Deformation rings which are not local complete intersections

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We study the inverse problem for the versal deformation rings R(Γ,V) of finite dimensional representations V of a finite group Γ over a field k of positive characteristic p. This problem is to determine which complete local commutative Noetherian rings with residue field k can arise up to isomorphism as such R(Γ,V). We show that for all integers n ≥ 1 and all complete local commutative Noetherian rings W with residue field k, the ring W[[t]]/(pⁿ t,t²) arises in this way. This ring is not a local complete intersection if pⁿW≠{0}, so we obtain an answer to a question of M. Flach in all characteristics.