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.

  • 2
    $\begingroup$ Not sure if there's a more methodical way, but by inspection of the divisors of $168$, $21+24+24+42+56=167$. $\endgroup$ – Ben West May 17 '17 at 0:38
  • $\begingroup$ Did you use any tricks to rule out some possibilities (e.g. rule out 84, or rule out smaller divisors)? The numbers look fairly large to just try everything. $\endgroup$ – Felipe Jacob May 17 '17 at 0:44
  • $\begingroup$ No, I just suspected the divisors would be in the range between 21 and 56, since 14 and below seem too small, and 84 seems too large since we can only make 5 choices. Luckily it worked out. $\endgroup$ – Ben West May 17 '17 at 0:48

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