how to classify all groups of order 66, up to isomorphism?
Firstly, we have 66=$3\times 11\times 2$, suppose the number of the Sylow-11 group of this group is $n_k$, since 11 divides $n_k-1$ we can know that $n_k$=1 since $n_k$ must divide 6. If so, we can know any group with order 66 can only have one Sylow-11 subgroup,but now I am not sure how to continue analyze this question. Can someone tell me how to determine all the groups of order 66?
Best Answer
You correctly deduced that such a group $G$ always has a normal cyclic subgroup $P$ of order $11$. Let us fix a generator $x\in P$. From that point on:
The choices for the two epsilons leave four possible combinations. All of them occur.