I'm currently working with a geometric object that has a symmetry group isomorphic to the direct sums of three cyclic groups of order two, i.e., $$\mathbb{Z}_2\oplus\mathbb{Z}_2\oplus\mathbb{Z}_2,$$ so it is generated by three reflections.
I'm wondering if any other, well-known and well-studied objects have this as their symmetry group as well?
Thank you.
A rectangular box will do.
The points $(a,b,c,d),(a,b,d,c),(a,b,-c,-d),(a,b,-d,-c)$ form a rectangle in the $c-d$ plane, as four points on a hyperbola $cd=constant$.
Another four points happen with $a$ and $b$ swapped; the vector from one rectangle to the other is $(b-a,a-b,0,0)$ Which is perpendicular to the rectangles.
The set $V=\{(w,x,y,z)|w=a,x=b\}$ is a plane in $\mathbb{R}^4$.
The intersection of $V$ with $S$ is a region bounded by the hyperbola $yz=ab$.
This hyperbola has the symmetries of a rectangle. It only has the symmetries of a square if $ab=0$.