I have the following questions but I don't totally understand them, so im not sure about it all:

Let H be the subgroup of

generated by the permutations

(1) Find the order of

. What's the order of

?

(2) Find the order of H.

(3) Find the conjugacy classes of H.

(4) Show that

is a normal subgroup of H

(5) To which well-known group is the quotient H/Z isomorphic?

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For (1) I know the orders are all 4 I think.

For (2) Is the fastest way of doing this to actually go through and compute every new element?

**Or is there an easier way of finding the order? **
For (3) I think the conjugacy classes are the sets in which the permutions all have the same cycle structure?

For (4) Z is normal iff Z is a union of conjugacy classes?

For (5) I think this is isomorphic to the Klein Four Group. Find the orders of elements in H/Z ?

Thanks for any help.