Let $S$ be the set of minor and major triads. Two sets of actions are defined on the set:
1) Musical transposition and inversion
2) P, L, R actions
$P(C-major) = c-minor,$
$L(C-major) = e-minor,$
$R(C-major) = a-minor.$
I already know that each action can be described as a homomorphism from our group into $Sym(S)$ ($S_n$). I just don't really know how to identify these 'distinguished copies'.
Apparently, each of these homomorphisms (of action 1 and 2) is an embedding so that we have two distinguished copies, H1 and H2, of the dihedral group of order 24 in Sym(S).
This is the duality in music described by David Lewin.
"The two group actions are dual in the sense that each of these subgroups H1 and H2 of Sym(S) is the centralizer of the other!"
These notions are defined in this paper: https://www.maa.org/sites/default/files/pdf/upload_library/22/Hasse/Crans2011.pdf