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Let $A$ be the generic abelian crossed product with respect to$\mathbb{Z}_3\times \mathbb{Z}_3$, in this note we show that $A$ is similar tothe tensor product of 4 symbol algebras (3 of degree 9 and one of degree 3) andif $A$ is of exponent $3$ it is similar to the product of 31 symbol algebras ofdegree $3$. We then use \cite{RS} to prove that if $A$ is any algebra of degree$9$ then $A$ is similar to the product of $35840$ symbol algebras ($8960$ ofdegree $3$ and $26880$ of degree $9$) and if $A$ is of exponent $3$ it issimilar to the product of $277760$ symbol algebras of degree $3$. We then showthat the essential $3$-dimension of the class of $A$ is at most $6$.

$\mathbb{Z}_3\times \mathbb{Z}_3$ crossed products