  By using high pressure and high temperature (3.7 GPa, 1120 $^{\circ}$C)
synthesis technique, we have grown (Cu,C)Ba$_2$Ca$_3$Cu$_4$O$_{11+\delta}$
single crystals. X-ray diffraction, scanning electron microscopy, resistivity
and magnetization measurements are carried out and all show that the samples
have good quality. The single crystal has onset and zero-resistance transition
temperatures of about 111 K and 109.6 K, indicating a very narrow transition
width, which is consistent with a rather sharp magnetization transition.
Magnetization hysteresis loops (MHLs) are also measured, showing a pronounced
second peak effect in the intermediate temperature region. The magnetic
critical current density calculated from the MHLs at 77 K and 1.5 T is about
6.4$\times$10$^4$ A/cm$^2$. By using a criterion of 1$\%$ normal state
resistivity, we have determined the irreversibility line which exhibits an
irreversibility field of about 8 T at 77 K. Compared with other layered
systems, it is easy to find that the irreversibility line is rather high and
could be further improved with the optimized transition temperature of about
118 K as previously discovered in polycrystalline samples.
