//Standard generators of L2(81) are a and b where a has order 2, b has order 3
//and ab has order 20.
//Standard generators of the double cover 2.L2(81) = SL2(81) are preimages A and
//B where B has order 3 (and AB has order 40).
_LR := rec < recformat< F: GrpFP, AI: SeqEnum, G: GrpMat > |
      F := FreeGroup(2) >;
    _LR`AI:=[[a,
       b*a*b*a*b*a*b^-1*a*b*a*b*a*b*a*b*a*b*a*b*a*b*a*b*a*b^-1*a*b^-1*a^-1],
                                                    //PSL(2,81).2=PGL(2,81)
   [a^-1, a*b*a*b*a*b^-1*a*b^-1*a*b^-1*a*b*a*b],//PSL(2,81).4=PSigmaL(2,81)
   [a^-1, b*a^-1*b^-1*a*b^-1*a*b*a*b*a*b*a*b^-1*a*b]]//square of AI[2]
             where a is (_LR`F).1 where b is (_LR`F).2;

//irreducible, mapped to inequiv rep by AI[1], fixed by AI[2]
_LR`G := sub<GL(41,Integers()) |
\[ 0,0,1,1,0,1,0,0,0,1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,
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0,0,0,0,0,-1,0,-1,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,
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0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,1,-1,0,0,
0,0,0 ],
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0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,
0,0,0,0,0,1,0,-1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,-1,0,0,0,0,-1,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,0,1,
0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,
0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,
0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,
0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,-1,
0,0,0,0,0,1,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,
0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,
0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,-1,0,
0,0,-1,0,0,-1,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,
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0,0,0,1,0,-1,0,0,1,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,
0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,
0,0,-1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,-1,
0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,-1,0,
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0,0,1,-1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,
0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,
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0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,1,0,0,0,-1,0,0,0,0,
0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,
0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,0,
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0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,1,0,0,
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0,0,0,0,0,1,0,0,0,-1,0,-1,-1,0,0,0,0,0,0,0,0,-1,-1,0,
0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,
1,0,0,0,0,0,0,0,0,0,1,1,0,-1,-1,0,0,0,0,0,1,0,0,0,1,
0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,1,0,
0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,-1,0,0,0,0,-1,0,
0,0,0,0,0,1,0,0,1,-2,-1,0,0,0,1,-1,0,0,-1,0,0,0,-1,0,
1,0,0,1,-1,0,0,0,1,0,1,0,-1,0,0,0,0,0,1,0,1,-1,-1,0,
0,0,1,0,0,0,0,1,-1,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,
1,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,-1,
1,0,0,1,0,1,0,0,0,-1,0,0,-1,1,-1,0,0,0,0,-1,0,1,0,0,
0,1,0,0,0,0,0,1,1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,
0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,
0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,
0,-1,1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,-1,1,0,0,0,0,0,
1,0,0,0,0,0,0,1,0,0,0,1,-1,0,0,0,0,-1,0,-1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,0,1,0,
0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1,0,0,
0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,0,0,0,-1,0,0,0,-1,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,-1,
0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,
0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,
0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,-1,
0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,
0,1,0,1,-1,0,0,0,0,0,0,1,-1,0,0,1,0,0,0,0,0,0,0,0,0,
0,0,0,0,1,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,
1,-1,0,0 ] >;

return _LR;
