The object of size 56

The object of size 56, stabilizer $C_3$ , one class of 28 subgroups · 2 automorphisms

Stabilizer $C_3$ and automorphism group $C_2$ : the points of contact of the bitangents, the triples of the projective line, the faces of Thurston’s link complement.

The object of size 56 in both lives: the antiflag

(1, 246)

with a cyclic order on its line, and the three-subset

\{0,1,3\}

\Proj^1(\F_7)

Its incarnations

The seam table names the object in each of the five theories that meet the group. Every entry there is built.

Fano plane: antiflags with a cyclic order on the line
Projective line: ordered pairs; 3-subsets
The group: elements of order 3
Klein quartic: points of contact of the bitangents
Graphs: Coxeter vertices with a cyclic order of their neighbours

The seams between two incarnations form a torsor under $N_G(H)/H$ , a group of order 2, so there are 2 of them.

The fifteen objects

The object on the line of sizes. In ink, the objects it maps onto; in blue, those that map onto it. Dots count each object’s automorphisms, and dashed brackets join the pairs with one permutation character.

Proposition

$N_G(C_3)=S_3$ , so the automorphism group is $C_2$ , and the power of an automorphism is $\pm1$ . A point of contact of a bitangent is the same thing as a bitangent with one of its two points of contact: the stabilizer of a bitangent is an $S_3$ whose subgroup of order 3 fixes both points of contact and whose involutions exchange them. So sending a point of contact to its bitangent is the natural map $G/C_3\to G/S_3$ , with fibres of size 2.

Example

Antiflags of the Fano plane with a cyclic order on the line; ordered pairs and three-subsets of $\Proj^1(\F_7)$ ; the elements of order 3; the points of contact of the bitangents, which are the vertices of Klein’s map; Coxeter vertices with a cyclic order of their neighbours; the 9-cycles of the Coxeter graph, the 12-cycles of the Heawood graph and one orbit of Coxeter 12-cycles; and the 56 ideal faces of Thurston’s congruence link complement.

Proposition(Natural seams into 3A and 4A)

Each of these is a seam onto the class $3A$ of elements of order 3: an ordered pair $(a,b)$ of points of $\Proj^1(\F_7)$ goes to the element of order 3 fixing $a$ and $b$ with multiplier 2 at $a$ ; a three-subset goes to the element rotating it in its cyclic order $(a,b,c)$ for which $[a,b][b,c][c,a]$ is a square in $\F_7$ ; an oriented antiflag goes to the element fixing the point and advancing the line in its cyclic order; a point of contact goes to the element of its stabilizer acting on the tangent line by $\omega=e^{2\pi i/3}$ , which at $(1:\omega:\omega^2)$ is $h$ . On the projective line the triangle of ordered pairs, three-subsets and $3A$ is coherent.

Remark

No natural seam between two different theories of this object is known except through $3A$ , so no cross-theory monodromy can be read off. This is a gap in knowledge, not an absence.

Example

In Thurston’s congruence link complement it is the faces, the edges with one of their ends, and, in two orbits each, the tetrahedra with a face and the tetrahedra with a cusp. It is the object of the split torus: the diagonal matrices of $\SL(2,7)$ map onto a group of class $C_3$ , whose two fixed points on $\Proj^1(\F_7)$ make the ordered pairs, and the automorphism exchanges them. Over it the double cover adds the new object of size 112, the pairs $(\bar v,\bar w)$ of square classes with $\det(v,w)$ a nonzero square; its automorphism $\sigma(\bar v,\bar w)=(\bar w,\overline{-v})$ lies over the exchange, has order 4 and squares to $-I$ .

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