  This paper deals with $\tilde{\chi}^{(6)}$, the six-particle contribution to
the magnetic susceptibility of the square lattice Ising model. We have
generated, modulo a prime, series coefficients for $\tilde{\chi}^{(6)}$. The
length of the series is sufficient to produce the corresponding Fuchsian linear
differential equation (modulo a prime). We obtain the Fuchsian linear
differential equation that annihilates the "depleted" series
$\Phi^{(6)}=\tilde{\chi}^{(6)} - {2 \over 3} \tilde{\chi}^{(4)} + {2 \over 45}
\tilde{\chi}^{(2)}$. The factorization of the corresponding differential
operator is performed using a method of factorization modulo a prime introduced
in a previous paper. The "depleted" differential operator is shown to have a
structure similar to the corresponding operator for $\tilde{\chi}^{(5)}$. It
splits into factors of smaller orders, with the left-most factor of order six
being equivalent to the symmetric fifth power of the linear differential
operator corresponding to the elliptic integral $E$. The right-most factor has
a direct sum structure, and using series calculated modulo several primes, all
the factors in the direct sum have been reconstructed in exact arithmetics.
