Geodesic
Two Nonconjugate Embeddings of the Group $S_3\times\mathbb Z_2$ into the Cremona Group
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and algebraic geometry, Tome 241 (2003), pp. 105-109
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A concrete example of two isomorphic nonconjugate subgroups in the two-dimensional Cremona group is presented. 
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V. A. Iskovskikh. Two Nonconjugate Embeddings of the Group $S_3\times\mathbb Z_2$ into the Cremona Group. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and algebraic geometry, Tome 241 (2003), pp. 105-109. http://geodesic.mathdoc.fr/item/TM_2003_241_a4/

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