f an m are two 1d vectors, they should not be towards each other <- <-(initial), -> ->, <- -> and unallowed f case -> <-. We define rotations of these vectors(or just change the arrow directions) from initial state by $f$ and $m$. (I know this is simply an "implication" but i want to try it with group theory.)
Edit: Vector context is added.
For the following group of symmetries, $f$ is not allowed;
$$D=\{e,m,m^2,fm,mf^2m | m^2=mf^2m=e\} f \notin D$$
How do we define it or is it even possible to define as a group other way?
This is not a group as it is not closed under multiplication. Consider the following product: $$fm * m = fm^2 = fe = f \not \in D$$
Note that this relies on the assumption than we have associativity, and that we can write $fm = f*m$. If we drop this assumption, then it isn't really clear how each of the elements in the structure are supposed to relate to one another. For example, if we write each of the elements in the structure in a form that doesn't look like a product (i.e. $fm = a$) then we get something like: $$D = \{e,m,a|m^2 = e\}$$ which isn't really enough information to define the structure. If we think of those more as generators (implying closure), i.e. we write: $$ D = \langle e,m,a|m^2 = e \rangle$$ Then we get a magma, but in order to get structure than that you need more assumptions, I believe.