The automorphism group of the Burkhardt quartic is the Burkhardt group U4(2) = PSp4(3), a simple group of order 25920, which is isomorphic to a subgroup of index 2 in the Weyl group of E6.
Anders Björner and Francesco Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, vol.
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As A. J. Coleman says, "He exhibited the characteristic equation of the Weyl group when Weyl was 3 years old and listed the orders of the Coxeter transformation 19 years before Coxeter was born."