I'm sure this is a really basic question and I'm missing something elementary but I've not been making any sort of headway with this.
So in a previous part to the question I was required to find $\alpha$ and $\beta$ such that ord($\alpha$) = 3 and ord($\beta$) = 3 and ord($\alpha \beta$) = 5. I managed to solve this mostly through trial and error with
$\alpha$ = (1 2 3) and
$\beta$ = (3 4 5),
both with ord = 3, and ord($\alpha \beta$) = ord(1 2 4 5 3) = 5.
I really am not sure if there is some process I should be able to see in order for me to generate a product with a specific order. I know that I should be attempting to make a product composed of a 2-cycle and a 5-cycle, but I don't have enough of an understanding of multiplying in cycle form for me to select inputs for a specific output form.
As $$ (a\,b\,c)(a\,b\,d)=(a\,c)(b\,d)$$ we can use $$ (1\,2\,3)(6\,7\,8)\;(3\,4\,5)(6\,7\,9)=(1\,2\,3\,4\,5)(6\,8)(7\,9)$$