  In this paper we present the full NLO QCD + NLO EW corrections to the
$Z$-boson pair production in association with a hard jet at the LHC. The
subsequent $Z$-boson leptonic decays are included by adopting both the naive
NWA and MadSpin methods for comparison. Since the $ZZ+{\rm jet}$ production is
an important background for single Higgs boson production and new physics
search at hadron colliders, the theoretical predictions with high accuracy for
the hadronic production of $ZZ+{\rm jet}$ are necessary. We present the
numerical results of the integrated cross section and various kinematic
distributions of final particles, and conclude that it is necessary to take
into account the spin correlation and finite width effects from the $Z$-boson
leptonic decays. We also find that the NLO EW correction is quantitatively
nonnegligible in matching the experimental accuracy at the LHC, particularly is
significant in high transverse momentum region.
