,23,1,pos); //Sollkonstituent cth[pos]
scp:=[x:x in scp];
deg,i:=Min([x[3]:x in scp]);
scp22:=scp[i];
sch:=Sort([x:x in scp22[1]]);
sce:=Sort([x:x in scp22[2]]);
scp22:=[sch,sce];
Mygrps:=[H,E];
Mygens:=[gensH,gensE];
MyConjClasReps:=[cchr,ccer];

////////////////////////////////////////////////////
//Presentation for these groups
//////////////////////////////////

Hp<r,s,c>,fH:=FPGroup(sub<H|rr,ss,cc>);
Ep<r,s,g>,fE:=FPGroup(sub<E|rr1,ss1,gg>);
ND:=NormalSubgroups(D);
ND:=[x`subgroup:x in ND];
bo:=[];
for x in ND do Dmodx,epi:=quo<D|x>;
  Append(~bo,HasComplement(Dmodx,epi(C)));
end for;

grps:=[Hp,Dp,Ep];
noms:=["H","D","E"];
PrintFile(home cat "Rels.tex","":Overwrite:=true);
for i in [1..#grps] do
  PrintFile(home cat "Rels.tex","Relations of " cat noms[i] cat " :=");
  PrintFile(home cat "Rels.tex", Relations(grps[i]));
  PrintFile(home cat "Rels.tex","\n");
end for;

hh1:=ss * rr^-1 * cc^2 * ss * rr^-1 * cc^-1 * rr * cc;
hh2:=cc * ss^-1 * rr^-1 * cc * ss^-3 * cc;
H1:=sub<H|hh1,hh2>;//is a core free subgroup of H od index 448 isomo to Alt(6)

//////////////////////////////////////
//Some automorphism groups
//////////////////////////////////////

AD:=AutomorphismGroup(D); 
pad:=PermutationRepresentation(AD);
ad:=Image(pad);
iad:=sub<ad|[x@pad:x in InnerGenerators(AD)]>;// ad/iad IsElementaryAbelian of order 8

AH:=AutomorphismGroup(H);
pah:=PermutationRepresentation(AH);
ah:=Image(pah);
iah:=sub<ah|[x@pah:x in InnerGenerators(AH)]>;

*/

////////////////////////////////////////////////
//Some elements ss,rr,cc,zz,uu,tt,zz2,tt2,uu2,
///////////////////////////////////////////////

