classification of lower autonilpotent dihedral groups

‎given a group $g$‎, ‎a subgroup $k(g)$ is defined as $k(g)=langle[g,alpha]=g^{-1}g^{alpha}|gin g,alphain aut(g)angle$‎.‎also $k_i(g)$ is defined inductively as $k_i(g)=[g,underbrace{aut(g),...,aut(g)}_{imbox{ times}}]$‎.‎a group $g$ is said to be lower autonilpotent of class n if $k_n(g)=1$ and $k_{n-1}(g)eq1$‎.‎in the present article we determine the structure of automorphism group of dihedral groups and it is shown that lower autonilpotent dihedral group of class n is $d_{2^n}$‎.