$D_6$ and cycle notation problem

Question

I have a hexagon with edges $A,B,C,D,E,F$ and I want to work with its symmetry group $D_6$ in cycle notation. My calculations don't yield consistent results.

For example, I correctly get $r^4 \cdot r^3 = (AEC)(BFD) \cdot (AD)(BE)(CF) = (BCDEFA) = (ABCDEF) = r$

But on the other hand, I wrongly get $r^2 \cdot r^3 = (ACE)(BDF) \cdot (AD)(BE)(CF) = (FABCDE) = (ABCDEF) = r$

Where is my error?

Answer 1

The right composition appears to be $$r^2 \cdot r^3 = (ACE)(BDF) \cdot (AD)(BE)(CF) = (AFEDCB) = r^5.$$

• A to C to F
• F to B to E
• E to A to D
• D to F to C
• C to E to B
• B to D to A