Sy:=Sym(480);
Sy1:=Sym(112);
ss:=Sy!
(1, 453, 382, 335, 155, 272, 27, 160, 203, 212, 457, 461, 353, 444, 45, 25)(2, 303, 420,    37, 68, 305, 438, 61, 189, 95, 381, 400, 273, 281, 141, 312)(3, 234, 226, 156, 24,
    121, 269, 46, 116, 214, 274, 255, 105, 85, 164, 210)(4, 302, 84, 346, 110, 304, 307,    334, 107, 40, 319, 124, 195, 280, 196, 379)(5, 366, 78, 391, 193, 474, 311, 270,
    176, 150, 460, 235, 52, 380, 336, 371)(6, 437, 408, 98, 190, 86, 329, 136, 149, 257,    476, 147, 9, 388, 470, 231)(7, 338, 264, 81, 368, 360, 275, 96, 64, 417, 162, 179,
    92, 434, 419, 109)(8, 367, 243, 89, 450, 249, 99, 35, 58, 154, 267, 152, 430, 261,
    11, 192)(10, 250, 350, 404, 276, 426, 240, 300, 286, 443, 260, 72, 357, 355, 343,
    19)(12, 251, 301, 292, 181, 283, 465, 227, 186, 159, 66, 73, 131, 112, 478, 442)(13,    32, 337, 284, 422, 424, 441, 129, 83, 57, 418, 282, 168, 130, 373, 445)(14, 268,
    327, 345, 401, 389, 146, 206, 33, 328, 94, 69, 38, 140, 97, 103)(15, 113, 222, 200,
    106, 137, 128, 239, 115, 295, 331, 395, 63, 217, 471, 349)(16, 464, 145, 132, 213,
    177, 354, 191, 29, 407, 290, 383, 93, 23, 365, 34)(17, 330, 252, 242, 428, 406, 259,    133, 108, 446, 262, 254, 394, 463, 266, 378)(18, 369, 475, 340, 201, 413, 315, 36,
    299, 158, 449, 142, 30, 77, 182, 376)(20, 209, 153, 399, 403, 218, 76, 409, 386,
    120, 67, 119, 277, 169, 325, 466)(21, 172, 313, 352, 451, 238, 207, 28, 387, 245,
    62, 339, 139, 291, 104, 65)(22, 363, 88, 101, 248, 219, 224, 39, 54, 87, 298, 204,
    51, 370, 293, 390)(26, 359, 358, 425, 180, 236, 287, 421, 148, 221, 285, 309, 314,
    374, 47, 41)(31, 427, 228, 414, 208, 317, 440, 135, 202, 324, 411, 297, 448, 477,
    165, 288)(42, 362, 318, 175, 173, 436, 118, 480, 458, 143, 127, 247, 198, 211, 56,
    456)(43, 455, 398, 171, 232, 372, 402, 205, 459, 429, 364, 59, 341, 423, 348,
    102)(44, 344, 323, 469, 361, 393, 183, 111, 100, 216, 138, 415, 296, 294, 184,
    412)(48, 279, 320, 342, 197, 479, 356, 431, 187, 126, 385, 75, 468, 310, 223,
    397)(49, 82, 91, 416, 117, 316, 215, 151, 90, 53, 161, 220, 55, 185, 439, 326)(50,
    144, 375, 447, 123, 253, 347, 199, 263, 289, 452, 157, 166, 396, 271, 256)(60, 114,
    467, 178, 230, 79, 244, 170, 306, 188, 134, 80, 241, 435, 332, 194)(70, 322, 351,
    125, 392, 71, 258, 225, 384, 167, 432, 233, 333, 462, 472, 278)(74, 174, 265, 321,
    377, 237, 405, 229, 454, 246, 163, 433, 122, 410, 308, 473) ;


