Table 1.

Isotropy subgroups of the actor-hoop configurations.

Geometry	Hoop symmetry	Actor symmetry	Actor-hoop symmetry (isotropy subgroup)
Triangle	D3	S3	D3 (or S3), Z3, D1 (or Z2), I
Square	D4	S3	D1 (or Z2), I
Pentagon	D5	S3	D1 (or Z2), I
Hexagon	D6	S3	D3 (or S3), Z3, D1 (or Z2), I