  We present a new approach to calculating the higher Frobenius-Schur
indicators for the simple modules over the Drinfeld double of a finite group.
In contrast to the formula by Kashina-Sommerh{\"a}user-Zhu that involves a sum
over all group elements satisfying a certain condition, our formula operates on
the level of conjugacy classes and character tables. It can be implemented in
the computer algebra system GAP, efficiently enough to deal, on a laptop, with
symmetric groups up to $S_{18}$ (providing further evidence that indicators are
non-negative in this case) or simple groups of order up to $2 \cdot 10^8$. The
approach also allows us to test whether all indicators over the double of a
given group are rational , without computing them. Among simple groups of order
up to about $5 \cdot 10^{11}$ an inspection yields exactly one example (of
order about $5 \cdot 10^9$) where irrational indicators occur.
