I don't know what you mean by "syntax group name"? Do you mean "scope"? In which case you could take a look into ScopeHunter available via Package Control.
If you just want to display the current scope, rather than print or copy, then use Ctrl-Alt-Shift-P to display it in the status bar.
"I'm here to save your life. But if I'm going to do that, I'll need total uninanonynymity." Me Myself & Irene.