  The $^9$C nucleus and related capture reaction,
${^8\mathrm{B}}(p,\gamma){^9\mathrm{C}}$, 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 (${^7\mathrm{Be}}+p+p$). Its
continuum structure is also important to describe reaction processes of $^9$C,
with which the reaction rate of the ${^8\mathrm{B}}(p,\gamma){^9\mathrm{C}}$
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.
