Show that G is Abelian and Determine its order Let $G=\langle(123)(4567), (89)(10\ 11), (8\ 11)(9\ 10)\rangle ≤ S(11).$
(a) Show that G is abelian and determine its order.
[I have a brief idea on this one, by using abelian property of commutativity, you can multiply the elements. However i don't know how to calculate the order of G. Is it 3? Since there are 3 elements in G]
(b) Give a complete list (without duplications) of all isomorphism classes of abelian groups having order |G|.
[For this i will need my answer from A to correctly answer i believe]
(c) Determine the isomorphism class of G.
[These below two i also have little idea]
Thanks, help would be much appreciated!
 A: With respect to part (a) for the most part:
From the comment

Okay, so if i find that the order of the permutations in $G$ are $12, 2$ and $2$ in order. This would mean that the order of $G$ would have to be 12?

Calling your generating permutations $\alpha, \beta,$ and $\gamma$ and assuming they commute (they do), then it's not quite so easy. It will turn out that $\alpha \beta \gamma$ has order $12$. But there are, as I mentioned, at least as many elements in $G$ as there are in $\langle \alpha \rangle = \{\alpha^k : k = 0, 1, \ldots, 11\}$ which has $12$ elements. But, wouldn't $ \alpha^k,\ \alpha^k \beta,\ \alpha^k \gamma$, and $\alpha^k \beta \gamma$ all be distinct elements of $G$, for each $k = 0, 1, \ldots, 11$?
For part (a), you should convince yourself that this is true (note that $\beta$ and $\gamma$ alone would generate a group isomorphic to the Klein four group, the non-cyclic abelian group of order $4$). Not only are these all distinct group elements, they're all the distinct group elements! (Again, make sure you're able to justify this). So your group has order $12 \cdot 2 \cdot 2$.
When you convince yourself that this is true, you should be able to express your group as a direct sum (or 'product', whatever we call it) of cyclic groups. Recall, for part (b), that all finite Abelian groups can be classified as such a direct sum, using only groups of prime-power order as summands. This will allow you to place you group on this list, if you understand how to arrive at the order in part (a).
