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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 ().

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  • Kleinsche Quartik
  • Klein quartic
  • Quartique de Klein
  • 클라인 4차 곡선
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  • En géométrie hyperbolique, la quartique de Klein, du nom du mathématicien allemand Felix Klein, est une surface de Riemann compacte de genre 3. Elle a le groupe d'automorphismes d'ordre le plus élevé possible parmi les surfaces de Riemann de genre 3, à savoir le groupe simple d'ordre 168. La quartique de Klein est en conséquence la (en) de genre le plus bas possible.
  • 대수기하학에서, 클라인 4차 곡선(Klein4次曲線, 영어: Klein’s quartic curve)은 종수 3의 리만 곡면 가운데 가장 대칭적인 것인 모듈러 곡선이다.
  • Die Kleinsche Quartik ist eine Kurve 4. Grades in der komplexen projektiven Ebene, die in homogenen Koordinaten durch die Gleichung gegeben ist. Sie wurde 1879 durch Felix Klein eingeführt und besitzt außergewöhnliche Symmetrieeigenschaften. Der algebraischen Kurve entspricht eine Riemannsche Fläche. Betrachtet man die hyperbolische Ebene als komplexe obere Halbebene , auf der wirkt, so ist die Riemannsche Fläche zur Klein-Quartik gegeben durch mit der Kongruenzuntergruppe Sie ist eine Modulkurve (mit Geschlecht 3 und 24 Spitzen, siehe Modulform).
  • 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 ().
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