From Halcrow et al. paper on pCf equilibria:
The isotropy subgroup is particularly important, as the equilibria belong to this conjugacy class, as do most of the solutions reported here. The NBC isotropy subgroup of Schmiegel and our S are conjugate to under quarter-cell coordinate transformations. In keeping with previous literature, we often represent this conjugacy class with rather than the simpler conjugate group .
Re. methods of visualizing the state-space portraits with the 4th-order isotropy subgroup quotiented out: the double-angle trick from Lorenz will not suffice here, since we have mirror symmetry as well as the rotation-about axis . The double-angle trick is suitable only for the latter. It would reduce the four quadrants to two, but unfortunately not in the way we would like: it would map and , leaving us with distinct . And it's we are most interested in equating. – John F. Gibson 2009-03-19