//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 sextuple cover 6.L3(4) are preimages A and B
//where AB has order 2, B has order 4, AB has order 21, ABB has order 5 and
//ABABABBB has order 5.
_LR := rec < recformat< F: GrpFP, AI: SeqEnum, G: GrpMat > |
      F := FreeGroup(2) >;
_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*a*(b^-1*a)^4*b^2*a*b*a*b^-1*a*b^-1]  ]
       //L34.2_3 = duality  - order 2q = _1 x _2
                  where a is (_LR`F).1 where b is (_LR`F).2;

//two constituents, fixed by AI[1], interchanged by AutIms[2].
_LR`G :=
MatrixGroup<6, ext<K|Polynomial(K, [1, 1, 1])> where K is RationalField() |
[[-1,0],[0,0],[
0,0],[0,0],[0,0],[
0,0],[0,0],[0,
1],[1,0],[0,-1],[
0,-1],[0,0],[
1,2],[0,0],[0,-1],[
1,2],[1,0],[0,
1],[2,1],[0,1],[
0,-1],[2,1],[
1,-1],[1,1],[-1,-2],[
1,1],[0,1],[
-1,-3],[-2,-1],[0,-1],[
0,0],[-1,-1],[
-1,0],[0,1],[0,1],[
-1,0]],
[[0,1],[0,-1],[
0,0],[0,1],[0,1],[
0,0],[-1,-1],[
0,0],[1,1],[-1,-2],[
-1,0],[0,-1],[
0,0],[-1,0],[0,1],[
1,0],[1,1],[0,
0],[-1,-1],[0,1],[
1,1],[-1,-2],[
0,0],[0,0],[1,1],[
0,0],[0,-1],[
0,1],[0,0],[0,0],[
0,1],[-1,-1],[
0,0],[0,2],[1,1],[
-1,0]]>;

return _LR;
