Question:
Consider two molecular symmetry groups, for example $C_s$ and $C_{2v}$.
- $C_s$ has one inversion plane, and two irreductible representations: the symmetric $A'$, and the antisymmetric $A''$.
- $C_{2v}$ has one inversion plane and one 2-fold axis; and four irreductible representations: $A_1$, $B_1$, $A_2$, $B_2$.
It is clear that if a molecule has a $C_{2v}$ symmetry, it also has a $C_s$ symmetry. It also stands (in a selected coordinate system), that an orbital of a molecule with $C_{2v}$ symmetry that belong to $A_1$ or $B_1$ also belong to $A'$, and similarly, orbitals belonging to $A_2$ and $B_2$ also belong to $A''$.
- What mathematical term describes the relationship between $C_s$ and $C_{2v}$? Is $C_{2v}$ a subgroup of $C_s$? Is it rather a subset of it? Should we rather say that $C_{2v}$ implies $C_s$?
- What's the correct term for the relationship between $A_1$/$B_1$ and $A'$, and between $A_2$/$B_2$ and $A''$? Are they subspecies? Does $A_1$ imply $A'$?
Background:
I'm comparing some orbitals of two similar molecules. One is a root with $C_{2v}$ symmetry, and the other is an alkylated version of it with $C_s$ symmetry. For certain reasons, symmetry classifications of the molecules play an important part of the writeup. I want to explicitly point out that the symmetry and the irreductable representations of the root correspond to the ones of the alkylated molecule, even though they're labeled differently.
While this sounds like a basic question, I found it surprisingly hard to find an answer. I've been struggling for almost a week at this point. All textbooks and online resources talk about how you can obtain $C_{2v}$ from $C_s$ with certain symmetry operations, and about what properties they have individually, but never mention the terms I'm looking for.
I don't really have a strong background in group theory, so if I used any other terms incorrectly, I'm glad to be corrected.