In this paper by Dijkgraaf, Vafa, Verlinde, Verlinde, orbifold CFT is discussed.
In that paper, it outlined that orbifold CFT provides a way to generate the new theories from the old known ones. (i.e. "... it would be very interesting to devise operations that act on the space of RCFT's and generate new theories from a given one. As an example one can think of the construction of nondiagonal modular invariant combinations of characters, coset models, etc. Another operation of that kind is the concept of an orbifold.")
In p.521, Table 6 of this paper, they provide an example of SU(2)/H$_8$ $\simeq$ SU(2)/($Z_2 \times Z_2$) orbifold. (here H$_8$ is a quaternion group.)
$\bullet$ Here I am interested in orbifold CFT of SU(2)/G and SO(3)/G kinds (with G as its normal subgroup) . Can someone provide some more examples and data (i.e. S and T matrix, fusions of quasi-particles) of SU(2)/G and SO(3)/G orbifold CFT?
$\bullet$ What are their central charges $c$ of this 1+1D CFT? (e.g. what is the central charge of this SU(2)/D$_2$ model?) What is their ground state degeneracy of 2+1D bulk TQFT on the spatial $T^2$ torus?
$\bullet$ If there is also information on its corresponding twisted quantum doubles, in the form of $D^\omega(G')$ of some certain group $G'$. It will be nicer.
Reference and books are welcome, but please offer summaries instead of just giving Reference. (ps. Francesco yellow-book CFT Sec 17.8.4 has some discussions, but I could not find orbifold CFT of $SO(3)/D_n$ of $D_n$ dihedral group or $SU(2)/H$ of $H$ as a quaternion.)