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Let  V  be a vector space over a field of characteristic  p . In this paper we complete the classification of all irreducible subgroups  G  ofGL  ⁢   (  V  ){\mathrm{GL}(V)}    that contain a  p -element whose Jordan normal form has exactly one non-trivial block, and possibly multiple trivial blocks. Broadly speaking, such a group acting primitively is a classical group acting on a symmetric power of a natural module, a 7-dimensional orthogonal group acting on the 8-dimensional spin module, a complex reflection group acting on a reflection representation, or one of a small number of other examples, predominantly with a self-centralizing cyclic Sylow  p -subgroup.

Groups with a p -element acting with a single non-trivial Jordan block on a simple module in characteristic p