  Based on the Riemann- and Caputo definition of the fractional derivative we
use the fractional extensions of the standard rotation group SO(3) to construct
a higher dimensional representation of a fractional rotation group with mixed
derivative types. An analytic extended symmetric rotor model is derived, which
correctly predicts the sequence of magic numbers in metal clusters. It is
demonstrated, that experimental data may be described assuming a sudden change
in the fractional derivative parameter $\alpha$ which is interpreted as a
second order phase transition in the region of cluster size with $200 \leq N
\leq 300$.
  Furthermore it is demonstrated, that the four different realizations of
higher dimensional fractional rotation groups may successfully be connected to
the four fundamental interaction types realized in nature and may be therefore
used for a prediction of magic numbers and binding energies of clusters with
gravitational force and weak force respectively bound constituents.
  The results presented lead to the conclusion, that mixed fractional
derivative operators might play a key role for a successful unified theoretical
description of all four fundamental forces realized in nature.
