Linear representation theory of Mathieu group:M12

Degrees of irreducible representations
The degrees of irreducible representations can be computed using the CharacterDegrees, CharacterTable, and MathieuGroup functions:

gap> CharacterDegrees(CharacterTable(MathieuGroup(12))); [ [ 1, 1 ], [ 11, 2 ], [ 16, 2 ], [ 45, 1 ], [ 54, 1 ], [ 55, 3 ], [ 66, 1 ], [ 99, 1 ], [ 120, 1 ], [ 144, 1 ], [ 176, 1 ] ]

Character table
The character table can be computed using the Irr, CharacterTable, and MathieuGroup functions:

gap> Irr(CharacterTable(MathieuGroup(12))); [ Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ),   [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6),      (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ), [ 11, -1, -1, -1, 0, 0, 0, 2, 3, 1, -1, -1, 1, 3, -1 ] ), Character( CharacterTable( Group(    [ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ), [ 11, -1, -1, -1, 0, 0, 0, 2, 3, 1, 1, 3, -1, -1, -1 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ),   [ 16, 1, 1, 4, E(11)^2+E(11)^6+E(11)^7+E(11)^8+E(11)^10, E(11)+E(11)^3+E(11)^4+E(11)^5+E(11)^9, 0, -2, 0, 1, 0, 0, 0, 0, -1 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ),   [ 16, 1, 1, 4, E(11)+E(11)^3+E(11)^4+E(11)^5+E(11)^9, E(11)^2+E(11)^6+E(11)^7+E(11)^8+E(11)^10, 0, -2, 0, 1, 0, 0, 0, 0, -1 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ),   [ 45, -1, 3, 5, 1, 1, 0, 0, -3, 0, -1, 1, -1, 1, 0 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6),      (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ), [ 54, 0, 0, 6, -1, -1, 0, 0, 6, -1, 0, 2, 0, 2, 1 ] ), Character( CharacterTable( Group(    [ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ), [ 55, 1, 1, -5, 0, 0, 1, 1, 7, 0, -1, -1, -1, -1, 0 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ),   [ 55, 1, 1, -5, 0, 0, -1, 1, -1, 0, -1, 3, 1, -1, 0 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6),      (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ), [ 55, 1, 1, -5, 0, 0, -1, 1, -1, 0, 1, -1, -1, 3, 0 ] ), Character( CharacterTable( Group(    [ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ), [ 66, 0, 0, 6, 0, 0, -1, 3, 2, 1, 0, -2, 0, -2, 1 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ),   [ 99, -1, 3, -1, 0, 0, 0, 0, 3, -1, 1, -1, 1, -1, -1 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6),      (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ), [ 120, 0, 0, 0, -1, -1, 1, 3, -8, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( Group(    [ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ), [ 144, 1, -3, 4, 1, 1, 0, 0, 0, -1, 0, 0, 0, 0, -1 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ),   [ 176, -1, -1, -4, 0, 0, 0, -4, 0, 1, 0, 0, 0, 0, 1 ] ) ]