Originally Posted by

**redsoxfan325** We are studying representation theory and we were asked to find the unirreps (unitary irreducible representations) for a group $\displaystyle G$ given that $\displaystyle |G|=p^3$, $\displaystyle p$ a prime.

We have the theorem that states that the number of unirreps is equal to the number of conjugacy classes of the group $\displaystyle G$.

There are five cases, three of which are abelian and therefore trivial. The other two groups, which we figured out in class, are:

1.) Let $\displaystyle z=xyx^{-1}y^{-1}$. Then $\displaystyle G=\langle x,y~|~x^p=y^p=z^p=1,~xz=zx,~yz=zy\rangle$

2.) $\displaystyle G=\langle x,y~|~x^{p^2}=y^p=1,~yxy^{-1}=x^{p+1}\rangle$

How do I go about finding the conjugacy classes of these groups, or is there a better way to find the number of unirreps for each group?