Aspects of the Segre variety S1, 1, 1(2)

We consider various aspects of the Segre variety S := S1,1,1(2) in PG(7, 2), whose stabilizer group GS < GL(8, 2) has the structure N o Sym(3), where N := GL(2, 2) × GL(2, 2) × GL(2, 2). In particular we prove that S determines a distinguished Z3-subgroup Z < GL(8, 2) such that AZA−1 = Z, for all A ∈ GS , and in consequence S determines a GS invariant spread of 85 lines in PG(7, 2). Furthermore we see that Segre varieties S1,1,1(2) in PG(7, 2) come along in triplets {S,S ′,S ′′} which share the same distinguished Z3-subgroup Z < GL(8, 2). We conclude by determining all fifteen GS -invariant polynomial functions on PG(7, 2) which have degree < 8, and their relation to the five GS -orbits of points in PG(7, 2). MSC2010: 51E20, 05B25, 15A69