<P> If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumscribing circle between B and C, then PE + PF = PA + PB + PC + PD . </P> <P> The regular hexagon has Dih symmetry, order 12 . There are 3 dihedral subgroups: Dih, Dih, and Dih, and 4 cyclic subgroups: Z, Z, Z, and Z . </P> <P> These symmetries express 9 distinct symmetries of a regular hexagon . John Conway labels these by a letter and group order . r12 is full symmetry, and a1 is no symmetry . d6, a isogonal hexagon constructed by four mirrors can alternate long and short edges, and p6, an isotoxal hexagon constructed with equal edge lengths, but vertices alternating two different internal angles . These two forms are duals of each other and have half the symmetry order of the regular hexagon . The i4 forms are regular hexagons flattened or stretched along one symmetry direction . It can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites . g2 hexagons, with opposite sides parallel are also called hexagonal parallelogons . </P> <P> Each subgroup symmetry allows one or more degrees of freedom for irregular forms . Only the g6 subgroup has no degrees of freedom but can seen as directed edges . </P>

How many lines of symmetry does a hexagon has