This paper employes variable function method and the technique of conformal mapping to discuss the anti-plane problem of a circular hole with three unequal cracks in a one-dimensional (1D) hexagonal piezoelectric quasicrystal. Based on the piezoelectricity fundamental equations of quasicrystal materials and the symmetry of 1D hexagonal quasicrystal and its linear piezoelectricity effect, 1D hexagonal quasicrystal control equations of anti-plane problem are derived. Applying Cauchy integral formula, the analytical expressions for the crack tip filed intensity factors are presented with the assumption that the crack are electrical impermeable and electrical permeable. With the variation of the hole-size and the crack length, some of the new model of crack are obtained. In the absence of the electric load, the results match with the classical ones. The numerical results indicate the effects of geometric parameters on the field intensity factors. It is verified that the horizontal crack length and the circle radius can easily promote crack growth. Research on such issues will provide reliable theoretical value for the engineering materials preparation and application.