  A large family of 4d N=2 SCFT's was introduced in arXiv:1210.2886. Its
elements $D_p(G)$ are labelled by a positive integer p\in N and a simply-laced
Lie group G; their flavor symmetry is at least G. In the present paper we study
their physics in detail. We also analyze the properties of the theories
obtained by gauging the diagonal symmetry of a collection of $D_{p_i}(G)$
models. In all cases the computation of the physical quantities reduces to
simple Lie-theoretical questions. To make the analysis more functorial, we
replace the notion of the BPS-quiver of the N=2 QFT by the more intrinsic
concept of its META-quiver. In particular: 1) We compute the SCFT central
charges a, c, k, and flavor group F for all $D_p(G)$ models. 2) We identify the
subclass of $D_p(G)$ theories which correspond to previously known SCFT's
(linear SU and SO-USp quiver theories, Argyres-Douglas models, superconformal
gaugings of Minahan-Nemeshanski E_r models, etc.), as well as to non-trivial IR
fixed points of known theories. The $D_p(E_r)$ SCFT's with $p\geq 3$ cannot be
constructed by any traditional method. 3) We investigate the finite BPS
chambers of some of the models. 4) As a by product, we prove three conjectures
by Xie and Zhao, and provide new checks of the Argyres-Seiberg duality.