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

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

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

//zz2:=P.1;
zz2:=Sy!
(1, 188, 203, 114)(2, 204, 189, 101)(3, 106, 116, 63)(4, 117, 107, 55)(5, 180, 176, 314)(6,    171, 149, 59)(7, 32, 64, 57)(8, 65, 58, 28)(9, 102, 190, 205)(10, 253, 286, 396)(11,
    291, 99, 238)(12, 201, 186, 30)(13, 109, 83, 96)(14, 16, 33, 29)(15, 24, 115, 105)(17,
    56, 108, 118)(18, 181, 299, 131)(19, 166, 300, 123)(20, 361, 386, 296)(21, 367, 387,
    154)(22, 305, 54, 281)(23, 140, 177, 389)(25, 60, 160, 306)(26, 193, 148, 52)(27, 194,
    45, 170)(31, 187, 202, 48)(34, 103, 191, 206)(35, 104, 192, 207)(36, 292, 376, 73)(37,
    224, 400, 293)(38, 93, 401, 213)(39, 68, 390, 273)(40, 316, 302, 185)(41, 235, 421,
    391)(42, 262, 458, 252)(43, 388, 459, 86)(44, 403, 100, 277)(46, 200, 210, 395)(47, 78,    287, 460)(49, 110, 90, 195)(50, 72, 263, 404)(51, 303, 248, 95)(53, 304, 82, 280)(61,
    298, 312, 88)(62, 152, 313, 89)(66, 182, 301, 315)(67, 184, 153, 183)(69, 132, 345,
    383)(70, 333, 384, 392)(71, 167, 462, 322)(74, 122, 454, 377)(75, 414, 342, 297)(76,
    138, 325, 323)(77, 159, 413, 251)(79, 155, 435, 353)(80, 457, 178, 382)(81, 168, 179,
    422)(84, 215, 319, 439)(85, 295, 121, 113)(87, 438, 363, 141)(91, 307, 161, 196)(92,
    130, 368, 424)(94, 290, 327, 145)(97, 365, 146, 354)(98, 348, 147, 402)(111, 399, 412,
    119)(112, 158, 283, 369)(120, 393, 209, 294)(124, 151, 346, 326)(125, 225, 233,
    278)(126, 427, 279, 324)(127, 428, 318, 394)(128, 274, 471, 226)(129, 419, 445,
    275)(133, 211, 378, 436)(134, 453, 467, 212)(135, 431, 288, 397)(136, 398, 231,
    364)(137, 234, 217, 214)(139, 261, 451, 249)(142, 227, 340, 442)(143, 254, 362,
    242)(144, 276, 289, 357)(150, 236, 366, 374)(156, 349, 255, 239)(157, 343, 447,
    240)(162, 282, 264, 284)(163, 405, 265, 308)(164, 222, 269, 331)(165, 356, 440,
    223)(169, 216, 218, 344)(172, 267, 245, 243)(173, 266, 198, 259)(174, 237, 246,
    410)(175, 406, 247, 463)(197, 208, 468, 448)(199, 260, 256, 350)(219, 420, 370,
    381)(220, 334, 416, 379)(221, 474, 359, 380)(228, 385, 411, 320)(229, 433, 473,
    321)(230, 461, 241, 335)(232, 437, 341, 257)(244, 444, 332, 272)(250, 271, 443,
    347)(258, 351, 472, 432)(268, 407, 328, 464)(270, 425, 371, 309)(285, 336, 358,
    311)(310, 317, 479, 477)(329, 429, 470, 455)(330, 480, 446, 456)(337, 338, 418,
    417)(339, 450, 352, 430)(355, 452, 426, 375)(360, 373, 434, 441)(372, 476, 423,
    408)(409, 469, 466, 415)(449, 478, 475, 465);//(r^-1 * c^-1)^3 or (s*r*s)^3

