What is the name of the ethnic group seeking an independent state of their own

An ethnic group seeking an independent state of its own is called a separatist group.

Explanation:

There are many separatist groups functional around the world. Most separatist groups have a rigid belief that their culture or their ethnicity is not given enough respect staying within the boundaries of the nation. That being one of the major reasons, these groups demand a separate state for themselves and even run local campaigns to gain support from the local people and to give a louder voice to their campaigns.