Using Characters to Resolve Reducible Representations

It was pointed out earlier that several irreducible representations can be combined into a larger-dimensional reducible representation. Our example was 

Furthermore, no matter how F is disguised by a similarity transformation, its character vector is unchanged. Now, suppose you were given the representation F, disguised through some similarity transformation so as to be nonblock diagonal and asked to tell which irreducible representations were present. How could you do it One way would be to find the similarity transformation that would return the representation to block 

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