  Elastic and inelastic alpha scatterings on $^{10}$C were measured using a
68-MeV/u radioactive $^{10}$C beam incident on the recently developed MAIKo
active target system. The phenomenological effective $\alpha$-$N$ interaction
and the point-nucleon density distribution in the ground state were determined
from the elastic scattering data. The cross sections of the inelastic alpha
scattering were calculated using this interaction and density distribution and
were compared with the experiment to determine the neutron quadrupole
transition matrix element $M_{n}$ between the ground state and the $2_{1}^{+}$
state at $E_{x} = 3.35$ MeV in $^{10}$C. The deduced neutron transition matrix
element is $M_{n} = 6.9\, \pm0.7\, \mathrm{(fit)}\, \pm1.2\, \mathrm{(sys)}$
fm$^{2}$. The ratio of the neutron transition strength to proton transition
strength was determined as $M_{n}/M_{p} = 1.05\, \pm0.11\, \mathrm{(fit)}\,
\pm0.17\, \mathrm{(sys)}$, which indicates that the quadrupole transition
between the ground state and the $2_{1}^{+}$ state in $^{10}$C is less neutron
dominant compared to that in $^{16}$C.
