_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 cover 12a.L3(4) are preimages A and B
//where A has order 2, B has order 4, AB has order 21 and ABB has order 5.

_LR`AI :=
    [[a,a*b^-1*a*b*a*b^-1*a*b*a*b*a*b^-1*a*b*a*b*a*b^-1*a*b*a*b^-1*(a*b)^7],
       //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;

//eight constituents, not fixed by any automorphism.
_LR`G :=
/*
Original group: c9Group("12al34p")
Direct integral method
Schur index: 1
Character: ( 384, -384, 0, -192, -192, 0, 0, 0, 0, 0, 0, 4, 4, 192, 192, 0, 0, 
0, -8, -8, 0, 0, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 8, -2, -2, -2, 
-2, 0, 0, 0, 0, 4, 4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, -4, -4, -4, -4, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 )
*/

MatrixGroup<384,IntegerRing() |
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-9,-39,62,85,-22,-15,-80,-37,-59,-57,18,4,-90,5,33])>;

return _LR;
