The dihedral group (D4) has a 2D representation:
<-- Sometimes -I or i. This is the inversion operator
<-- Sometimes -R
<-- In your notation. Usually
<-- Reflection over y = x
<-- Reflection over y = -x
We are interested in the left cosets of the subgroup , where is one of the reflections.
So, for example, consider the subgroup . The left cosets are:
where the represents the elements of D4.
So
So there are 4 distinct left cosets: , , , and .
You take a look at the other reflection subgroups.
-Dan