Every participant automatically gets an admin party. This isn’t a party you should be using for your workflows. Think of it being a bit like a “root” account in a unix system. It exists, but you normally don’t run applications on it, only the system does.
The party id used is the same identifier as used by the participant. So given that you control the name of the participant, you also control the id of its party.
And you can set a display name for that party too. However, beware that display names are participant local. They are not shared among the participants.
I hope this helps!