tt2:=Sy!
(1, 11, 435, 172)(2, 250, 390, 375)(3, 9, 115, 171)(4, 140, 90, 407)(5, 168, 148, 109)(6,
    63, 205, 24)(7, 403, 424, 296)(8, 457, 352, 170)(10, 398, 289, 98)(12, 169, 299,
    294)(13, 314, 81, 193)(14, 198, 213, 262)(15, 59, 116, 190)(16, 259, 38, 458)(17, 145,
    318, 97)(18, 393, 186, 218)(19, 438, 50, 219)(20, 32, 100, 92)(21, 60, 249, 461)(22,
    295, 95, 214)(23, 110, 268, 55)(25, 451, 230, 154)(26, 96, 176, 422)(27, 65, 178,
    430)(28, 80, 450, 45)(29, 266, 401, 42)(30, 344, 181, 209)(31, 53, 468, 40)(33, 173,
    93, 252)(34, 215, 345, 196)(35, 134, 313, 272)(36, 150, 227, 221)(37, 395, 298,
    255)(39, 452, 189, 443)(41, 283, 371, 413)(43, 240, 437, 256)(44, 368, 386, 57)(46,
    312, 239, 224)(47, 418, 311, 434)(48, 280, 208, 185)(49, 464, 107, 389)(51, 137, 305,
    121)(52, 83, 180, 179)(54, 113, 303, 234)(56, 94, 394, 365)(58, 382, 339, 194)(61, 349,    293, 210)(62, 444, 192, 467)(64, 277, 130, 361)(66, 412, 475, 466)(67, 419, 323,
    284)(68, 426, 101, 347)(69, 307, 191, 439)(70, 308)(71, 167, 462, 322)(72, 420, 166,
    363)(73, 374, 142, 380)(74, 432)(75, 143, 135, 378)(76, 162, 184, 445)(77, 421, 112,
    270)(78, 417, 285, 441)(79, 245, 203, 99)(82, 197, 302, 202)(84, 132, 161, 206)(85,
    248, 217, 281)(86, 447, 232, 260)(87, 404, 381, 123)(88, 156, 400, 200)(89, 244, 104,
    453)(91, 103, 319, 383)(102, 105, 149, 106)(108, 290, 127, 146)(111, 449, 409,
    301)(114, 238, 155, 243)(117, 177, 195, 328)(118, 327, 428, 354)(119, 465, 415,
    182)(120, 201, 216, 131)(122, 258)(124, 356, 220, 411)(125, 225, 233, 278)(126, 406,
    317, 456)(128, 408, 164, 429)(129, 325, 264, 183)(133, 342, 362, 288)(136, 276, 402,
    396)(138, 282, 153, 275)(139, 241, 367, 160)(141, 263, 370, 300)(144, 147, 286,
    364)(151, 440, 334, 320)(152, 332, 207, 212)(157, 341, 350, 388)(158, 425, 159,
    391)(163, 333)(165, 379, 385, 326)(174, 237, 246, 410)(175, 310, 446, 324)(187, 304,
    448, 316)(188, 291, 353, 267)(199, 459, 343, 257)(204, 271, 273, 355)(211, 297, 242,
    397)(222, 470, 274, 372)(223, 416, 228, 346)(226, 423, 331, 329)(229, 433, 473,
    321)(231, 357, 348, 253)(235, 369, 309, 251)(236, 340, 474, 292)(247, 479, 330,
    427)(254, 431, 436, 414)(261, 335, 387, 306)(265, 392)(269, 455, 471, 476)(279, 463,
    477, 480)(287, 337, 336, 360)(315, 399, 478, 469)(338, 358, 373, 460)(351, 454)(359,
    376, 366, 442)(377, 472)(384, 405);//r^-1 * s^-1 * r * s^-1 * r^-1 * s^-2
uu2:=Sy!
(2, 471, 102, 123)(3, 370, 426, 470)(4, 415, 436, 180)(5, 55, 466, 378)(6, 72, 39, 164)(7,
    17, 73, 304)(8, 450, 58, 430)(9, 300, 101, 274)(10, 113, 395, 343)(12, 191, 418,
    310)(13, 464, 478, 397)(14, 323, 320, 235)(15, 438, 271, 423)(16, 76, 228, 421)(18,
    345, 360, 427)(19, 204, 226, 190)(20, 365, 221, 202)(21, 60, 249, 461)(22, 400, 43,
    147)(23, 66, 75, 168)(24, 363, 443, 408)(25, 451, 230, 154)(26, 195, 111, 362)(27, 457,    45, 382)(28, 339, 65, 352)(29, 325, 411, 41)(30, 103, 338, 477)(31, 386, 354, 359)(32,
    56, 36, 82)(33, 138, 385, 391)(34, 337, 479, 186)(35, 244, 62, 212)(37, 459, 98,
    54)(38, 184, 223, 270)(40, 368, 428, 442)(42, 129, 220, 413)(44, 327, 366, 468)(46,
    447, 253, 85)(47, 175, 294, 307)(48, 361, 146, 474)(49, 399, 242, 193)(50, 273, 331,
    59)(51, 61, 341, 136)(52, 90, 119, 254)(53, 57, 118, 376)(63, 420, 452, 429)(64, 108,
    292, 280)(67, 440, 309, 213)(68, 222, 171, 263)(69, 434, 324, 299)(70, 351, 308,
    454)(71, 225, 321, 410)(74, 384, 432, 405)(77, 458, 445, 416)(78, 406, 120, 161)(80,
    194, 178, 170)(81, 389, 315, 297)(83, 407, 465, 431)(84, 285, 456, 216)(86, 348, 281,
    224)(87, 250, 476, 105)(88, 437, 364, 95)(89, 467, 207, 272)(91, 460, 463, 209)(92,
    394, 227, 302)(93, 153, 165, 425)(94, 150, 197, 100)(96, 328, 449, 288)(97, 380, 187,
    296)(104, 444, 152, 134)(106, 381, 375, 455)(107, 469, 211, 314)(109, 268, 475,
    135)(110, 412, 143, 148)(112, 259, 162, 346)(115, 141, 347, 372)(116, 219, 355,
    329)(117, 409, 133, 176)(121, 210, 157, 396)(122, 392, 258, 265)(124, 283, 266,
    264)(125, 473, 246, 322)(126, 131, 132, 441)(127, 340, 185, 130)(128, 205, 166,
    189)(137, 349, 350, 276)(139, 241, 367, 160)(140, 182, 414, 179)(142, 316, 424,
    318)(144, 214, 156, 256)(145, 374, 448, 403)(149, 404, 390, 269)(151, 369, 198,
    284)(158, 173, 282, 326)(159, 252, 275, 379)(163, 377, 333, 472)(167, 233, 229,
    174)(169, 439, 311, 446)(177, 301, 342, 422)(181, 383, 373, 279)(183, 356, 371,
    401)(192, 332, 313, 453)(196, 287, 247, 393)(199, 289, 234, 255)(200, 240, 286,
    295)(201, 206, 417, 317)(208, 277, 290, 236)(215, 336, 330, 218)(217, 239, 260,
    357)(231, 248, 312, 232)(237, 462, 278, 433)(251, 262, 419, 334)(257, 398, 303,
    298)(261, 335, 387, 306)(293, 388, 402, 305)(319, 358, 480, 344);//(r^-1 * s^-1)^2

