Very good article in that it taught me something I didn't know existed, and explained it very well.<p>However, I wish the article gave an example on how to actually construct a mapping between e.g. a small finite group and the <i>actual</i> matrices in the representation, to - sort of - get a feel for why that actually works.<p>I suspect there must be some sort of canonical method/algorithm to get from the "multiplication table" of a finite group to each matrix in a representation, but I haven't been able to find a reference.<p>Would anyone have pointers?<p>Also, jumping from the group to the character table (which seem to imply that there is indeed an algorithm to compute all possible representations) without having been told <i>how</i> the mapping is constructed feels like a rather big mental jump (and what makes the trace of the matrices important, btw - rather than, say, the determinant?).