# What is the representation of the different classes of nontrivial textures in an ordered field of biaxial nematics in terms of $SU(2)$-like rotations?

I was reading Mermin's classic review on Topological Defects in Ordered Media, in which he describes ordered media with non-Abelian Fundamental Groups by taking the example of biaxial nematics with the order parameter space $$SU(2)/Q$$. Free homotopy classes in non-Abelian media are in correspondence with the conjugacy classes of the fundamental group, so essentially, defects in biaxial nematics can be classified according to the conjugacy classes of $$Q$$.
$$Q$$ has five conjugacy classes, corresponding to 180-degree disclinations in the $$xy$$, $$yz$$ and $$zx$$ planes, the trivial defect and finally the 360-degree disclination.
I was trying to think of these disclinations by describing each one with a function $$\phi:\Bbb R^n\to SU(2)/Q$$, which associates the appropriate group transformation to every point in space to create the correct texture for the disclination in question. Moreover, the appropriate functions $$\phi$$ have to satisfy the algebra of the group $$Q$$, i.e. when the composition $$\phi_1 \circ \phi_2$$ must yield the appropriate texture.
I have tried to define the functions $$\phi$$ in a few different ways, but none of them satisfy the algebraic identities mentioned above. What is the correct way to define this set of functions? Is it even possible to think of the disclinations as being represented by functions $$\phi:\Bbb R^n\to SU(2)/Q$$?