_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.
//Standard generators of the quadruple cover 4a.L3(4) are preimages A and B
//where 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)^3*(b*a)^3*b ],
       //L34.2_2 = field  - order 2 - not same as in Online ATLAS.
            [ 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[3], swapped by AI[1], AI[2].

_LR`G :=
/*
Original group: c9Group("4al34p")
Direct induction from degree 1
Schur index: 1
Character: ( 112, -112, 0, 4, 0, 0, 0, 0, 0, 2, 2, -4, 0, 0, 0, 0, -2, -2, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0 )
*/

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

return _LR;
