  The structure and decay properties of $d^*$ have been detailedly investigated
in both the chiral SU(3) quark model and the extended chiral SU(3) quark model
that describe the energies of baryon ground states and the nucleon-nucleon (NN)
scattering data satisfactorily. By performing a dynamical coupled-channels
study of the system of $\Delta\Delta$ and hidden-color channel (CC) with
quantum numbers $I(J^P)=0(3^+)$ in the framework of the resonating group method
(RGM), we find that the $d^*$ has a mass of about $2.38-2.42$ GeV and a
root-mean-square radius (RMS) of about $0.76-0.88$ fm. The channel wave
function is extracted by a projection of the RGM wave function onto the
physical basis, and the fraction of CC component in the $d^*$ is found to be
about $66\%-68\%$, which indicates that the $d^*$ is a hexaquark-dominated
exotic state. Based on this scenario the partial decay widths of $d^*\to d
\pi^0 \pi^0$ and $d^*\to d \pi^+\pi^-$ are further explicitly evaluated and the
total width is then obtained by use of the branching ratios extracted from the
measured cross sections of other possible decay channels. Both the mass and the
decay width of $d^*$ calculated in this work are compatible with the data
($M\approx 2380$ MeV, $\Gamma\approx 70$ MeV) reported by WASA-at-COSY
Collaboration.
