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Symmetry

The regular pentagon has Dih symmetry, order 10. Since 5 is a prime number there is one subgroup with dihedral symmetry: Dih, and 2 cyclic group symmetries: Z, and Z.

These 4 symmetries can be seen in 4 distinct symmetries on the pentagon. John Conway labels these by a letter and group order. Full symmetry of the regular form is r10 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g5 subgroup has no degrees of freedom but can be seen as directed edges.