Scott Murray asked

Is their a command to tell if two character tables
are the same, without computing the entire group
of transforming permutations (section 65.18 of the
manual)?

It depends on what ``the same'' means.

If one is interested in equality of the lists of irreducible
characters (and power maps) then it is sufficient to
compare these bits of information for the two tables.

If one is interested in equality up to permutations of
rows and columns then the only GAP command for comparing
character tables is `TransformingPermutationsCharacterTables'.
If transforming permutations exist then they are encoded
via one transformation of columns and rows plus the
group of table automorphisms of the second argument;
this automorphism group is computed as a byproduct of the
current implementation, it does not cause extra work.

In particular, there is no non-constructive algorithm in GAP
for deciding the equality of character tables up to permutations
of columns and rows.

Kind regards,
Thomas