Why is the number of the conjugacy classes of $D_4 \rtimes Aut(D_4)$ $16$?

Currently, I am reading Representation Theory of Semi-Direct Products by Reyes.

In section $6$, the author mentions that the number of the conjugacy classes of $D_4 \rtimes Aut(D_4)$ is $16$. My question is why $16$?

I don't even know where to start.

• If it helps with searching, the group $G \rtimes \operatorname{Aut}(G)$ is called the holomorph of $G$; so you want to know about the holomorph of dihedral groups (something I know nothing about, personally). – pjs36 Jan 10 '17 at 0:02
• Since I'd like to know more about these things, I'm giving it a shot. I'm starting by figuring out what $\operatorname{Aut}(D_4)$ is (the internet knows) to find the order of $\operatorname{Hol}(G)$; then, I'd like to know what the inverse of $(g, \varphi) \in \operatorname{Hol}(G)$ is so I can start conjugating. I'm sure there are results about $\operatorname{Hol}(G)$, but in lieu of knowing such things, one can always compute -- it's pretty interesting so far. – pjs36 Jan 10 '17 at 7:35
• @pjs36, well, $Aut(D_n) = C_n \rtimes_\phi Aut(C_n)$. – Omar Shehab Jan 10 '17 at 7:38