  We find an infinite family of $4D$ $\mathcal{N}=2$ interacting superconformal
field theories which enter the description of the strong-coupling limit of
$SU(2N+1)$ gauge theories with hypermultiplets in the
$\wedge^2(\square)+\text{Sym}^2(\square)$ . These theories arise from the
compactification of the $6D$ $(2,0)$ theory of type $A_{2N}$ on a sphere with
two full twisted punctures and one minimal untwisted puncture. For $N=1$, this
theory is the "new" rank-1 SCFT with $\Delta(u)=3$ of Argyres and Wittig. Using
the superconformal index, we finally pin down the properties of this theory.
