Linear representation theory of Mathieu group:M23

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

gap> CharacterDegrees(MathieuGroup(23)); [ [ 1, 1 ], [ 22, 1 ], [ 45, 2 ], [ 230, 1 ], [ 231, 3 ], [ 253, 1 ], [ 770, 2 ], [ 896, 2 ], [ 990, 2 ], [ 1035, 1 ], [ 2024, 1 ] ]

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

gap> Irr(CharacterTable(MathieuGroup(23))); [ Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21, 16) ]) ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( Group(   [ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21,16) ]) ),    [ 22, 0, 0, 0, 2, 6, -1, -1, -1, -1, 1, 1, -1, -1, 2, 4, 0 ] ), Character( CharacterTable( Group(    [ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21,16) ]) ),    [ 45, 1, 1, -1, 1, -3, -1, -1, -E(7)-E(7)^2-E(7)^4, -E(7)^3-E(7)^5-E(7)^6, E(7)+E(7)^2+E(7)^4, E(7)^3+E(7)^5+E(7)^6, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21, 16) ]) ), [ 45, 1, 1, -1, 1, -3, -1, -1, -E(7)^3-E(7)^5-E(7)^6, -E(7)-E(7)^2-E(7)^4, E(7)^3+E(7)^5+E(7)^6, E(7)+E(7)^2+E(7)^4, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21, 16) ]) ), [ 230, -1, -1, 0, 2, 22, 0, 0, 1, 1, -1, -1, 0, 0, 0, 5, 1 ] ), Character( CharacterTable( Group(   [ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21,16) ]) ),    [ 231, 0, 0, -1, -1, 7, 1, 1, 0, 0, 0, 0, 1, 1, 1, 6, -2 ] ), Character( CharacterTable( Group(    [ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21,16) ]) ),    [ 231, 0, 0, -1, -1, 7, 1, 1, 0, 0, 0, 0, -E(15)-E(15)^2-E(15)^4-E(15)^8, -E(15)^7-E(15)^11-E(15)^13-E(15)^14, 1, -3, 1 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21, 16) ]) ), [ 231, 0, 0, -1, -1, 7, 1, 1, 0, 0, 0, 0, -E(15)^7-E(15)^11-E(15)^13-E(15)^14, -E(15)-E(15)^2-E(15)^4-E(15)^8, 1, -3, 1 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21, 16) ]) ), [ 253, 0, 0, -1, 1, 13, 0, 0, -1, -1, 1, 1, 1, 1, -2, 1, 1 ] ), Character( CharacterTable( Group(   [ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21,16) ]) ),    [ 770, 0, 0, 0, -2, -14, E(23)^5+E(23)^7+E(23)^10+E(23)^11+E(23)^14+E(23)^15+E(23)^17+E(23)^19+E(23)^20+E(23)^21+E(23)^22,      E(23)+E(23)^2+E(23)^3+E(23)^4+E(23)^6+E(23)^8+E(23)^9+E(23)^12+E(23)^13+E(23)^16+E(23)^18, 0, 0, 0, 0, 0, 0, 0, 5, 1 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21, 16) ]) ), [ 770, 0, 0, 0, -2, -14, E(23)+E(23)^2+E(23)^3+E(23)^4+E(23)^6+E(23)^8+E(23)^9+E(23)^12+E(23)^13+E(23)^16+E(23)^18,     E(23)^5+E(23)^7+E(23)^10+E(23)^11+E(23)^14+E(23)^15+E(23)^17+E(23)^19+E(23)^20+E(23)^21+E(23)^22, 0, 0, 0, 0, 0, 0, 0, 5, 1 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21, 16) ]) ), [ 896, 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, 0, 0, -1, -1, 0, 0, 0, 0, 1, 1, 1, -4, 0 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21, 16) ]) ), [ 896, 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, 0, 0, -1, -1, 0, 0, 0, 0, 1, 1, 1, -4, 0 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21, 16) ]) ), [ 990, 0, 0, 0, 2, -18, 1, 1, E(7)^3+E(7)^5+E(7)^6, E(7)+E(7)^2+E(7)^4, E(7)^3+E(7)^5+E(7)^6, E(7)+E(7)^2+E(7)^4, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21, 16) ]) ), [ 990, 0, 0, 0, 2, -18, 1, 1, E(7)+E(7)^2+E(7)^4, E(7)^3+E(7)^5+E(7)^6, E(7)+E(7)^2+E(7)^4, E(7)^3+E(7)^5+E(7)^6, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21, 16) ]) ), [ 1035, 1, 1, 1, -1, 27, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( Group(   [ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21,16) ]) ),    [ 2024, 0, 0, 0, 0, 8, 0, 0, 1, 1, 1, 1, -1, -1, -1, -1, -1 ] ) ]