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Let $\mathrm{k}=\mathbb{Q}\left(\sqrt[3]{p},\zeta_3\right)$, where $p$ is aprime number such that $p \equiv 1 \pmod 9$, and let $C_{\mathrm{k},3}$ be the$3$-component of the class group of $\mathrm{k}$. In \cite{GERTH3}, Frank GerthIII proves a conjecture made by Calegari and Emerton \cite{Cal-Emer} whichgives necessary and sufficient conditions for  $C_{\mathrm{k},3}$ to be of $\operatorname{rank}\,$ two. The purpose of thepresent work is to determine generators of $C_{\mathrm{k},3}$, whenever it isisomorphic to $\mathbb{Z}/9\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$.

The generators of $3$-class group of some fields of degree $6$ over $\mathbb{Q}$