  Non-Hermitian quasicrystal forms a unique class of matter with
symmetry-breaking, localization and topological transitions induced by gain and
loss or nonreciprocal effects. In this work, we introduce a non-Abelian
generalization of the non-Hermitian quasicrystal, in which the interplay
between non-Hermitian effects and non-Abelian quasiperiodic potentials create
mobility edges and rich transitions among extended, critical and localized
phases. These generic features are demonstrated by investigating three
non-Abelian variants of the non-Hermitian Aubry-Andr\'e-Harper model. A unified
characterization is given to their spectrum, localization, entanglement and
topological properties. Our findings thus add new members to the family of
non-Hermitian quasicrystal and uncover unique physics that can be triggered by
non-Abelian effects in non-Hermitian systems.
