2 added 1616 characters in body

ADDED: I'm still somewhat doubtful about the value of the question itself. Classical character theory shows that (in principle) each irreducible complex character determines uniquely an irreducible matrix representation, up to equivalence. As David points out in his answer, there is (in principle) an algorithm for working out this matrix representation in the context of the group algebra, assuming you have complete knowledge of the group and its multiplication table. Plus lots of time, patience, computing power.

In practice, characters have been developed partly to shortcut the need for such algorithms, which are usually impractical for interesting groups like SL$(n,q)$ or other finite groups of Lie type: Lusztig's work over several decades has shown how much can be known about the characters even while many of the key representations remain elusive.

Only in rare cases like symmetric groups do you find a "natural" construction of the representations themselves. And rarely do you know the given group completely enough to carry out David's algorithm based on a given character. If you have that kind of omniscience, it seems you might just construct all the irreducible representations within the group algebra without first knowing the character values.

Computational methods have been most used in recent decades to study the more complicated representation theory of finite groups in prime characteristic when the prime divides the group order. Here even the Brauer characters fail to capture enough information about indecomposable representations, etc.

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If I recall correctly, it's quite possible for two nonisomorphic finite groups to have the same character table. (The dihedral and quaternion groups of order 8 should be an example, but I don't have references at hand.) So you can't expect to recover a group or its representations from a knowledge of characters alone.