The fourth Painlevé equation

${P}_{\text{IV}}$ is known to have the symmetry of the affine Weyl group of type

${A}_{2}^{\left(1\right)}$ with respect to the Bäcklund transformations. Here, a new representation of

${P}_{\text{IV}}$, called the symmetric form, is introduced by taking the three fundamental invariant divisors as the dependent variables. A complete description of the symmetry of

${P}_{\text{IV}}$ is given in terms of this representation. Through the symmetric form, it is shown that

${P}_{\text{IV}}$ 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

${P}_{\text{IV}}$, called Okamoto polynomials, are expressed in terms of the 3-reduced Schur functions.