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Groups : Third Sylow Theorem

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Suppose a simple group G of order 660 is a subgroup of the symmetric group S11 (here S is with 11 as a subscript ) and x = {1,2,3,4,5,6,7,8,9,10,11}( a permutation like 1 goes to 2 and so on, I think ) in G. If P equals the span of x ( or <x> ) determine the permutations which generate NGP ( here G is written as the subscript of N (the normalizer of P in G, I think).
Hint: if you know the order of the normalizer N of <x> can you find the generators for N ?
Give a solid math argument why this is correct and please keep the answer as simple as possible at the same time.

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Solution Summary

Permutations are found using the third Sylow theorem. The solution is detailed and well presented.

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