On the total character of a finite group

The total character $\tau_G$ of a finite group $G$ is the sum of all irreducible complex characters of $G$, and the total degree of $G$ is $T(G) := \tau_G(1)$. A proper subgroup $H$ of $G$ is rich if $\tau_G$ is ''contained'' in the permutation character $(1_H)^G$. In the first part of this paper, we investigate rich subgroups whose index is a product of two primes. We also consider rich subgroups of symmetric and alternating groups. In the second part we establish a formula for $T(G)$ in the case where the order of $G$ is a prime power. This result is analogous to a formula for the class number of $G$ proved by P. Hall, and it confirms a conjecture by Heffernan and MacHale from 2008. In the last part of the paper, we investigate finite groups $G$ where $T(G)$ is small, in a certain sense.