Find the centralizer of each element of the dicyclic group of order $4n$.
My try:
Consider the dicyclic group given by $Q_{4n}=\{\langle a, b\rangle : a^{2n}=b^4=1, ab=ba^{-1},a^2=b^2\}$. The elements of $Q_{4n}$ are $\{a,a^2,a^3,\ldots, a^{2n}=1, ba,ba^2, \ldots, ba^{2n}=b\}$.
Now the center of $Q_{4n}$ is $Z(Q_{4n})=\{1,a^n\}$.
Centralizer of $a^i=C(a^i)=\{1, a,a^2,\ldots, a^{n-1}\}.$
I think that Centralizer of $ba^i=C(ba^i)=\{1, ba^i, ba^{i+n},a^n\}.$
I am stuck on $C(ba^i)$. I can see that $\{1, ba^i, ba^{i+n},a^n\}\subseteq C(ba^i)$ but I don't understand how to find $C(ba^i)$.
Is there any way to find the centralizer apart from checking each element of $Q_{4n}$?
Can someone please help me to find $C(ba^i)$? Any help will be greatly appreciated.