  We compute the full probability distribution of the spectral form factor in
the self-dual kicked Ising model by providing an exact lower bound for each
moment and verifying numerically that the latter is saturated. We show that at
large enough times the probability distribution agrees exactly with the
prediction of Random Matrix Theory if one identifies the appropriate ensemble
of random matrices. We find that this ensemble is not the circular orthogonal
one - composed of symmetric random unitary matrices and associated with
time-reversal-invariant evolution operators - but is an ensemble of random
matrices on a more restricted symmetric space (depending on the parity of the
number of sites this space is either ${Sp(N)/U(N)}$ or
${O(2N)/{O(N)\!\times\!O(N)}}$). Even if the latter ensembles yield the same
averaged spectral form factor as the circular orthogonal ensemble they show
substantially enhanced fluctuations. This behaviour is due to a recently
identified additional anti-unitary symmetry of the self-dual kicked Ising
model.
