Inertial quotient of a block ($E(B)=N_G(D,b_D)/(D{C}_G(D))$)

For a maximal Brauer pair $(D,b_D)$ of a block of a group algebra, the inertial quotient is the displayed quotient of its pair stabilizer by $D{C}_G(D)$. It has order coprime to $p$, and its order is the inertial index of a block. For abelian $D$, conjugation embeds it in $\mathrm{Aut}(D)$.

Choose a maximal Brauer pair $(D,b_D)$ for the block. Its inertial quotient is $N_G(D,b_D)/(D{C}_G(D))$, and its order is the inertial index. For cyclic $D$ of order $q$, this is a prime-to-$p$ divisor of $p-1$. The number of edges of the block's Brauer tree is this index, and the exceptional multiplicity is $(q-1)/e(B)$.