The $^9$C nucleus and related capture reaction, ${^8mathrm{B}}(p,gamma){^9mathrm{C}}$, have been intensively studied with an astrophysical interest. Due to the weakly-bound nature of $^9$C, its structure is likely to be described as the three-body (${^7mathrm{Be}}+p+p$). Its continuum structure is also important to describe reaction processes of $^9$C, with which the reaction rate of the ${^8mathrm{B}}(p,gamma){^9mathrm{C}}$ process have been extracted indirectly. We perform three-body calculations on $^9$C and discuss properties of its ground and low-lying states via breakup reactions. We employ the three-body model of $^9$C using the Gaussian-expansion method combined with the complex-scaling method. This model is implemented in the four-body version of the continuum-discretized coupled-channels method, by which breakup reactions of $^9$C are studied. The intrinsic spin of $^7$Be is disregarded. By tuning a three-body interaction in the Hamiltonian of $^9$C, we obtain the low-lying $2^+$ state with the resonant energy 0.781 MeV and the decay width 0.137 MeV, which is consistent with the available experimental information and a relatively high-lying second $2^+$ wider resonant state. Our calculation predicts also sole $0^+$ and three $1^-$ resonant states. We discuss the role of these resonances in the elastic breakup cross section of $^9$C on $^{208}$Pb at 65 and 160 MeV/A. The low-lying 2$^+$ state is probed as a sharp peak of the breakup cross section, while the 1$^-$ states enhance the cross section around 3 MeV. Our calculations will further support the future and ongoing experimental campaigns for extracting astrophysical information and evaluating the two-proton removal cross-sections.
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