The fourth Painlevé equation
is known to have the symmetry of the affine Weyl group of type
with respect to the Bäcklund transformations. Here, a new representation of
, called the symmetric form, is introduced by taking the three fundamental invariant divisors as the dependent variables. A complete description of the symmetry of
is given in terms of this representation. Through the symmetric form, it is shown that
is obtained as a similarity reduction of the 3-reduced modified KP hierarchy. It is proved in particular that the special polynomials for rational solutions to
, called Okamoto polynomials, are expressed in terms of the 3-reduced Schur functions.