  We consider symmetric separately radial (with corresponding group $S_n\rtimes
\mathbb{T}^n$) and alternating separately radial (with corresponding group
$A_n\rtimes \mathbb{T}^n$) symbols, as well as the associated Toeplitz
operators on the weighted Bergman spaces on the unit ball on $\mathbb{C}^n$.
Using a purely representation theoretic approach we obtain that the
$C^*$-algebras generated by each family of such Toeplitz operators is
commutative. Furthermore, we show that the symmetric separately radial Toeplitz
operators are more general than radial Toeplitz operators, i.e., every radial
Toeplitz operator is a symmetric separately radial.
