Linear representation theory of symmetric group:S8

GAP implementation
The degrees of irreducible representations can be computed using GAP's CharacterDegrees and SymmetricGroup functions:

gap> CharacterDegrees(SymmetricGroup(8)); [ [ 1, 2 ], [ 7, 2 ], [ 14, 2 ], [ 20, 2 ], [ 21, 2 ], [ 28, 2 ], [ 35, 2 ], [ 42, 1 ], [ 56, 2 ], [ 64, 2 ], [ 70, 2 ], [ 90, 1 ] ]

The characters of irreducible representations can be computed using GAP's CharacterTable function:

gap> Irr(CharacterTable(SymmetricGroup(8))); [ Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 1, -1, 1, -1, 1, 1, -1, 1, 1, -1, -1, 1, -1, -1, 1, 1, -1, 1, -1, 1, 1, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 7, -5, 3, -1, -1, 4, -2, 0, 1, 1, -3, 1, 1, 0, -1, 2, 0, -1, -1, -1, 0, 1 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 20, -10, 4, -2, 4, 5, -1, 1, -1, -1, -2, 0, -2, 1, 0, 0, 0, 0, 1, 1, -1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 28, -10, 4, -2, -4, 1, -1, 1, 1, -1, 2, 0, 2, -1, 0, -2, 0, 1, 1, -1, 0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 14, -4, 2, 0, 6, -1, -1, -1, 2, 2, 2, 0, -2, -1, 2, -1, 1, -1, 0, 0, 0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 21, -9, 1, 3, -3, 6, 0, -2, 0, 0, -3, -1, 1, 0, 1, 1, 1, 1, 0, 0, 0, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 64, -16, 0, 0, 0, 4, 2, 0, -2, 2, 0, 0, 0, 0, 0, -1, -1, -1, 0, 0, 1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 70, -10, 2, 2, -2, -5, -1, -1, 1, -1, 4, 0, 0, 1, -2, 0, 0, 0, -1, 1, 0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 56, -4, 0, -4, 8, -4, 2, 0, -1, -1, 0, 0, 0, 0, 0, 1, 1, 1, -1, -1, 0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 42, 0, 2, 0, -6, -6, 0, 2, 0, 0, 0, -2, 0, 0, 2, 2, 0, -1, 0, 0, 0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 35, -5, -5, 3, 3, 5, 1, 1, 2, -2, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 1 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 90, 0, -6, 0, -6, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, -1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 56, 4, 0, 4, 8, -4, -2, 0, -1, 1, 0, 0, 0, 0, 0, 1, -1, 1, 1, -1, 0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 70, 10, 2, -2, -2, -5, 1, -1, 1, 1, -4, 0, 0, -1, -2, 0, 0, 0, 1, 1, 0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 14, 4, 2, 0, 6, -1, 1, -1, 2, -2, -2, 0, 2, 1, 2, -1, -1, -1, 0, 0, 0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 35, 5, -5, -3, 3, 5, -1, 1, 2, 2, 1, -1, 1, 1, -1, 0, 0, 0, 0, 0, 0, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 64, 16, 0, 0, 0, 4, -2, 0, -2, -2, 0, 0, 0, 0, 0, -1, 1, -1, 0, 0, 1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 28, 10, 4, 2, -4, 1, 1, 1, 1, 1, -2, 0, -2, 1, 0, -2, 0, 1, -1, -1, 0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 21, 9, 1, -3, -3, 6, 0, -2, 0, 0, 3, -1, -1, 0, 1, 1, -1, 1, 0, 0, 0, 1 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 20, 10, 4, 2, 4, 5, 1, 1, -1, 1, 2, 0, 2, -1, 0, 0, 0, 0, -1, 1, -1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 7, 5, 3, 1, -1, 4, 2, 0, 1, -1, 3, 1, -1, 0, -1, 2, 0, -1, 1, -1, 0, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 8 ] ) ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ) ]