[GAP Forum] first example towards GAP, need some explanation

[Jon]

Dear All,

The first example I tried on GAP is about the symmetric group of 4
elements(?). I tried to get its irreducible matrix representation. The
outcome I got from GAP is

gap> List(g,g->g^reps[3]);
[ [ [ 1, 0 ], [ 0, 1 ] ], [ [ 0, E(3) ], [ E(3)^2, 0 ] ], [ [ E(3)^2, 0 ],
[ 0, E(3) ] ], [ [ E(3), 0 ], [ 0, E(3)^2 ] ], [ [ 0, E(3)^2 ], [ E(3), 0 ]
],
  [ [ E(3), 0 ], [ 0, E(3)^2 ] ], [ [ 0, 1 ], [ 1, 0 ] ], [ [ 0, E(3) ], [
E(3)^2, 0 ] ], [ [ E(3)^2, 0 ], [ 0, E(3) ] ], [ [ 0, 1 ], [ 1, 0 ] ], [ [
E(3), 0 ], [ 0, E(3)^2 ] ], [ [ 1, 0 ], [ 0, 1 ] ],
  [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(3)^2, 0 ], [ 0, E(3) ] ], [ [ 0, E(3) ], [
E(3)^2, 0 ] ], [ [ 0, E(3)^2 ], [ E(3), 0 ] ], [ [ E(3), 0 ], [ 0, E(3)^2 ]
],
  [ [ 0, E(3)^2 ], [ E(3), 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ], [ [ E(3)^2, 0 ],
[ 0, E(3) ] ], [ [ 0, E(3) ], [ E(3)^2, 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ], [ [
0, E(3)^2 ], [ E(3), 0 ] ], [ [ 0, 1 ], [ 1, 0 ] ] ]

My question is:

(1) How do I know which matrix corresponds to which group element?
(2) What does E(3) mean?
(3) There can be different representations which has all matrix elements
real, how can I find a similarity transformation which can do this?
(4) Can the output be set in a way that these 24 matrices can be read in
directly by say Fortran?

Thank you very much,

Sincerely,
Jon