Dihedralizing the Quaternions

In this paper, we will take the classic dihedral and quaternion groups and explore questions like "what if we replace i=e2πi/4 in Q8 with a larger root of unity?" and "what if we add a reflection to Q8?" The delightful answers reveal lesser-known families like the dicyclic, diquaternion, semidihedral, and semiabelian groups, which come to life with visuals such as Cayley graphs, cycle graphs, and subgroup lattices.