Is the size of the normalizer of a Sylow

p-subgroup determined by the ordinary character table of the group?

And if so, how does one calculate it?

In a solvable group, apparently one can compute the prime divisors of the Sylow normalizers from the character table (Isaacs–Navarro, 2002), but I don't see any discussion of the entire order. I suppose it must be harder to compute the order, and I somewhat hope it is *too* hard, that is, the character table does not determine the Sylow normalizer's order. If it helps to prove you *can* find the order, then I am happy to assume one also knows the power maps (and so element orders).

Isaacs, I. M.; Navarro, Gabriel. "Character tables and Sylow normalizers." Arch. Math. (Basel) 78 (2002), no. 6, 430–434. MR1921731 DOI:10.1007/s00013-002-8267-4