Calculation of Group cohomology of $\mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}$ over $\mathbb{C}^{\times}$

Question

Is there any explicit way to compute the cohomology groups $H^{4}( \mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z},\mathbb{C}^{\times})$?. If it is nontrivial then how to produce a non trivial element in this group.

Answer 1

https://groupprops.subwiki.org/wiki/Group_cohomology_of_elementary_abelian_group_of_prime-square_order

yields that your group is isomorphic to $\mu_3^2$, and that the $H^2$ is isomorphic to $\mu_3$. So to cosntruct a non trivial element of $H^4$, I would try first to take the non-trivial element of $H^2$ and take the cup-product with itself...