  The procedure that was used in an earlier paper on the generalized Ces\`{a}ro
matrices of order two is adapted here to show that generalized Ces\`{a}ro
matrices of order three are supraposinormal on $\ell^2$. This leads to
information about their posinormality, coposinormality, and hyponormality. The
reader is then invited to formulate a conjecture regarding generalized
Ces\`{a}ro matrices of other orders.
