Question:
Let $G$ be a non-abelian group of order $p^3$ ($p$ is a prime).
(a) Determine the number of irreducible complex representations of $G$, and find their dimensions.
(b) Which of the irreducible complex representations of $G$ are faithful?
Answer:
In part (a), by using class equation, I showed that there are exactly $p^2+p-1$ many irreducible representations. Then using abelianization, I managed to show that there are $p^2$-many $1$-dimensional representations and $p-1$-many $p$-dimensional representations.
However, in the part $(b)$, I couldn't relate those findings above with a representation being faithful. Any help/hint would be appreciated. Thanks in advance...