  Our starting point is a very simple one, namely that of a set L_4 of four
mutually skew lines in PG(7,2): Under the natural action of the stabilizer
group G(L_4) < GL(8,2) the 255 points of PG(7,2) fall into four orbits omega_1,
omega_2, omega_3 omega_4; of respective lengths 12, 54, 108, 81: We show that
the 135 points in omega_2 \cup omega_4 are the internal points of a hyperbolic
quadric H_7 determined by L_4; and that the 81-set omega_4 (which is shown to
have a sextic equation) is an orbit of a normal subgroup G_81 isomorphic to
(Z_3)^4 of G(L_4): There are 40 subgroups (isomorphic to (Z_3)^3) of G_81; and
each such subgroup H < G_81 gives rise to a decomposition of omega_4 into a
triplet of 27-sets. We show in particular that the constituents of precisely 8
of these 40 triplets are Segre varieties S_3(2) in PG(7,2): This ties in with
the recent finding that each Segre S = S_3(2) in PG(7,2) determines a
distinguished Z_3 subgroup of GL(8,2) which generates two sibling copies S'; S"
of S.
