Asymmetric pattern: The identity is the only symmetry of P.

Bilateral pattern: The only symmetries of P are the identity and reflection with mirror L.
Example: A human's picture is almost bilateral symmetry.

Radial pattern(Wheel pattern): The only possible symmetries of P are the identity, rotations with center C, and reflections with mirror through C.
C_n ( Cyclic group) and D_n( Dihedaral group) are only possible infinite classes.

Strip pattern:  There are seven types of strip patterns.

Wallpaper pattern: There are seventeen types of wallpaper patterns. 

The set of symmetries of a pattern P is a group because the symmetries of a plane pattern P have the following four group axioms:

1. Identity: The identity isometry is a symmetry of P.

2. Product ( closed ): If the isometries S and T are symmetries of P, then their product isometry ST is a symmetry of P.

3. Inverse: If the isometry S is a symmetry of P, then the inverse isometry S-1 is a symmetry of P.

4. Associativity: If the isometries S, T, U are symmetries of P, then (ST)U = S(TU). 

D₅  has 10 symmetries, i.e., 5 reflections and 5 rotations;

R₀  -------- rotation by 0
R₁  -------- rotation by 72
R₂  -------- rotation by 144
R₃  -------- rotation by 216
R₄  -------- rotation by 288
F₀  -------- reflection  mirror 0
F₁  --------  reflection  mirror 36
F₂  -------- reflection  mirror 72
F₃  -------- reflection  mirror 108
F₄  -------- reflection  mirror 144