In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 336 automorphisms if orientation may be reversed. As such, the Klein quartic is the Hurwitz surface of lowest possible genus; see Hurwitz's automorphisms theorem. Its (orientation-preserving) automorphism group is isomorphic to PSL(2,7), the second-smallest non-abelian simple group. The quartic was first described in (Klein 1878b).
Klein's quartic occurs in many branches of mathematics, in contexts including representation theory, homology theory, octonion multiplication, Fermat's last theorem, and the Stark–Heegner theorem on imaginary quadratic number fields of class number one; see (Levy 1999) for a survey of properties.
Read more about Klein Quartic: Closed and Open Forms, As An Algebraic Curve, Quaternion Algebra Construction, Tiling, 3-dimensional Models, Dessin D'enfants, Related Surfaces