Representing group by permutaions

Question

Let the group is given by this relations $<a,b\ |\ a^5=b^4=e,bab^{-1}=a^2>$. I am asked to find the cycle index of this group. In order to find cycle index I need to represent the group with permutations. Can you please suggest a hint how it can be done? If permutations are given it should be easy to get cycle index.

Answer 1

With the hint that $G\cong \mathbb {Z}_5\rtimes \mathbb {Z}_4$ its actually not too hard to find it inside $\mathbf S_5$. Take any 5-cycle for a, compute its square (this will also be a 5-cycle), and try to find a 4-cycle so that they are conjugated.

In particular: consider the elements $a = (1\ 2\ 3\ 4\ 5)$ and $b = (2\ 4\ 5\ 3)$. Then $a^5=b^4=1$ and $bab^{-1} = (1\ 3\ 5\ 2\ 4) = a^2$. (If you multiply from left to right.)