Geodesic
Representations of dihedral groups over a field of characteristic 2
Sbornik. Mathematics, Tome 25 (1975) no. 1, pp. 58-68 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper there is given a complete classification of the representations of the group $(2)*(2)$ (where $*$ denotes the free product) and the finite dihedral groups over a field of characteristic 2. These results follow from the solution of the problem of classifying to within similarity pairs of matrices $A,B$ (over a field) for which $A^2=0$, $B^2=0$. Bibliography: 5 titles. 
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V. M. Bondarenko. Representations of dihedral groups over a field of characteristic 2. Sbornik. Mathematics, Tome 25 (1975) no. 1, pp. 58-68. http://geodesic.mathdoc.fr/item/SM_1975_25_1_a2/

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[2] V. A. Bashev, “Predstavlenie gruppy $Z_2\times Z_2$ v pole kharakteristiki 2”, DAN SSSR, 141:5 (1961), 1015–1018 | Zbl

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[5] L. A. Nazarova, A. V. Roiter, V. V. Sergeichuk, V. M. Bondarenko, “Primenenie modulei nad diadoi dlya klassifikatsii konechnykh $p$-grupp, obladayuschikh abelevoi podgruppoi indeksa $r$, i par vzaimno annuliruyuschikh operatorov”, Zapiski nauch. seminarov LOMI, 28 (1972), 69–92 | MR | Zbl