# character tables for groups of order $pq^2$

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.

• 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. – Noah Snyder Sep 8 '11 at 19:12
• 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. – 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.