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We study the action of the Klein simple group G consisting of 168 elements ontwo rational threefolds: the three-dimensional projective space and a smoothFano threefold X of anticanonical degree 22 and index 1. We show that theCremona group of rank three has at least three non-conjugate subgroupsisomorphic to G. As a by-product, we prove that X admits a Kahler-Einsteinmetric, and we construct a smooth polarized K3 surface of degree 22 with anaction of the group G.

Three embeddings of the Klein simple group into the Cremona group of rank three