Is it permissible for a girl to go out with her brother, with her mother's permission, in her father's absence, especially if her father never allows her to leave the house at all, and this causes her psychological distress?

Obeying one's father is among the greatest acts of devotion and the best deeds in the sight of Allah Almighty. Allah has linked His right to the rights of parents in the Holy Quran, as in His saying: "And your Lord has decreed that you not worship except Him, and to parents, good treatment" and His saying: "Worship Allah and do not associate anything with Him, and to parents, good treatment." It is obligatory, based on the Prophet's (peace be upon him) saying: "The pleasure of Allah is in the pleasure of the father, and the displeasure of Allah is in the displeasure of the father." Disobeying parents is one of the greatest major sins.

Therefore, it is obligatory to obey your father in his insistence on not going out to see friends, especially since a woman's staying in her home is better for her, as in His saying: "And abide in your houses and do not display yourselves as [was] the display of the former times of ignorance." Thus, it is not permissible to disobey his command unless he consents and permits it.