The object of size 21

The flags of the Fano plane, and its quadrangles with an undirected 4-cycle; the bisections of $\Proj^1(\F_7)$ in the orbit of $\{0,1,2,3\}|\{4,5,6,\infty\}$, and three orbits of perfect matchings; the involutions, and the subgroups $C_2$, $C_4$ and $D_8$; the centres and the axes of involutions in Klein’s plane; the edges of the Heawood graph; the 8-cycles of the Heawood and Coxeter graphs; and the edges of the link of a vertex of the building of $\PGL(3,\Q_2)$.

An undirected 4-cycle on a quadrangle has its two diagonals in one parallel class of the affine plane, which is a point of the complementary line: undirected 4-cycles are flags, a seam that holds for every marking.

In Thurston’s congruence link complement it is the bisections of the cusps into two sets of four that span no tetrahedron, and in the Fano plane of the cells the flags. In Klein’s lattice the 21 pairs of vectors of norm 2 reduce at the prime 2 to the flags, the triangles of the building at a vertex; at $\sqrt{-7}$ to the 21 points inside the conic of the sky; and in Klein’s plane to the centres of the 21 involutions $s_v(x)=-x+h(x,v)v$.

At Klein’s lattice each flag $(p,\ell)$ is a vector $\bar\alpha u_k$ of norm 2, with $u_k$ an axis of the cube at the neighbour $M_p$, and its involution $s_v$ is the half-turn about that axis; modulo $(\alpha)$ it is the transvection with centre $p$ and axis $\ell$.