  In this article, we study Griess algebras and vertex operator subalgebras
generated by Ising vectors in a moonshine type VOA such that the subgroup
generated by the corresponding Miyamoto involutions has the shape $3^2{:}2$ and
any two Ising vectors generate a 3C subVOA $U_{3C}$. We show that such a Griess
algebra is uniquely determined, up to isomorphisms. The structure of the
corresponding vertex operator algebra is also discussed. In addition, we give a
construction of such a VOA inside the lattice VOA $V_{E_8^3}$, which gives an
explicit example for Majorana representations of the group $3^2{:}2$ of 3C-pure
type.
