It's very hard to just randomly join someone else's group because it means that you not only have to be friends with that person but also with the rest of the group. My advice to you is to set small goals and go for this one step at a time. First set a goal that you want to be friends with this girl. Then the next goal is to become friends with maybe two of her other friends. Then finally you can set a goal to be incorporated into the group.
Making friends is not always easy. When I entered college, I had to relearn how to make friends again since I was always with a similar group of pals essentially throughout middle school and high school (some people left the group and some people entered the group over time but it was pretty much the same) thus I never had to make new ones for a long time. What really helped was to of course initiate conversation. Just simply say hi to her and get a conversation going based on the class. For instance ask her how she thought of the homework before class starts. And it's okay if you just randomly go up to her although don't make it appear as if she's your target. Just pretend as if she was just the first person you happen to see and go for it! I know that it's really awkward at first but sometimes the most random people you meet and talk to may end up being your friends.
Once you're friends with her, it won't be as difficult to befriend her friends. You guys may hang out together with your common friend (the girl) and you can try to get to know each other through that.
I can't get into any specific details because each situation is always different. You just have to play it by ear and hope that things go well. And if you end up not being friends with this girl, then try to find someone who sits closer to you in class or who you pass by in the hallways often. It's a bit less random and awkward if you start to talk to people who you are often near anyway. Well I hope this helps you. Good luck!