My child complained about being bullied at school. I feel like complaining to the school but I am good friends with those kid's mums and do not know how will they react or do I talk to them directly?

who is more important here - the mums who are your friends or your child? i understand you don't want to suddenly take a step that could harm those kids in an academic area, but you have to do something about it. please speak to the school, and if required, withhold the names for now, or inform the staff about it but ask them to keep a watch without disclosing anything yet. that way, when the kids are caught bullying, the staff will have direct proof and will take the required action. also, i think if you are friends with the moms you should have a direct talk with them. they would understand if they really are friends.