I'm very confused since I've always seen the Dihedral Group $D_8$ given as
$$D_8 = \langle a, b \mid a^4 = b^2 =1, ab= a^3b\rangle.$$
I'm given the Definition
$$D_8 = \langle a, b \mid a^2 = b^2 = (ab)^4=1\rangle.$$
I want to figure out the elements based on this definition but I don't understand how these definitions differ. I would be incredibly grateful for your help!
Tomato