How can I get this relationship off the ground?

I've met a girl through a friend who I like and I think she likes me. After we met for the first time my friend asked us both what we thought of each other, obviously doing some match making, but she didn't need to because I had already decided I wanted to get to know her better. She said that she would like to see more of me but perhaps a date was a little too formal and maybe we should just hang out with our mutual bunch of friends and get to know each other that way. Well I met her again last weekend when we were out for our mutual friend's birthday and we got on brilliantly. I'd love to see more of her but she doesn't really come out with my group of friends regularly so I'm heavily reliant on my friend inviting her out. I thought about asking my friend for her number and inviting her out myself but am worried that this may be seen as a little too forward, like I was pushing for that date. I've been guilty of appearing too keen with women in the past which has put them off so I really don't want to make the same mistake again but she's in my head 24/7 and it's driving me crazy.