Suppose $G$ is a finite group and suppose that complex character table of $G$ is given.It is well known that from character table we cannot determine the Group uniquely (For example $Q_8$ and $D_8$ have the same character tables but obviously they are not isomorphic.So the question arises How much we know about the Group from its Complex character table? I know that from character table we can tell normal subgroups,commutator subgroup and center of group.Are there some other things which also can be determine from character table?Are there some conditions on character table which ensures the existence of unique group?

Note:This question is motivated by one problem of Artin's Algebra book in which Artin asks to find group in terms of generators and relation by its complex character table.

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    $\begingroup$ Only a bibliography reference here: I would suggest 'Character Theory of Finite Groups' by Martin Isaacs. When I was studying rep theory and character theory this was the best source, in my opinion. $\endgroup$ – rafforaffo Mar 5 '15 at 9:57

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