Create Plane Figures w/ Non-Isomorphic 12 Element Symmetry Groups

[MikeDietrich]

Homework Statement

Draw two plane figures, each having a 12 element group of symmetries, such that the two groups are NOT isomorphic. Demonstrate that they are not isomorphic.

Homework Equations

I know that every finite group of isometries of the plane is isomorphic to either Z_n or to the dihedral group D_n.

The Attempt at a Solution

I drew a regular hexagon (D_6) but now I am stuck as to what to draw for a figure to represent Z_12. Would a 12 bladed windmill (pinwheel) type shape with pronged ends work?

 

[ystael]

Yes, this will work, since it is not preserved by a reflection.