Pedagogical examples of distinguishing "types of symmetry"

Consider the group of symmetries of a brick $P$ (a right-angled parallelepiped with side-lengths $a, b, c$, where $a> b> c$; as an option you can also consider $a> b=c$, let's call it $P'$) and the group of symmetries of a cube $Q$.

Most people will immediately identify reflection symmetries of both. With a bit help from you, they will discover order 2 rotational symmetries of the brick (or an order 4 rotational symmetry of $P'$) and the central symmetry of both $P$ and $P'$. Most will have harder time believing (even with your help) that cube has a rotational symmetry of order 3. Then some even might ask you "how do you know that $P'$ has no order 3 symmetry".

Then you explain that all symmetries of $P$ commute, but they will need your help to see noncommuting symmetries of $P'$ and $Q$. Before explaining the noncommutativity phenomenon, give them the following example: $A=$ put on socks; $B=$ put on shoes. Then demonstrate that $AB\ne BA$.

Then notice that the central symmetries of $P, P'$ and $Q$ commute with all other symmetries. After you are done demonstrating this fact, ask them

"can you think of a highly symmetric solid which has no nontrivial symmetries commuting with the rest of the symmetries?"

(The regular tetrahedron is an answer.)

A uniform triangular prism has as rotational group the dihedral group of order $6$, whereas a right hexagonal pyramid with regular hexagonal base has a cyclic rotational group of order $6$.

I made this answer community wiki in the hope someone nice will add some images for illustration, as this question deserves it (and remove this comment).