Geodesic
On a~method of constructing $C$-groups
Izvestiya. Mathematics , Tome 59 (1995) no. 4, pp. 765-783

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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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     author = {Yu. V. Kuz'min},
     title = {On a~method of constructing $C$-groups},
     journal = {Izvestiya. Mathematics },
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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/