You might want to try Karl Rosenfeld or Felix Crousse, they are red flowering doubles that look a lot like that one and should be easy to find. I don't think you can take a cutting from it, but a section of the root could be dug. Depends on the size of the root as to whether it would bloom next year or not. If you purchased from Walmart or Lowe's, it is not likely-usually theirs are cheap, but often small. Peonies can take a couple of years to bloom from a small root. Plant in sun, and be sure to keep the root close to the surface of the soil. If they are planted more than an inch or 2 down, they just will not bloom. Peonies can seem particular, but once they are settled you will have a plant from now on. They are very hardy & long lived, and require very little care, other than to cut back the foliage in the fall. The foliage can look pretty ratty in late summer though. I like to plant them behind other plants that get a bit taller later in the year. Double flowering forms also tend to have very heavy flowers, so may need staking. I love peonies & tend to get a new one every year.

You certainly wouldn't want to trust digging up her plant just so you can take a part of it by dividing.
That would cost her of that season's bloom...and possibly the next as well.

Division though is the only method by which peony is further propagated.
Any nursery that sells peony will be happy to advise you on the many varieties. And don't just plant one...or two.....plant many.