We consider various aspects of the Segre variety S := S_{ 1,1,1}(2) in PG(7,2), whose stabilizer group G_S < GL(8, 2) has the structure N { \rtimes} Sym(3), where N := GL(2,2)\times GL(2,2)\times GL(2,2). In particular we prove that S determines a distinguished Z_3-subgroup Z < GL(8, 2) such that AZA^{ -1} = Z, for all A in G_S, and in consequence S determines a G_S-invariant spread of 85 lines in PG(7,2). Furthermore we see that Segre varieties S_{ 1,1,1}(2) in PG(7,2) come along in triplets { S,S',S"} which share the same distinguished Z_3-subgroup Z < GL(8,2). We conclude by determining all fifteen G_S-invariant polynomial functions on PG(7,2) which have degree < 8, and their relation to the five G_S-orbits of points in PG(7,2).
Categories: math.CO