The object of size 168

The object of size 168, stabilizer 1, one class of 1 subgroup · 168 automorphisms

The group acting on itself: trivial stabilizer, and the whole group of order 168 as its automorphisms.

The object of size 168 as pairs of Coxeter vertices at distance 3: the base

d_0

and the twelve vertices three steps from it, each pair one element of a single regular orbit of 168.

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: frames (ordered triangles)
Projective line: ordered triples (two orbits, of $(0,1,2)$ and $(0,1,3)$ )
The group: elements under left translation
Klein quartic: a regular orbit of points, e.g. of $(1:2:5)$
Graphs: Coxeter pairs at distance 3

The seams between two incarnations form a torsor under $N_G(H)/H$ , a group of order 168, so there are 168 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

The stabilizer is trivial and $N_G(1)/1=G$ , so the object is not rigid: between any two of its incarnations there are 168 seams.

Example

Its incarnations in the seam table: the frames of the Fano plane, ordered triangles; the ordered triples of $\Proj^1(\F_7)$ , in two orbits, of $(0,1,2)$ and $(0,1,3)$ ; the elements of $G$ under left translation; a regular orbit of points of the Klein quartic, for example of $(1:2:5)$ ; and the 168 pairs of vertices at distance 3 in the Coxeter graph, which form one regular orbit.

Remark

No conjugacy class of elements or of subgroups is an incarnation of it, since the centralizers and normalizers are never trivial. On the Klein quartic every orbit other than those of sizes 24, 56 and 84 is regular.

Example

In the closed table it is also the faces of Thurston’s congruence link complement $M$ with one of their edges, or its cusps with a face through them. Over it the double cover $\SL(2,7)$ adds its largest new object, the bases $(v,w)$ of $\F_7^2$ with $\det(v,w)=1$ , on which the automorphisms form $\SL(2,7)$ itself; $\SL(2,7)$ has no subgroup of order 168, so it does not split over the group.

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