Conversation so far:
What is the more progressive Amish group called?
The more progressive Amish members, comprising roughly two-thirds of the group, became known by the name Amish Mennonite.
Other more traditionally minded Amish groups name is?
The more traditionally minded Amish groups became known as the Old Order Amish.
What is the amish old order called now?
In the late 1920s, the more change minded faction of the Old Order Amish broke away from the mainstream and organized under the name Beachy Amish.
What is another Amish order besides the Beachy Amish?
Another Amish order besides Beachy Amish is Old Beachy Amish.

Question: What is this faction known for
Self-contained rewrite: What is the Old beachy amish faction known for?