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.

More things to try:

Weisstein, Eric W. "Dihedral Group D_5." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DihedralGroupD5.html