Make use of the connection you already have with them: your best friend!
Ask your best friend first about your intentions and desire to become closer to her friends. The fact that she was the one to invite you in the first place for hangout indicates that she is likely to welcome the fact that you are interested in getting to know them better. She may help you sort out the logistics of hangouts.
Another reason for asking your best friend/connection is to prevent any possible misunderstanding when you do end up spending more time with her friend group. Although your description of her makes this unlikely, it is possible she might become jealous or confused when you choose to hang out with a friend group without keeping them in the loop.
How to actually contact them
Next time you meet this friend group, take the courage to ask them about exchanging contact information. If you feel awkward about a simple, direct approach of saying:
"Hey, I really enjoy spend time with you (you guys). Can I have your number so we can hang out more often?"
Then I suggest you to use some sort of a transition or crutch during an ongoing conversation. For example:
Friend A: "I have never tried bubble tea but a lot of people told me they like it."
You: "Oh, I think you will enjoy it! I know a good spot downtown. Do you want to go there and try it out sometime? You can text me at xxx-xxxx and I will add you so we can set a time up soon."
As you mentioned in comments, you have a few from this friend group on Instagram which allows direct message depending on the privacy settings. If you are shy about saying this out loud in person, using social media to contact them can be useful.
After all, I want to re-emphasize the former section about utilizing your best friend as a great resource in breaking the ice and opening your path to become friends with her friend group. I feel this will make the whole process easier for everyone involved.
Source: I have been in a similar situation of wanting to know people better in someone's friend group. For my case, it was my SO's friends from her distant hometown which made the logistics difficult. I asked my SO to relay my intentions to them because I did not have the contact information of her friends. This worked well since she was actually worried about my impression of them - she gladly let her friends know, and we made time to hang out when it was convenient for us as in passing by each other's area, for example.
