friendlymum
I agree with others that it sounds like you and her brothers had the situation under control.
I suspect teachers are aware that your DD is coming to school upset - but I'm not clear whether you or they understand why.
If it is for serious reasons (bullying/ intimidation) at school - raise it immediately with the school (Form Tutor/ Head of Year).
I wonder whether this is just a going into school thing - or whether she's upset all day. It may be worth asking her or her brothers whether she's upset at other times during the school day.
I know eating on your own or being at a loose end (the odd one out) - can be really awkward to the point of being upsetting - but it happens to everybody. The point is not to allow it.
If she isn't upset when she's busy in classes - maybe she needs to be reminded that she can be busy at lunch.
Join lunchtime clubs (helps to make friends).
Get lunch and then ask to sit at a table with someone - don't feel you have to join the conversation, but don't just sit there silently looking miserable either.
At break - get a snack and eat it on the way to the library - then go and see what books they have to read. You may even meet other people there.
I think you need to be working on getting her to attempt more positive behaviour at school-
encouraging her to talk about what she liked at school that day - maybe something as simple as dessert was nice. I liked the joke Mr X told. Get her to find something positive about her day at school each evening.
keep herself busy (so she doesn't feel overwhelmed/ upset)
encourage her to be proactive: introduce herself to people/ say hello/ ask questions - join some clubs.
If the tears are about feeling stressed by the new expectations on her as a student - maybe she needs to be honest with her teachers that she's struggling. Most schools have sessions to support Maths/ English/ MFL - that are at lunch or after school - encourage your DD to go along and get help if that's the problem. She can get the help she needs and may well make friends there too.
HTH