Linear representation theory of Mathieu group:M22

GAP implementation
The degrees of irreducible representations can be computed using the CharacterDegrees function:

gap> CharacterDegrees(MathieuGroup(22)); [ [ 1, 1 ], [ 21, 1 ], [ 45, 2 ], [ 55, 1 ], [ 99, 1 ], [ 154, 1 ], [ 210, 1 ], [ 231, 1 ], [ 280, 2 ], [ 385, 1 ] ]

The full character table can be printed out using the Irr and CharacterTable functions:

gap> Irr(CharacterTable(MathieuGroup(22))); [ Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,4,5,9,3)(2,8,10,7,6)(12,15,16,20,14)(13,19,21,18,17),     (1,21)(2,10,8,6)(3,13,4,17)(5,19,9,18)(11,22)(12,14,16,20) ]) ), [ 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)(12,13,14,15,16,17,18,19,20,21,22), (1,4,5,9,3)(2,8,10,7,6)(12,15,16,20,14)(13,19,21,18,17),      (1,21)(2,10,8,6)(3,13,4,17)(5,19,9,18)(11,22)(12,14,16,20) ]) ), [ 21, -1, -1, 1, 5, 1, -1, 1, 0, 0, -1, 3 ] ), Character( CharacterTable( Group(    [ (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,4,5,9,3)(2,8,10,7,6)(12,15,16,20,14)(13,19,21,18,17),      (1,21)(2,10,8,6)(3,13,4,17)(5,19,9,18)(11,22)(12,14,16,20) ]) ), [ 45, 1, 1, 1, -3, 1, -1, 0, E(7)^3+E(7)^5+E(7)^6, E(7)+E(7)^2+E(7)^4, 0, 0 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,4,5,9,3)(2,8,10,7,6)(12,15,16,20,14)(13,19,21,18,17),     (1,21)(2,10,8,6)(3,13,4,17)(5,19,9,18)(11,22)(12,14,16,20) ]) ), [ 45, 1, 1, 1, -3, 1, -1, 0, E(7)+E(7)^2+E(7)^4, E(7)^3+E(7)^5+E(7)^6, 0, 0 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,4,5,9,3)(2,8,10,7,6)(12,15,16,20,14)(13,19,21,18,17),     (1,21)(2,10,8,6)(3,13,4,17)(5,19,9,18)(11,22)(12,14,16,20) ]) ), [ 55, 0, 0, -1, 7, 3, 1, 0, -1, -1, 1, 1 ] ), Character( CharacterTable( Group(    [ (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,4,5,9,3)(2,8,10,7,6)(12,15,16,20,14)(13,19,21,18,17),      (1,21)(2,10,8,6)(3,13,4,17)(5,19,9,18)(11,22)(12,14,16,20) ]) ), [ 99, 0, 0, -1, 3, 3, -1, -1, 1, 1, 0, 0 ] ), Character( CharacterTable( Group(    [ (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,4,5,9,3)(2,8,10,7,6)(12,15,16,20,14)(13,19,21,18,17),      (1,21)(2,10,8,6)(3,13,4,17)(5,19,9,18)(11,22)(12,14,16,20) ]) ), [ 154, 0, 0, 2, 10, -2, 0, -1, 0, 0, 1, 1 ] ), Character( CharacterTable( Group(    [ (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,4,5,9,3)(2,8,10,7,6)(12,15,16,20,14)(13,19,21,18,17), (1,21)(2,10,8,6)(3,13,4,17)(5,19,9,18)(11,22)(12,14,16,20) ]) ), [ 210, 1, 1, -2, 2, -2, 0, 0, 0, 0, -1, 3 ] ), Character( CharacterTable( Group( [ (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,4,5,9,3)(2,8,10,7,6)(12,15,16,20,14)(13,19,21,18,17),     (1,21)(2,10,8,6)(3,13,4,17)(5,19,9,18)(11,22)(12,14,16,20) ]) ), [ 231, 0, 0, -1, 7, -1, -1, 1, 0, 0, 1, -3 ] ), Character( CharacterTable( Group( [ (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,4,5,9,3)(2,8,10,7,6)(12,15,16,20,14)(13,19,21,18,17),     (1,21)(2,10,8,6)(3,13,4,17)(5,19,9,18)(11,22)(12,14,16,20) ]) ), [ 280, 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, -8, 0, 0, 0, 0, 0, 1, 1 ] ), Character( CharacterTable( Group( [ (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,4,5,9,3)(2,8,10,7,6)(12,15,16,20,14)(13,19,21,18,17),     (1,21)(2,10,8,6)(3,13,4,17)(5,19,9,18)(11,22)(12,14,16,20) ]) ), [ 280, 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, -8, 0, 0, 0, 0, 0, 1, 1 ] ), Character( CharacterTable( Group( [ (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,4,5,9,3)(2,8,10,7,6)(12,15,16,20,14)(13,19,21,18,17),     (1,21)(2,10,8,6)(3,13,4,17)(5,19,9,18)(11,22)(12,14,16,20) ]) ), [ 385, 0, 0, 1, 1, 1, 1, 0, 0, 0, -2, -2 ] ) ]