PSL(2,7)

The general linear group GL(2, 7) consists of all invertible 2×2 matrices over F7, the finite field with 7 elements.

Then PSL(2, 7) is defined to be the quotient group obtained by identifying I and −I, where I is the identity matrix.

It is a general result that PSL(n, q) is simple for n, q ≥ 2 (q being some power of a prime number), unless (n, q) = (2, 2) or (2, 3).

Note that the classes 7A and 7B are exchanged by an automorphism, so the representatives from GL(3, 2) and PSL(2, 7) can be switched arbitrarily.

G = PSL(2, 7) acts via linear fractional transformation on the projective line P1(7) over the field with 7 elements: Every orientation-preserving automorphism of P1(7) arises in this way, and so G = PSL(2, 7) can be thought of geometrically as a group of symmetries of the projective line P1(7); the full group of possibly orientation-reversing projective linear automorphisms is instead the order 2 extension PGL(2, 7), and the group of collineations of the projective line is the complete symmetric group of the points.

The Klein quartic is the projective variety over the complex numbers C defined by the quartic polynomial It is a compact Riemann surface of genus g = 3, and is the only such surface for which the size of the conformal automorphism group attains the maximum of 84(g − 1).

For the Klein quartic this yields a tiling by 24 heptagons, and the order of G is thus related to the fact that 24 × 7 = 168.

Klein's quartic arises in many fields of mathematics, including representation theory, homology theory, octonion multiplication, Fermat's Last Theorem, and Stark's theorem on imaginary quadratic number fields of class number 1.