  The automorphism groups of the three known extremal even unimodular lattices
of dimension 48 and the one of dimension 72 are computed using the
classification of finite simple groups. Restrictions on the possible
automorphisms of 48-dimensional extremal lattices are obtained. We classify all
extremal lattices of dimension 48 having an automorphism of order $m$ with
$\varphi(m) > 24$. In particular the lattice $P_{48n}$ is the unique extremal
48-dimensional lattice that arises as an ideal lattice over a cyclotomic number
field.
