Sp(1)
Intuitive
$Sp(1)$ is the group of quaternions of unit length, also known as the compact symplectic group of order one, hence the name $Sp(1)$. (See https://math.ucr.edu/home/baez/symplectic.html for an explanation of the name.)
The group $Sp(1)$ is isomorphic to $SU(2)$.
Concrete
Representations
The diagram below shows the defining representation of $Sp(1)$ in its upper branch and the (3-dimensional) adjoint representation of the same group in the lower branch. The latter representation can be used to rotate objects in 3D space and has applications in computer graphics. For a more detailed explanation of this diagram and representations of other Lie groups see Fun with Symmetry.
Abstract
The motto in this section is: the higher the level of abstraction, the better.