The group is one of the two groups of order 10. Unlike the cyclic group , is non-Abelian. The molecule ruthenocene belongs to the group , where the letter indicates invariance under a reflection of the fivefold axis (Arfken 1985, p. 248).
has cycle index given by
Its multiplication table is illustrated above.
The dihedral group has conjugacy classes , , , and . It has 8 subgroups: , , , , , , , and , of which , , and , , , , , , , , , are normal.