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This is the first in a series of papers in which I investigate the orbifoldsof permutation-type as candidates for {\it new physical string systems} atmultiples of critical closed-string central charges. Examples include thebosonic orientation orbifolds, the bosonic permutation orbifolds and others, aswell as superstring extensions. In this paper I use the extended (twisted)Virasoro algebras of these orbifolds to construct the corresponding extendedaction formulations of the twisted open- and closed-string sectors of all thebosonic orbifolds at $\hatc=26K$. The extended actions exhibit a large set of{\it new twisted world-sheet gravities}, whose extended diffeomorphism groupsclearly indicate that the associated operator-string theories can be free ofnegative- norm states at higher central charge.Comment: 54 page

The orbifolds of permutation-type as physical string systems at multiples ofc= 26: I. Extended actions and new twisted world-sheet gravities