© 2016 American Physical Society. We study the complex electronic band structure of low In content InGaAs/GaP quantum dots. A supercell extended-basis tight-binding model is used to simulate the electronic and the optical properties of a pure GaAs/GaP quantum dot modeled at the atomic level. Transitions between hole states confined into the dots and several XZ-like electronic states confined by the strain field in the GaP barrier are found to play the main role on the optical properties. Especially, the calculated radiative lifetime for such indirect transitions is in good agreement with the photoluminescence decay time measured in time-resolved photoluminescence in the μs range. Photoluminescence experiments under hydrostatic pressure are also presented. The redshift of the photoluminescence spectrum with pressure is also in good agreement with the nature of the electronic confined states simulated with the tight-binding model.