//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 3.L3(4) are preimages A, B where A has order 2
//and B has order 4.
_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*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[1],AI[2], swapped by AI[3].
_LR`G :=
/*
Original group: c9Group("3l34p")
Direct induction from degree 8
Schur index: 1
Character: ( 168, 8, -84, -84, 0, 0, 0, 0, -2, -2, -4, -4, 0, 0, 0, 0, 0, 0, 0, 
0, 1, 1, 1, 1, 0, 0, 0, 0 )
*/

MatrixGroup<168,IntegerRing() |
Matrix(SparseMatrix(168,168,\[
2,5,-1,7,1,2,6,-1,8,1,2,3,1,5,-1,2,4,1,6,-1,1,5,-1,
1,6,-1,2,1,1,5,-1,2,2,1,6,-1,1,17,1,1,18,1,1,19,1,
1,20,1,1,21,1,1,22,1,1,23,1,1,24,1,1,9,1,1,10,1,1,
11,1,1,12,1,1,13,1,1,14,1,1,15,1,1,16,1,1,41,1,1,
42,1,1,43,1,1,44,1,1,45,1,1,46,1,1,47,1,1,48,1,1,
57,1,1,58,1,1,59,1,1,60,1,1,61,1,1,62,1,1,63,1,1,
64,1,1,25,1,1,26,1,1,27,1,1,28,1,1,29,1,1,30,1,1,
31,1,1,32,1,1,81,1,1,82,1,1,83,1,1,84,1,1,85,1,1,
86,1,1,87,1,1,88,1,1,33,1,1,34,1,1,35,1,1,36,1,1,
37,1,1,38,1,1,39,1,1,40,1,1,65,1,1,66,1,1,67,1,1,
68,1,1,69,1,1,70,1,1,71,1,1,72,1,1,105,1,1,106,1,1,
107,1,1,108,1,1,109,1,1,110,1,1,111,1,1,112,1,1,49,1,
1,50,1,1,51,1,1,52,1,1,53,1,1,54,1,1,55,1,1,56,1,
2,123,1,124,1,1,123,-1,4,123,1,124,1,125,-1,126,-1,2,
123,-1,125,1,4,123,1,124,1,127,-1,128,-1,2,123,-1,127,1,
4,121,-1,122,-1,123,1,124,1,2,121,1,123,-1,1,137,1,
1,138,1,1,139,1,1,140,1,1,141,1,1,142,1,1,143,1,1,
144,1,1,73,1,1,74,1,1,75,1,1,76,1,1,77,1,1,78,1,1,
79,1,1,80,1,2,159,1,160,1,1,159,-1,4,157,-1,158,-1,
159,1,160,1,2,157,1,159,-1,4,153,-1,154,-1,159,1,160,1,
2,153,1,159,-1,4,155,-1,156,-1,159,1,160,1,2,155,1,
159,-1,2,90,-1,96,1,4,89,1,90,1,95,-1,96,-1,1,90,-1,
2,89,1,90,1,2,90,-1,92,1,4,89,1,90,1,91,-1,92,-1,2,
90,-1,94,1,4,89,1,90,1,93,-1,94,-1,1,133,1,1,134,1,
1,135,1,1,136,1,1,129,1,1,130,1,1,131,1,1,132,1,1,
97,1,1,98,1,1,99,1,1,100,1,1,101,1,1,102,1,1,103,1,
1,104,1,1,145,-1,1,146,-1,2,145,-1,151,1,2,146,-1,
152,1,2,145,-1,149,1,2,146,-1,150,1,2,145,-1,147,1,
2,146,-1,148,1,2,114,-1,118,1,4,113,1,114,1,117,-1,
118,-1,2,114,-1,120,1,4,113,1,114,1,119,-1,120,-1,2,
114,-1,116,1,4,113,1,114,1,115,-1,116,-1,1,114,-1,2,
113,1,114,1,1,161,-1,1,162,-1,2,161,-1,165,1,2,162,-1,
166,1,2,161,-1,163,1,2,162,-1,164,1,2,161,-1,167,1,2,
162,-1,168,1
])),Matrix(SparseMatrix(168,168,\[
1,9,1,1,10,1,1,11,1,1,12,1,1,13,1,1,14,1,1,15,1,
1,16,1,1,25,1,1,26,1,1,27,1,1,28,1,1,29,1,1,30,1,
1,31,1,1,32,1,1,33,1,1,34,1,1,35,1,1,36,1,1,37,1,
1,38,1,1,39,1,1,40,1,1,49,1,1,50,1,1,51,1,1,52,1,
1,53,1,1,54,1,1,55,1,1,56,1,1,65,1,1,66,1,1,67,1,
1,68,1,1,69,1,1,70,1,1,71,1,1,72,1,1,73,1,1,74,1,
1,75,1,1,76,1,1,77,1,1,78,1,1,79,1,1,80,1,1,1,1,1,
2,1,1,3,1,1,4,1,1,5,1,1,6,1,1,7,1,1,8,1,1,89,1,1,
90,1,1,91,1,1,92,1,1,93,1,1,94,1,1,95,1,1,96,1,1,
97,1,1,98,1,1,99,1,1,100,1,1,101,1,1,102,1,1,103,1,
1,104,1,1,113,1,1,114,1,1,115,1,1,116,1,1,117,1,1,
118,1,1,119,1,1,120,1,2,81,1,87,-1,2,82,1,88,-1,2,
85,1,87,-1,2,86,1,88,-1,2,83,1,87,-1,2,84,1,88,-1,1,
87,-1,1,88,-1,4,129,1,130,1,135,-1,136,-1,2,129,-1,
135,1,4,129,1,130,1,133,-1,134,-1,2,129,-1,133,1,2,
129,1,130,1,1,129,-1,4,129,1,130,1,131,-1,132,-1,2,
129,-1,131,1,1,17,1,1,18,1,1,19,1,1,20,1,1,21,1,1,
22,1,1,23,1,1,24,1,2,145,1,147,-1,2,146,1,148,-1,1,
147,-1,1,148,-1,2,147,-1,151,1,2,148,-1,152,1,2,147,-1,
149,1,2,148,-1,150,1,1,121,1,1,122,1,1,123,1,1,124,1,
1,125,1,1,126,1,1,127,1,1,128,1,1,41,1,1,42,1,1,
43,1,1,44,1,1,45,1,1,46,1,1,47,1,1,48,1,1,166,-1,2,
165,1,166,1,2,166,-1,168,1,4,165,1,166,1,167,-1,168,-1,
2,164,1,166,-1,4,163,-1,164,-1,165,1,166,1,2,162,1,
166,-1,4,161,-1,162,-1,165,1,166,1,1,153,1,1,154,1,
1,155,1,1,156,1,1,157,1,1,158,1,1,159,1,1,160,1,
1,105,1,1,106,1,1,107,1,1,108,1,1,109,1,1,110,1,1,
111,1,1,112,1,2,139,-1,143,1,2,140,-1,144,1,1,139,-1,
1,140,-1,2,139,-1,141,1,2,140,-1,142,1,2,137,1,139,-1,
2,138,1,140,-1,1,57,1,1,58,1,1,59,1,1,60,1,1,61,1,
1,62,1,1,63,1,1,64,1
]))>;

return _LR;
