  The Fibonacci cube $\Gamma_n$ is obtained from the $n$-cube $Q_n$ by removing
all the vertices that contain two consecutive 1s. If, in addition, the vertices
that start and end with 1 are removed, the Lucas cube $\Lambda_n$ is obtained.
The number of vertex and edge orbits, the sets of the sizes of the orbits, and
the number of orbits of each size, are determined for the Fibonacci cubes and
the Lucas cubes under the action of the automorphism group. In particular, the
set of the sizes of the vertex orbits of $\Lambda_n$ is $\{k \ge 1;\ k \divides
n\} \cup\, \{k \ge 18;\ k \divides 2n\}$, the number of the vertex orbits of
$\Lambda_n$ of size $k$, where $k$ is odd and divides $n$, is equal to
$\sum_{d\divides k}\mu\left(\frac{k}{d}\right) F_{\lfloor \frac{d}{2}\rfloor +
2}$, and the number of the edge orbits of $\Lambda_n$ is equal to the number of
the vertex orbits of $\Gamma_{n-3}$. Dihedral transformations of strings and
primitive strings are essential tools to prove these results.
