Let q and p be prime with q = a 2 + b 2 ≡ 1 (mod 4), a ≡ 1 (mod 4), and p = qf + 1. In the nineteenth century Cauchy ( Mém. Inst. France 17 (1840) , 249–768) and Jacobi ( J. für Math. 30 (1846) , 166–182) generalized the work of earlier authors, who had determined certain binomial coefficients (mod p) (see H. J. S. Smith, “Report on the Theory of Numbers,” Chelsea, 1964), by determining two products of factorials given by Π k kf! (mod p = qf + 1) where k runs through the quadratic residues and the quadratic non-residues (mod q), respectively. These determinations are given in terms of parameters in representations of p h or of 4 p h by binary quadratic forms. A remarkable feature of these results is the fact that the exponent h coincides with the class number of the related quadratic field. In this paper C. R. Mathews' ( Invent. Math. 54 (1979) , 23–52) recent explicit evaluation of the quartic Gauss sum is used to determine four products of factorials (mod p = qf + 1, q ≡ 5 (mod 8) > 5), given by Π k kf! where k runs through the quartic residues (mod q) and the three cosets which may be formed with respect to this subgroup. These determinations appear to be considerably more difficult. They are given in terms of parameters in representations of 16 p h by quaternary quadratic forms. Stickelberger's theorem is required to determine the exponent h which is shown to be closely related to the class number of the imaginary quartic field Q( i√2 q + 2 a√ q), q = a 2 + b 2 ≡ 5 (mod 8), a odd.