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Recently, hexagonal image processing has attracted attention. The hexagonal lattice has several advantages in comparison with the rectangular lattice, the conventionally used lattice for image sampling and processing. For example, a hexagonal lattice needs less sampling points; it has better consistent connectivity; it has higher symmetry; and its structure is plausible to human vision systems. The multiresolution analysis method has been used for hexagonal image processing. Since the hexagonal lattice has high degree of symmetry, it is desirable that the hexagonal filter banks designed for multiresolution hexagonal image processing also have high order of symmetry, which is pertinent to the symmetry structure of the hexagonal lattice. The orthogonal or prefect reconstruction (PR) hexagonal filter banks that are available in the literature have only threefold symmetry. In this paper, we investigate the construction of orthogonal and PR finite impulse response (FIR) hexagonal filter banks with sixfold symmetry. We obtain block structures of 7-size refinement (seven-channel two-dimensional) orthogonal and PR FIR hexagonal filter banks with sixfold rotational symmetry. radic7-refinement orthogonal and biorthogonal wavelets based on these block structures are constructed. In this paper, we also consider FIR hexagonal filter banks with axial (line) symmetry, and we present a block structure of FIR hexagonal filter banks with pseudo-sixfold axial symmetry.