ON CONJUGACY CLASSES OF THE KLEIN SIMPLE GROUP IN CREMONA GROUP

We consider countably many three-dimensional PSL2(F-7)-del Pezzo surface fibrations over P-1. Conjecturally, they are all irrational except two families, one of which is the product of a del Pezzo surface with P-1. We show that the other model is PSL2(F-7)-equivariantly birational to P-2 x P-1. Based on a result of Prokhorov, we show that they are non-conjugate as subgroups of the Cremona group Cr-3(C).