As you already discovered, GAP can easily compute this. The character table looks like this:
gap> G:=SymmetricGroup(7);
Sym( [ 1 .. 7 ] )
gap> ct := CharacterTable(G);;
CharacterTable( Sym( [ 1 .. 7 ] ) )
gap> Display(ct);
CT2
2 4 4 4 4 3 2 3 1 3 3 2 1 1 1 .
3 2 1 1 1 2 1 1 2 1 . 1 . . 1 .
5 1 1 . . . . . . . . . 1 1 . .
7 1 . . . . . . . . . . . . . 1
1a 2a 2b 2c 3a 6a 6b 3b 4a 4b 12a 5a 10a 6c 7a
2P 1a 1a 1a 1a 3a 3a 3a 3b 2b 2b 6b 5a 5a 3b 7a
3P 1a 2a 2b 2c 1a 2a 2b 1a 4a 4b 4a 5a 10a 2c 7a
5P 1a 2a 2b 2c 3a 6a 6b 3b 4a 4b 12a 1a 2a 6c 7a
7P 1a 2a 2b 2c 3a 6a 6b 3b 4a 4b 12a 5a 10a 6c 1a
X.1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 1
X.2 6 -4 2 . 3 -1 -1 . -2 . 1 1 1 . -1
X.3 14 -6 2 -2 2 . 2 -1 . . . -1 -1 1 .
X.4 14 -4 2 . -1 -1 -1 2 2 . -1 -1 1 . .
X.5 15 -5 -1 3 3 1 -1 . -1 -1 -1 . . . 1
X.6 35 -5 -1 -1 -1 1 -1 -1 1 1 1 . . -1 .
X.7 21 -1 1 3 -3 -1 1 . 1 -1 1 1 -1 . .
X.8 21 1 1 -3 -3 1 1 . -1 -1 -1 1 1 . .
X.9 20 . -4 . 2 . 2 2 . . . . . . -1
X.10 35 5 -1 1 -1 -1 -1 -1 -1 1 -1 . . 1 .
X.11 14 4 2 . -1 1 -1 2 -2 . 1 -1 -1 . .
X.12 15 5 -1 -3 3 -1 -1 . 1 -1 1 . . . 1
X.13 14 6 2 2 2 . 2 -1 . . . -1 1 -1 .
X.14 6 4 2 . 3 1 -1 . 2 . -1 1 -1 . -1
X.15 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
In order to understand how to read this, one needs to know (e.g. by a quick look at the documentation) that the rows labelled X.1 to X.15 correspond to the irreducible characters, while the columns correspond to the conjugacy classes of the group. So, let's ask GAP for the classes and a representative of each:
gap> cc:=ConjugacyClasses(ct);
[ ()^G, (1,2)^G, (1,2)(3,4)^G, (1,2)(3,4)(5,6)^G, (1,2,3)^G, (1,2,3)(4,5)^G, (1,2,3)(4,5)(6,7)^G,
(1,2,3)(4,5,6)^G, (1,2,3,4)^G, (1,2,3,4)(5,6)^G, (1,2,3,4)(5,6,7)^G, (1,2,3,4,5)^G, (1,2,3,4,5)(6,7)^G,
(1,2,3,4,5,6)^G, (1,2,3,4,5,6,7)^G ]
gap> for i in [1..15] do Print(i, ": ", Representative(cc[i]), "\n"); od;
1: ()
2: (1,2)
3: (1,2)(3,4)
4: (1,2)(3,4)(5,6)
5: (1,2,3)
6: (1,2,3)(4,5)
7: (1,2,3)(4,5)(6,7)
8: (1,2,3)(4,5,6)
9: (1,2,3,4)
10: (1,2,3,4)(5,6)
11: (1,2,3,4)(5,6,7)
12: (1,2,3,4,5)
13: (1,2,3,4,5)(6,7)
14: (1,2,3,4,5,6)
15: (1,2,3,4,5,6,7)
Of course you can get a lot more information from this, too. More on this can be found in the GAP manual.