I. INTRODUCTION Although the standard model (SM) is a very successful theory, there are still some mysteries and problems. One of them is the flavor structure of fermions. We currently know that quarks mix with each other through the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1] and that there is a hierarchy among the three mixing angles of the CKM matrix: θ 12 ≃ λ ≃ 0 . 22, θ 23 ≃ λ 2 and θ 13 ≃ λ 4 [2]. Yet another hierarchy exists among quark masses and it can be expressed in terms of λ as m$_{u}$/m$_{c}$ ≃ m$_{c}$/m$_{t}$ ≃ λ 4 and m$_{d}$/m$_{s}$ ≃ m$_{s}$/m$_{b}$ ≃ λ 2 [3]. In particular, θ 12 and √ m$_{d}$/m$_{s}$ surprisingly coincide with each other. However, the origin of the hierarchies is still unclear. This is because the Yukawa sector in the SM contains a huge number of unknown parameters. Consequently, fermion masses and mixing angles remain free parameters in the SM. Thus, it is interesting to extend the SM with a family symmetry. In Ref. [4], authors introduce a family U (1) symmetry and try to explain the quark mass hierarchies and small mixing angles via hierarchically suppressed nonrenormalizable Yukawa interactions. A finite group could also be the prime candidate of a family symmetry, which can reduce the number of parameters in the Yukawa sector and make the theory predictive [5–9]. Recently, Friedberg and Lee proposed a (hidden) translational family symmetry called Friedberg-Lee (FL) symmetry [10] and tried to relate quark mixing angles with the sym- metry. A more detailed analysis is given in Ref. [11] and an application to the fourth generation quark model is discussed in Ref. [12]. Furthermore, many attempts for the lepton sector are performed in Refs. [13, 14], some possible origins of the symmetry have been discussed in Ref. [15], and a new spontaneous CP violation mechanism with the symmetry is considered in Ref. [16]. Moreover, one of the motivations to consider the FL symmetry is that the smallness of up- and down-quark masses as well as the electron mass can be naturally understood as it always makes one of three generation fermions massless [10, 14]. However, the original approach [10] with the FL symmetry cannot explicitly reveal the hierarchies of quark masses and mixing angles, which are treated as input parameters. In this paper, we extend the discussion to connect both quark masses and mixing angles with the FL symmetry. In particular, as will be explained in the next section, we propose the FL symmetry which translates the three generation quarks hier- archically and also show that the hierarchical patterns can be responsible for the quark 2