We establish a new group-theoretic realization of
the basic representations of the twisted affine and twisted toroidal algebras
of ADE types in the same spirit of our new approach
(with I. Frenkel and W. Wang)
to the McKay correspondence
Our vertex operator construction provides
a unified description to the character tables for
the spin cover of the wreath product of
the twisted hyperoctahedral groups and an arbitrary finite group.
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