_LR := rec < recformat< F: GrpFP, AI: SeqEnum, G: GrpMat > |
      F := FreeGroup(2) >;
//Standard generators of L3(4) are a and b where a has order 2, b has
//order 4, ab has order 7 and abb has order 5.
_LR`AI := [ [ a, b^2*a*b^2*a*b*a*b*a*b^-1*a*b^2*a*b^2 ],
       //L34.2_1 = field x duality - order 2 - not same as in Online ATLAS.
            [ a, b^-1*a*b^2*a*b^-1*a*b^2*a*b*a*b^2*a*b^-1],
       //L34.3 = diagonal - order 3
            [ a, b^-1 ] ]
       //L34.2_3 = duality  - order 2
                  where a is (_LR`F).1 where b is (_LR`F).2;

//two constituents, fixed by AI[2], AI[2], interchanged by AI[1]
_LR`G :=
/*
Original group: c9Group("l34p")
Direct induction from degree 1
Schur index: 1
Character: ( 126, -2, 0, -2, -2, -2, 1, 1, 0, 0 )
*/

MatrixGroup<126,IntegerRing() |
Matrix(SparseMatrix(126,126,\[
1,2,1,1,1,1,1,5,1,1,7,1,1,3,1,1,10,1,1,4,1,1,13,1,
1,9,-1,1,6,1,1,17,1,1,18,1,1,8,1,1,14,1,1,21,1,1,
23,1,1,11,1,1,12,1,1,27,-1,1,29,1,1,15,1,1,32,1,1,
16,1,1,34,1,1,36,1,1,38,1,1,19,-1,1,41,-1,1,20,1,1,
43,1,1,44,1,1,22,1,1,47,-1,1,24,1,1,50,1,1,25,1,1,
52,1,1,26,1,1,55,1,1,57,1,1,28,-1,1,59,-1,1,30,1,1,
31,1,1,61,1,1,62,1,1,33,-1,1,65,-1,1,67,1,1,35,1,1,
69,1,1,37,1,1,72,1,1,73,1,1,39,1,1,74,1,1,40,1,1,
76,-1,1,42,-1,1,60,1,1,45,1,1,46,1,1,64,1,1,63,1,1,
48,-1,1,82,1,1,49,1,1,84,1,1,51,1,1,87,1,1,89,1,1,
53,1,1,54,1,1,56,1,1,92,-1,1,58,-1,1,77,-1,1,95,1,1,
96,1,1,80,1,1,81,-1,1,66,1,1,83,-1,1,68,1,1,100,1,
1,86,1,1,70,1,1,104,-1,1,71,1,1,106,-1,1,107,1,1,
75,-1,1,110,-1,1,112,1,1,78,1,1,79,1,1,114,1,1,103,1,
1,116,-1,1,85,1,1,102,1,1,101,1,1,98,1,1,88,-1,1,
119,1,1,90,-1,1,91,1,1,108,1,1,122,-1,1,93,-1,1,
111,-1,1,94,1,1,113,-1,1,97,1,1,117,1,1,99,-1,1,
115,1,1,118,1,1,105,1,1,123,1,1,124,1,1,109,-1,1,
120,1,1,121,1,1,125,-1,1,126,-1
])),Matrix(SparseMatrix(126,126,\[
1,3,1,1,4,1,1,6,1,1,8,-1,1,9,1,1,11,1,1,12,-1,1,
14,-1,1,15,1,1,16,1,1,1,1,1,19,1,1,20,1,1,2,1,1,
22,1,1,24,1,1,25,1,1,26,1,1,28,-1,1,30,1,1,31,1,
1,5,1,1,33,1,1,35,1,1,37,1,1,39,1,1,40,1,1,7,1,1,
42,1,1,44,1,1,45,1,1,46,1,1,48,-1,1,49,-1,1,10,1,1,
51,1,1,53,1,1,54,1,1,56,1,1,27,1,1,58,1,1,41,1,1,
60,1,1,13,1,1,55,1,1,63,1,1,64,-1,1,66,-1,1,43,1,1,
68,1,1,70,1,1,71,1,1,17,1,1,38,1,1,21,1,1,18,1,1,
75,1,1,29,1,1,77,1,1,34,-1,1,78,1,1,79,1,1,80,1,1,
47,1,1,81,-1,1,23,1,1,83,-1,1,85,1,1,86,-1,1,88,1,
1,61,1,1,90,1,1,72,-1,1,91,1,1,93,-1,1,94,1,1,
59,-1,1,52,1,1,97,1,1,32,1,1,98,1,1,67,-1,1,99,1,
1,76,1,1,101,1,1,102,-1,1,103,1,1,36,1,1,105,1,1,
96,-1,1,108,1,1,109,1,1,111,-1,1,113,1,1,65,-1,1,
73,1,1,115,1,1,95,1,1,82,1,1,87,-1,1,50,1,1,106,-1,
1,117,-1,1,118,-1,1,89,-1,1,69,-1,1,120,1,1,121,1,
1,123,-1,1,104,1,1,57,1,1,110,-1,1,84,1,1,114,-1,
1,62,1,1,116,-1,1,100,1,1,112,1,1,92,1,1,107,-1,1,
74,1,1,124,-1,1,119,-1,1,125,1,1,126,1,1,122,-1
]))>;

return _LR;
