  In this paper, we present a general construction of 3-transposition groups as
automorphism groups of vertex operator algebras. Applying to the moonshine
vertex operator algebra, we establish the Conway-Miyamoto correspondences
between Fischer 3-transposition groups $\mathrm{Fi}_{23}$ and
$\mathrm{Fi}_{22}$ and $c=25/28$ and $c=11/12$ Virasoro vectors of subalgebras
of the moonshine vertex operator algebra.
