  The $\beta$-decay rates of $^{60}$Ca have been studied within a microscopic
model, which is based on the Skyrme interaction T45 to construct
single-particle and phonon spaces. We observe a redistribution of the
Gamow-Teller strength due to the phonon-phonon coupling, considered in the
model. For $^{60}$Sc, the spin-parity of the ground state is found to be $1^+$.
We predict that the half-life of $^{60}$Ca is 0.3 ms, while the total
probability of the $\beta x n$ emission is 6.1%. Additionally, the random
matrix theory has been applied to analyse the statistical properties of the
$1^+$ spectrum populated in the $\beta$-decay to elucidate the obtained
results.
