How to display specific exposed filters of a view in a block
It's very simple.

Create a new view as a block using the Add button on top of your view admin page (Add > Block).

In your new view Exposed Form section, click on "Exposed form in a block" and press yes. In this way a block of your exposed filters is created in your Blocks admin page (admin/structure/block).

Using the blocks admin page enable your block and place it to the region of your liking.

Doing the above steps, you will get one exposed filter group on top of your view and one exposed filters block wherever you like in your site.