When you look at a figure $F$ with some symmetries you don't see a group but certain obvious symmetries: reflections with respect to various axes, rotational symmetries centered on various points, or translations, all of them being maps $\phi:\ F\to F$ respecting incidences, distances or colorings present in $F$.
Such a figure $F$ is erected on a "ground set" $E$, e.g. the euclidean plane, and any symmetry $\phi:\ F\to F$ carries with it a bijection of $E$. It follows that such a $\phi$ has an inverse which is also a symmetry.
Now comes the second step, namely the composition $\psi\circ\phi$ of symmetries. It follows from general principles about maps that any finite composition of symmetries $\phi$ is again a symmetry in the sense that it leaves incidences, distances, and colorings invariant, and that the set of all symmetries of $F$ obtained in this way forms a group $S_F\ $.
If $F$ is a simple figure, e.g., a rosetta with a center $O$ and $n$ leaves, then it is easy to describe $S_F\ $: It is either the cyclic group ${\mathbb Z}_n$ of $n$ elements generated by a rotation of ${2\pi\over n}$ around $O$, or it is the dihedral group $D_n$ which in addition contains $n$ reflections.
If $F$ is a complicated figure, say a tessellation of ${\mathbb R}^2$ by equilateral triangles, then you can immediately see a lot of symmetries, but you don't get a quick overview over $S_F$. In particular it is not obvious whether finite products of rotations, reflections, etc. again can be viewed as such "elementary" symmetries (with other pivots or axes), or whether there are new kinds of motions present in $S_F$. It turns out that $S_F$ may contain "glide reflections" which maybe you didn't see at the outset.
What I'm trying to say is that the connection between a figure $F$ with some symmetries and a particular group is not at all trivial. In many treatments of the subject (or of examples, like the symmetries of the $3$-cube) the question whether any product of "obvious" symmetries is again an "obvious" symmetry is not sufficiently addressed.