  Inspired by recent progresses in observations of the fully charmed tetraquark
states by LHCb, CMS, and ATLAS Collaborations, we perform a systematic study of
the ground states and the first radial excitations of the $S$- and $P$-wave
$\mathrm{cc}\bar{\mathrm{c}}\bar{\mathrm{c}}$ system. Their mass spectra, root
mean square(r.m.s.) radii and radial density distributions are studied with the
relativized quark model. The calculations show that there is no stable bound
states for the full-charmed tetraquark states, and the r.m.s. radii of these
tetraquark states are smaller than 1 fm. Our results support assigning X(6600)
structure, $M_{X(6600)}=6552\pm10\pm12$ MeV, as one of the $0^{++}$(1$S$) and
$2^{++}$(1$S$) states or their mixtures. Another structure also named as
X(6600) by CMS Collaboration, $M_{X(6600)}=6.62\pm0.03^{+0.02}_{-0.01}$ GeV,
may arise from the lowest 1$P$ states with $J^{PC}$=$0^{-+}$, $1^{-+}$, and
$2^{-+}$. The possible assignments for X(6900) include the $0^{++}$(2$S$),
$2^{++}$(2$S$) states, and the highest 1$P$ state with $J^{PC}=0^{-+}$. As for
X(7200), it can be interpreted as one of the highest 2$P$ states with
$J^{PC}=0^{-+}$, $1^{-+}$, and $2^{-+}$, and the 3$S$ states can not be
completely excluded from the candidates.
