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We show that every affine permutation has a unique maximal decomposition into cyclically decreasing elements. This decomposition coincides with the Lehmer code, and contains a great deal of information about the permutation, such as length, descent set, and whether the permutation is fully-commutative. We will also demonstrate a new tableau model on the Lehmer code of the affine permutation, in which tableaux are in bijection with reduced expressions for the permutation.