Geodesic
On a method of constructing $C$-groups
Izvestiya. Mathematics, Tome 59 (1995) no. 4, pp. 765-783 Cet article a éte moissonné depuis la source Math-Net.Ru

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The groups in question are the metabelian groups which can be given by defining relations $x^{-1}_\alpha x_\beta x_\alpha=x_\gamma$. We give a characterization and constructions of such groups under the assumption that all generators are conjugate. 
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Yu. V. Kuz'min. On a method of constructing $C$-groups. Izvestiya. Mathematics, Tome 59 (1995) no. 4, pp. 765-783. http://geodesic.mathdoc.fr/item/IM2_1995_59_4_a5/

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[3] Gildenhuys D., “Classification of soluble groups of cohomological dimension two”, Math. Z., 166:1 (1979), 21–25 | DOI | MR | Zbl