  We calculate the group of dualization operations for triple vector bundles,
showing that it has order 96 and not 72 as given in Mackenzie's original
treatment. The group is a nonsplit extension of S4 by the Klein group.
Dualization operations are interpreted as functors on appropriate categories
and are said to be equal if they are naturally isomorphic. The method set out
here will be applied in a subsequent paper to the case of n-fold vector
bundles.
