Dear Raman,
To answer your question, both in a legal and conventional way, very few information you have given us. Even so, I try to answer your query.
1. Whether your factory runs 24x7, then there is no question of a common weekly off day. There will be a rotational weekly off days to group of workers (called staggered weekly Off Days)
2. If you are working 6 days in a week, you can have any day as your common weekly off.
In both the cases, you are not supposed to take permission of Factory Inspector, but you have to notify him.
3. Normally, common weekly off day is decided on the basis of the local customs,usage, bazaar day or the day on which electric supply is generally shut-down. For example, in Pune, Maharashtra, Thursday is a weekly common off day. Hence, if you decide some other day as weekly holiday in Pune, it may not work. You will have to fix the weekly off day as per the prevalent practice in your factory area.
I hope this will clear your questions.