_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 12b.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 cycled by automorphisms
_LR`G := 
/*
Original group: c9Group("12bl34p")
Direct integral method
Schur index: 1
Character: ( 480, -480, 0, -240, -240, 0, 0, 0, 0, 16, 0, -16, 0, 0, 0, 240, 
240, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, -8, 0, 0, 0, 8, 0, 8, 0, 
0, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 )
*/

MatrixGroup<480,IntegerRing() |
Matrix(480,480,\[37,275,-340,146,-50,50,38,263,-539,-49,
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return _LR;
