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In this paper, we study the structure of the rational cohomology groups of the IA-automorphism group $\mathrm {IA}_3$
of the free group of rank three by using combinatorial group theory and representation theory. In particular, we detect a nontrivial irreducible component in the second cohomology group of $\mathrm {IA}_3$
, which is not contained in the image of the cup product map of the first cohomology groups. We also show that the triple cup product of the first cohomology groups is trivial. As a corollary, we obtain that the fourth term of the lower central series of $\mathrm {IA}_3$
has finite index in that of the Andreadakis–Johnson filtration of $\mathrm {IA}_3$
.
