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 gener
ation 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.
