Well, I am afraid if I will kill your learning process by giving away complete answers. But, nonetheless. I believe I should emphasize that you'll have to give all these questions a thought before you think you can post them here. For one, I am sure you can answer all your questions yourself.
Here are the hints.
A group $G$ is said to be direct product if and only if there exists subgroups $H$ and $K$ such that,
a) $H$ and $K$ are normal in $G$.
b) $H \cap K= \{e_G\}$
c) $HK=G$
If (a), (b) and (c) are met, we write, $G=H \times K$
Now, can you find an element of order $2$ in your groups? of order $3$? Can the groups they generate intersect non-trivially $^\dagger$ ? What is the order of the product of the groups generated by these elements? So, what can you conclude?
$\dagger$ Non-trivially means, in a "set" that contains some elements other than identity. I am not sure if you have proved, if $H$ and $K$ are subgroups of $G$ such that $(|H|,|K|)=1$, then $H \cap K= \{e_G\}$. Further, the intersection of subgroups is a subgroup and this means I can replace "set" by subgroup. (The lasst fact is required to prove the previous fact about trivial intersection.)