The geometry of the six quaternionic equiangular lines in ${\mathbb{H}}^2$

We give a simple presentation of the six quaternionic equiangular lines in ${\mathbb{H}}^2$ as an orbit of the primitive quaternionic reflection group of order 720 (which is isomorphic to 2.A_6 the double cover of $A_6)$. Other orbits of this group are also seen to give optimal spherical designs (packings) of 10, 15 and 20 lines in ${\mathbb{H}}^2$, with angles { 1/3, 2/3 }, { 1/4, 5/8 } and { 0, 1/3, 2/3 }, respectively. We consider the origins of this reflection group as one of Blichfeldt’s “finite collineation groups” for lines in ${\mathbb{C}}^4$, and general methods for finding nice systems of quaternionic lines.