Finite group schemes of essential dimension one are exactly those that act generically freely on the projective line, with explicit classification for infinitesimal groups.
II: Groupes de type multiplicatif, et s tructure des sch´ emas en groupes g´ en´ eraux, S´ eminaire de G´ eom´ etrie Alg´ ebrique du Bois Marie 1962/64 (SGA 3)
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Finite group schemes of essential dimension one
Finite group schemes of essential dimension one are exactly those that act generically freely on the projective line, with explicit classification for infinitesimal groups.