  We consider the exchange statistics of vortices, each of which traps an odd
number ($N$) of Majorana fermions. We assume that the fermions in a vortex
transform in the vector representation of the SO(N) group. Exchange of two
vortices turns out to be non-Abelian, and the corresponding operator is further
decomposed into two parts: a part that is essentially equivalent to the
exchange operator of vortices having a single Majorana fermion in each vortex,
and a part representing the Coxeter group. Similar decomposition was already
found in the case with N=3, and the result shown here is a generalization to
the case with an arbitrary odd $N$. We can obtain the matrix representation of
the exchange operators in the Hilbert space that is constructed by using Dirac
fermions non-locally defined by Majorana fermions trapped in separated
vortices. We also show that the decomposition of the exchange operator implies
tensor product structure in its matrix representation.
