In preparing for a Representation Theory exam, a past question asked to complete the character table of a group $G$ using row orthogonality.
The question however doesn't say how many elements are in each conjugacy class. The only things I know are that $|G| = 168$ and that there are 6 conjugacy classes.
Since there's always the class of the identity element, this means that I have to solve an equation of the form $$c_2+c_3+c_4+c_5+c_6 = 167$$ where $c_i \mid 168$.
Here I think I'm supposed to use tricks like Sylow's theorems to impose constraints on the $c_i$, but I haven't so far been able to spot the right idea.