I'm so sorry your DD is going through this. If it didn't bother her, that wouldn't be as big of an issue but she doesn't seem to be enjoying herself
Have you considered putting her in an extra day? Maybe the more regular routine would make her able to engage more with other children, and they would be more familiar with her too? We started DS1 off at daycare for 2 days and he hated it. We switched to 3 days and it made an instant difference.
from memory, I think my DS1 only had one "friend" in childcare when he was 3 until about 4. He did have a few other friends closer to 4 but some days they were BFFs and other days they couldn't stand each other and he would come home saying "such-and-such isn't my friend" and act really sad. He's turning 5 this weekend and still at the same centre with mostly the same kids and he has lots of friends now.
Regarding the carers at the centre, I would make a really big point of wanting your DD to make friends and socialise more and I would ask for a detailed run down of her day at every pick up. Try to pick her up at a time that isn't too busy.
I'm not sure if this is standard practice but at our boys' centre, they know who everyone's friends are and can tell us who they played with, what they did, and whether there were any issues with toilet, food, sleep, play etc. I'm not sure if it would be considered a big centre.. They have 2 rooms and about 25 kids in each room.
We also have pockets for parents to get notices, invitations and letters etc. you could maybe organise a little play date at a park or even if she has her bday party coming up, invite everyone and then hope enough people come to be able to get contact details and make connections.
Good luck x