I am working on an exercise which is asking to find the elements of the symmetry group of the following figure given below:

Note that the rectangular sides of the box all have the exact same pattern where the left half is grey and the right half is white.

If I consider $c$ to be the rotation of the figure around the z-axis (axis being perpendicular to the square base and top of the rectangular box) and $b$ to be the reflection on the $x-y$ plane (plane formed by the $x$ and $y$ axis which are both perpendicular to the vertical rectangular sides), I have figured out that $$c^4=e, b^2=e, (bc^2)^2=e$$.

I thought the element of my group would be $${e,c,c^2,c^3,b,bc,bc^2,bc^3}$$ which is the group $D_4$, but knowing that $bc^2bc^2=e$ I can only find $cb=c^3bc^2$ and I cannot find an expression to commute $c$ and $b$. That mean that I am missing elements on the group.

What elements would form my group and which group would it be?