What is the character table for groups or order $pq^2$? The classification of order $pq^2$ groups has already been discussed in relation to Sylow theory.

For the Abelian groups, $\mathbb{Z}_p \oplus \mathbb{Z}_{q^2}$ and $\mathbb{Z}_p \oplus \mathbb{Z}_q \oplus \mathbb{Z}_q$, all the irreducible representations are 1-dimensional.

According to some group theory lecture notes I found online (bottom of page 8), there is only one other group when $q \not\equiv 1 (\mod p)$ and two when $q \equiv 1 (\mod p)$. I am asking for the character table in any or all of these cases.

  • 2
    $\begingroup$ Try looking at the "method of little groups" as explained in Serre's book. This gives a way of calculating all irreps of any group that has a large normal abelian subgroup as certain induced representations. $\endgroup$ – Noah Snyder Sep 8 '11 at 19:12
  • $\begingroup$ There are more groups of order pqq than you indicate. For instance, there are 8 groups of order 5887 up to isomorphism, not just four. $\endgroup$ – Jack Schmidt Sep 8 '11 at 19:42

Any group of order $pq^2$ is a semidirect product by an abelian group. Indeed, by Sylow theory either the Sylow $p$-subgroup or the Sylow $q$-subgroup is normal. The Schur-Zassenhaus theorem says that if a normal subgroup has order coprime to its index, then it has a complement.

To classify all irreducible character of semidirect products by abelian groups is a very nice exercise in character theory. See another answer of mine for details. This is from Serre's representation theory book Part II, Section 8.2.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.