Show that 2 representations are not equivalent and find all the irreducible representations of $G$.
The group $G=T_{16}$ has order 16 and presentation given by $G=\langle a,b : a^8=b^2=1, b^{-1}ab=a^{-1} \rangle.$
Take $n = e^{2πi/8}$. For $1 ≤ j ≤ 3$, there is a representation $p_j(a) : G → \operatorname{GL}(2,\mathbb{C})$ determined by
$$p_j(a)=\begin{bmatrix}n^j & 0 \\ 0 & n^{-j}\end{bmatrix}$$
and
$$p_j(b)=\begin{bmatrix}0 & 1 \\ n^{4j} & 0\end{bmatrix}$$
(b.) If $1≤j,k≤3$ and $j$ is not equal to $k$, show that $p_j$ and $p_k$ are not equivalent. Hint: You may wish to consider characters.
So I am confused because does this mean I consider $p_1(a)$ with $p_2(b)$, $p_3(b)$ and $p_2(a)$ with $p_1(b)$, $p_3(b)$ and $p_3(a)$ with $p_1(b)$ and $p_2(b)$?
Also, I know that the definition of two representations being equivalent is that, $\sigma$ and $p$ are equivalent if there exists $T$ such that $\sigma(g)=T^{-1}p(g)T$, but the only example of have is where the lecturer just plucks the matrix $T$ out of thin air, saying she knows it works because she wrote the question and this doesn't help me at all!
The hint says to look at the characters, so I assume this question is answered easiest by doing that, so I just calculate the trace of each matrix and if the traces are not equal then the representations are not equivalent? I would appreciate guidance on answering a similar question using the matrices approach though in case I am asked to do that in my exam.
(c.) Find all the irreducible representations of $G$, briefly explaining why you have discovered them all.
I haven't tried this part yet, but I thought I would ask it now and can get help in it when I have finished the previous part.
Thanks!