S4 is not cyclic because it's not generated by any single element. A good cue is that all cyclic groups are abelian -- S4 is not abelian, so can't be cyclic.

S4 also has 24 elements, because Sn has n! elements for any n. But you want a subgroup with only 4 elements, so we can't be talking about S4.

You need a group generated by an element that has order 4.

Try the group generated by the element <(a b c d)>.

I hope you are familiar with cycle notation.

(a b c d)^1 = (a b c d) (obviously)

(a b c d)^2 = (a b c d)(a b c d) = (a c)(b d)

(a b c d)^3 = (a b c d)(a b c d)^2 = (a b c d)(a c)(b d) = (a d c b)

(a b c d)^4 = (a b c d)(a b c d)^3 = (a b c d)(a d c b) = (I), the identity element.

So this group has four elements in it, and it's cyclic. Of course its contained in S4, so it'll do for what you're looking for.