Groups of symmetries What are the symmetries of a solid rectangular box whose length, width and height are all different?
I get a group of order 4 by rotation 180, flipping along a vertical and horizontal axis and itself.

Your contention that you'll have cyclic subgroup of order $4$ is quite natural but needs a little rethinking. Since, you consider the symmetries of a mattress like object (where symmetry is to mean any rigid motion in 3-space which will move a copy of the mattress in any fashion and place the copy back on the original), a rotation of a "scalene" object by $\frac{\pi}{2}$ will not keep this object in the same place. So, rotation by $\pi$ is a valid symmetry.
Technically, the mattress group is called the Klein $4$-group.
$\hskip 2.5in$ 
$H$ stands for a horizontal flip about the axis parallel to $13$ through the midpoint of $12$; $V$ for a vertical flip about the axis parallel to $12$ and through the mid point of $13$ and $R$ for rotation through the mid-point of the mattress.
