From the following image I know that the first of group is the symmetric group of rank $n$ and the second is known as the Hyperoctahedral group.
I want to know if someone knows the name of the third family of groups. Wikipedia does not have links to see anything about this groups. But I'm pretty sure that it's stored in the web just with the original name.
I've worked extensively with these groups and mostly I just refer to them by a symbol or casually say "type D." I've used "demihyperoctahedral group" before, and I believe I saw it in the literature, but I don't recall where. I don't think anyone will tell you you're wrong if you use that name.