twylah, I had to check your location. Your description sounds identical to a family I worked with. In their case the six year old was male, not female.
there are three children in the family. The father's daughter from a previous marriage. Sounds like your boyfriend's kids, quiet and compliant. She also was raised knowing she would get hit if she didn't do what Dad said. The boy is Mom's son from a previous marriage. He is very active and strong willed. In working with Mom I helped her learn to encourage the behavior she wanted from the boy and avoid getting into power struggles with him. The step-father is the kind that wants to say jump and have the kid ask "how high" on the way up. I worked with Mom to encourage step-dad to change the way he interacted with the boy. He refused to change. Ultimately he progressed from yelling at the boy until he hit him and child protection had to step in. Please don't let this happen to your daughter.