What do we learn about a non-abelian simple group from the numbers "inside" its character table? 
What do we learn about a non-abelian simple group $G$ from the numbers "inside" its character table?

Clearly there is a lot of useful information in the "header" of the character table, like the order and size of the conjugacy classes and the power-up maps between conjugacy classes. But what additional information can we extract about $G$ from the numbers "inside" the character table? If I lookup the character table of some simple group in the ATLAS, how do I make sense of that array of numbers to learn something about $G$?
The usual answers I have seen about extracting information from a character table have to do with the normal structure of a group, but that isn't useful in the case where $G$ is non-abelian simple.
 A: Typically the individual values don't tell you much, but the table as a whole allows you to calculate (by Linear Algebra or basic Combinatorics) information about the group that would be much harder to obtain in other ways.
The character values allow you for example to deduce relations amongst representations (are they Galois conjugate? How do tensor products decompose?).
They allow you to calculate the class multiplication coefficients (how often does a product of elements in class $x$ and class $y$ lie in class $z$), and thus help to determine by what pairs of elements the group can be generated. [This is e.g. how it has been shown that many of the simple groups occur as Galois groups over the Rationals without ever writing down polynomials.]
The characters let you reconstruct the central idempotents $e_i$ (and thus decompose a representation into its irreducible constitutents).
By searching for combinations of characters that could be permutation characters, it is possible to get (an approximation of) the subgroup structure of $G$.
