$\pi_1=\pmatrix{1 & 2 & 3 & 4 & 5 \\ 5 & 3 & 1 & 4 & 2}$ and $\pi_2=\pmatrix{1 & 2 & 3 & 4 & 5 \\ 3 & 5 & 1 & 2 & 3}$. Let $\pi_3$ denote the result after using $\pi_1$ and $\pi_2$ after one another. Using $\pi_2(\pi_1(1))=\pi_2(5)=3, ..., \pi_2(\pi_1(5))=\pi_2(5)=5$, I get $\pi_3=\pmatrix{1 & 2 & 3 & 4 & 5 \\ 3 & 1 & 3 & 2 & 5}$. I have to now represent this in the permutation cycle form. $2$ of the cycles are $(3)$ and $(5)$ but what about the rest? I don't seem to see a cycle for the remaining elements.
However, the answer is $(5)(214)(3)$ and I'm not as to why.
What you named $\pi_3$ is not a permutation of five elements. As you can see there is not a element that is mapped to 4, that's a problem because $\pi_3$ is a bijection!