What is the group $O(4)/H$?
Here $O(4)$ is the group of orthogonal matrices and H is the center of $O(4)$.
What is the group $O(4)/H$? Here $O(4)$ is the group of orthogonal matrices and H is the center of $O(4)$. 


This group is isomorphic to $(SO(3) \times SO(3)) \rtimes \{\pm 1\}$, as is discussed in this question and its comments. (The order two factor comes from the fact that $O(4) = SO(4) \rtimes \{\pm 1\},$ and that $H \subset SO(4)$.) [Added as per Scott Carnahan's comment below: The order two factor shouldn't be a direct factor, but a semidirect factor. I leave it as an exercise to determine its action on $SO(3) \times SO(3)$. If someone reading this knows the correct LaTeX command for a semidirect product, feel free to make the appropriate edit.] 

