  Family gauge boson production at the LHC is investigated according to a
$U(3)$ family gauge model with twisted family number assignment. In the model
we study, a family gauge boson with the lowest mass, $A_1^{\ 1}$, interacts
only with the first generation leptons and the third generation quarks. (The
family numbers are assigned, for example, as $(e_1, e_2, e_3)= (e^-, \mu^-,
\tau^-)$ and $(d_1, d_2, d_3)=(b, d, s) $[or $(d_1, d_2, d_3)=(b, s, d)$]). In
the model, the family gauge coupling constant is fixed by relating to the
electroweak gauge coupling constant. Thus measurements of production cross
sections and branching ratios of $A_1^{\ 1}$ clearly confirm or rule out the
model. We calculate the cross sections of inclusive $A_1^{\ 1}$ production and
$b \bar{b} \, (t \bar{t})$ associated $A_1^{\ 1}$ production at $\sqrt{s} =
14~\text{TeV}$ and $100~\text{TeV}$. With the dielectron production cross
section, we discuss the determination of diagonalizing matrix of quark mass
matrix, $U_{u}$ and $U_{d}$, respectively.
