We investigate the mechanical, electronic, and optical properties of monolayer GeX (X= S, Se, and Te) with γ structure based on density-functional theory calculations. We find that the mechanical anisotropy of γ-GeS is higher than those of γ-GeSe and γ-GeTe, because of its strongest ionic bond. In the unstrained case, γ-GeX is an indirect-gap semiconductor with the Mexican-hat (MH) dispersion in the valence band. By applying tensile and biaxial strains, both energy band gap and valley positions are controlled. On the other hand, by applying an external electric field, the energy band gap is decreased to zero because of the downward interlayer band. We also find that the in-plane optical absorptions of monolayer γ-GeX in the visible-light region are comparable to that of the monolayer transition metal dichalcogenides such as MoS2. Due to the unique structure in the z direction, the monolayer γ-GeX also shows a high value of the out-of-plane optical absorptions. The strain engineering significantly modifies the optical absorption in the visible light, while the effect of the external electric field on the optical properties is weak. Our results will be helpful to design the electro-optical devices based on monolayer materials with MH band. Display omitted •Monolayer GeX (X = S, Se, and Te) shows an indirect-gap semiconductor with the Mexican-hat dispersion in the valence band.•Both energy band gap and valley positions can be controlled by applying strains.•Energy band gap is decreased to zero by applying an external electric field.•Strain engineering significantly modifies the optical absorption in visible light.