The dihedral group $D_{2n} = \{x, y \mid x^2=y^n=yxyx=1\}$ is tied with the symmetries of the regular polygon on a plane. What is the natural extension to higher dimension? For instance, in $3$D, does the extension correspond to the group of symmetries of a regular polyhedron with $n$ vertices or $n$ edges? What is the representation? For instance, what does the following group correspond to $\{x, y \vert x^3=y^n=(yx)^3=1\}$? I think this again corresponds to some symmetries of a two dimensional object, since we only have two generators. In general, is the number of generators $d$ tied to the dimension of the underlying space of the $d$-dimensional object.
Thanks
I believe that the irreducible representations of the dihedral group are all one and two dimensional, so as far as being the complete group of symmetries of a polyhedron in higher dimensions, I don't think they generalize that way.
However, there are other groups that are sort of like the dihedral groups, that are defined analogously. For instance the Tetrahedral group is the orientation preserving symmetries of a regular tetrahedron in space. It is isomorphic to the alternating group on $4$ letters.
More interesting is the icosahedral group. So interesting that Felix Klein wrote a whole book on it. The regular icosahedron has 12 faces that are pentagons, that meet at 20 vertices of valence three. It is generated by a counterclockwise rotations in a face $a$, vertex $b$ and edge $c$ with relations $a^5=b^3=c^2=e$ and $abc=e$.
It is a member of a family of groups called triangle groups that are symmetry groups in different geometries in dimension two. For instance the icosahedral group has to do with spherical geometry.
I want to recommend two books. The first is a little old fashioned, by Wilhelm Magnus called "Discrete Groups" that I really enjoyed. An even more old fashioned book by Hilbert and Cohn-Vosson called "Geometry and the Imagination" that I learned an awful lot from. These books are a really great introduction to this sort of math.