rr1:=Sy1!
(1, 92, 27)(2, 26, 102)(3, 103, 14)(4, 91, 21)(5, 71, 22)(6, 34, 40)(7, 11, 101)(8, 12, 83)(9, 13, 90)(10, 94, 67)(15, 58, 
    17)(16, 55, 85)(18, 29, 60)(19, 30, 61)(23, 87, 112)(24, 100, 69)(25, 104, 46)(28, 64, 97)(31, 95, 37)(32, 74, 86)(33, 
    77, 62)(35, 76, 84)(41, 88, 108)(42, 89, 111)(43, 106, 68)(44, 66, 105)(45, 109, 47)(48, 63, 110)(49, 65, 107)(50, 78, 
    99)(51, 82, 56)(52, 75, 54)(53, 93, 57)(59, 98, 72)(70, 80, 79)(73, 81, 96)
;

ss1:=Sy1!
(1, 101, 43, 111)(2, 27, 23, 69, 8, 21, 42, 67)(3, 100, 26, 106)(4, 103, 25, 112)(5, 83, 24, 108)(6, 55, 59, 82, 17, 74, 78,
    93, 18, 77, 81, 75, 19, 76, 80, 95)(7, 22, 41, 68, 14, 9, 44, 46)(10, 105, 13, 102)(11, 94, 12, 104)(15, 31, 37, 70, 29,
    51, 54, 73, 30, 53, 57, 50, 34, 52, 56, 72)(16, 85, 61, 79, 35, 86, 40, 96, 33, 62, 58, 99, 32, 84, 60, 98)(20, 64, 48, 
    110, 36, 45, 47, 107, 38, 63, 28, 97, 39, 65, 49, 109)(66, 91, 89, 71)(87, 90, 88, 92)
;

gg:=Sy1!
(1, 72, 58)(3, 95, 36)(4, 73, 61)(5, 70, 60)(6, 88, 97)(9, 41, 68)(10, 85, 15)(11, 93, 39)(12, 82, 38)(13, 50, 40)(16, 111, 
    96)(17, 87, 109)(18, 66, 110)(19, 89, 107)(20, 26, 75)(21, 42, 69)(22, 44, 46)(23, 67, 27)(24, 62, 30)(25, 86, 29)(28, 
    71, 74)(31, 57, 103)(32, 108, 99)(33, 112, 98)(34, 43, 84)(35, 105, 79)(37, 101, 53)(45, 81, 94)(47, 91, 77)(48, 90, 
    76)(49, 92, 55)(51, 54, 83)(52, 56, 102)(59, 100, 65)(63, 78, 104)(64, 80, 106)
;

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