  We consider the Lyndon-Hochschild-Serre spectral sequence with mod-p
coefficients for a central extension with kernel cyclic of order a power of p
and arbitrary discrete quotient group. For this spectral sequence the second
and third differentials are known, and we give a description for the fourth
differential. Using this result we deduce a similar formula for the Serre
spectral sequence for a principal fibration with fibre the classifying space of
a cyclic p-group. The differential from odd rows to even rows involves a Massey
triple product, so we describe the calculation of such products in the
cohomology of a finite abelian group. As an example we determine the Poincare
series for the mod-3 cohomology of various 3-groups.
  Remarks. 1) My definition of the higher differentials $d_i$ for $i\geq 2$ in
the spectral sequence for a double chain complex differs from the usual one by
a factor of $(-1)^{i+1}$. Both conventions are consistent, but the usual
definition has the advantage of agreeing with the ``obvious'' definition of the
differentials in the spectral sequence for the associated filtered chain
complex. All of the theorems in this paper remain true exactly as stated if the
more usual definition of $d_i$ is taken.
  2) Carles Broto found a small mistake in this paper: the result for
fibrations with fibre the classifying space of a cyclic group is stated for
arbitrary fibrations, although it is only proved for principal fibrations.
Since it is apparent from the first sentence of the proof that only principal
fibrations are being considered, I have not bothered to publish an erratum.
