What is the smallest number n such that $A_{n}$ contains a permutation of order 2004?
I calculated it to 334, but the answer is 176 and I can't see why? I first noticed that $2004 = 2^{2}\cdot 3 \cdot 167$ then I looked at the elements in $A_{n}$ as they were written in a form of disjoint cycles. Then I realized that in order for an element to have order 2004 the element must have disjoint cycles in such a way that the least common multiple of the orders of the disjoint cycles is 2004. I then tried to figure out how small I can pick a number n so $x,y .. \leq n$ and $lcm(x,y..) = 2004$ and still have a permutation with disjoint cycles of orders $x,y ..$ that is even.
How many permutations in $S_{6}$ commute with $(1 2) (3 4)$?
Is there any other way than using bruteforce?