In 1961, Solomon demonstrated that the sum of all entries in the character table of a finite group does not exceed the cardinality of the group. His key insight in estimating this sum was a representation-theoretic interpretation of the rows in the character table. We conjecture that the total sum is at most twice the first column sum, the latter being the sum of degrees of the irreducible characters. In this talk, we will establish the validity of this conjecture for symmetric, hyperoctahedral and demihyperoctahedral groups. Our approach relies on a classic result concerning column sums attributed to Frobenius and Schur. Time permitting, we will also derive the generating function for total sums for generalized symmetric groups. This is ongoing work with Hiranya Kishore Dey and Digjoy Paul.