quotient class

Let $a$ be an automorphism of an incarnation $Y$ of an object. The orbits of $\langle a\rangle$ on $Y$ form an object $Y/\langle a\rangle$, and its stabilizer class is the quotient class of $a$. If $a(y)=ny$ with $n\in N_G(G_y)$, the stabilizer of the orbit of $y$ is $\langle G_y,n\rangle$.

The name uses only the stabilizer class, applied to the quotient object.

The quotient class of $a$ depends only on $a$ up to conjugation in $\Aut_G(Y)$, and a seam $s\colon Y\to Y'$ gives $sas^{-1}$ the same quotient class.

The object $G/C_2$ has automorphism group $N_G(C_2)/C_2\cong C_2\times C_2$. The dihedral group $N_G(C_2)$ of order 8 has exactly three subgroups of order 4 containing its centre, one in each of the classes $C_4$, $V_4^a$, $V_4^b$, so the three involutions have these three quotient classes, and the quotient class names the involution. The natural ones are:

(1) on Coxeter arcs, reversing the arc: $C_4$, the quotient being the set of edges;

(2) on centres with one of the four bitangents through them, re-pairing the four bitangents at the centre: the pairing into Coxeter edges has class $C_4$, the two other pairings $V_4^a$ and $V_4^b$;

(3) on involutions $t$ with a Sylow 3-subgroup $P$ normalized by $t$, replacing $P$ by the $P'$ with $\langle P,P'\rangle=G$: $C_4$; by the $P'$ with $\langle P,P'\rangle\cong A_4$ of class $a$ or $b$: $V_4^a$ or $V_4^b$;

(4) on quadrangles with an ordered pair of vertices, exchanging the pair: $V_4^a$;

(5) on pairs of disjoint pairs of $\Proj^1(\F_7)$ with cross-ratio $\{2,4\}$, passing to the other such pairing of the same four points: $C_4$.

Sending a centre $c$ with a bitangent $\ell$ to the Coxeter arc from the vertex of $\ell$ to its neighbour fixed by the involution with centre $c$ is a seam, and it carries the Coxeter pairing of (2) to the reversal of (1).