"have..[group] ..so designated" means "designate ....[group] as [undesirable]"
So it means the government officials can designate ...without any [prior] trial or open process. That is, declaring a group "undesirable" can be done secretly, without giving any reason.
It does NOT mean that the groups, or those who associate with them, won't have a "trial or open process" after they are arrested (although that might also be the case, but the article does not say that).
I hope this clears that up. The phrase is adverbial to the verb "designate", not modifying the noun "groups".