For a graph $G$, let $P(G,\lambda)$ denote the chromatic polynomial of $G$. Two graphs $G$ and $H$ are chromatically equivalent if they share the same chromatic polynomial. A graph $G$ is chromatically unique if for any graph chromatically equivalent to $G$ is isomorphic to $G$. In this paper, the chromatically unique of a new family of 6-bridge graph $\theta(a,a,a,b,b,c)$ where $2\le a\le b\le c$ is investigated.