  We investigate the molecular gas properties of the deeply obscured luminous
infrared galaxy NGC 4418. We address the excitation of the complex molecule
HC3N to determine whether its unusually luminous emission is related to the
nature of the buried nuclear source. We use IRAM 30m and JCMT observations of
rotational and vibrational lines of HC3N to model the excitation of the
molecule by means of rotational diagrams. We report the first confirmed
extragalactic detection of vibrational lines of HC3N. We detect 6 different
rotational transitions ranging from J=10-9 to J=30-29 in the ground vibrational
state and obtain a tentative detection of the J=38-37 line. We also detect 7
rotational transitions of the vibrationally excited states v6 and v7, with
angular momenta ranging from J=10-9 to 28-27. The energies of the upper states
of the observed transitions range from 20 to 850 K. In the optically thin
regime, we find that the rotational transitions of the vibrational ground state
can be fitted for two temperatures, 30 K and 260 K, while the vibrationally
excited levels can be fitted for a rotational temperature of 90 K and a
vibrational temperature of 500 K. In the inner 300 pc of NGC 4418, we estimate
a high HC3N abundance, of the order of 10^-7. The excitation of the HC3N
molecule responds strongly to the intense radiation field and the presence of
warm, dense gas and dust at the center of NGC 4418. The intense HC3N line
emission is a result of both high abundances and excitation. The properties of
the HC3N emitting gas are similar to those found for hot cores in Sgr B2, which
implies that the nucleus (< 300 pc) of NGC 4418 is reminiscent of a hot core.
The potential presence of a compact, hot component (T=500 K) is also discussed.
