Element structure of unitriangular matrix group of degree three over a finite discrete valuation ring

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This article gives specific information, namely, element structure, about a family of groups, namely: unitriangular matrix group of degree three.
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This article describes the element structure of the unitriangular matrix group of degree three over a finite discrete valuation ring. It builds on the discussion at element structure of unitriangular matrix group of degree three over a finite field.

We assume that the residue field has size q and characteristic p, with r = \log_pq. We denote by l the length of the discrete valuation ring. The size of the ring is thus q^l = p^{rl} and the order of the field is q^{3l} = p^{3rl}.