Geodesic
Two Nonconjugate Embeddings of the Group $S_3\times\mathbb Z_2$ into the Cremona Group
Informatics and Automation, 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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     author = {V. A. Iskovskikh},
     title = {Two {Nonconjugate} {Embeddings} of the {Group} $S_3\times\mathbb Z_2$ into the {Cremona} {Group}},
     journal = {Informatics and Automation},
     pages = {105--109},
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     volume = {241},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2003_241_a4/}
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V. A. Iskovskikh. Two Nonconjugate Embeddings of the Group $S_3\times\mathbb Z_2$ into the Cremona Group. Informatics and Automation, Number theory, algebra, and algebraic geometry, Tome 241 (2003), pp. 105-109. http://geodesic.mathdoc.fr/item/TRSPY_2003_241_a4/