How to generalize the

**Fano" = Klein **equation ?

there is a transitivity hierarchy (stick-n-stuck-all structures)

1....... groups, affine spaces with translations

1+ ... affine spaces with translations and dilatations

2...... lines, fields

3...... projective lines

3+.... also projective lines

4,5.... the sporadic Mathiew groups.

After groups and fields, the next symmetric objects are the projective lines.

Reading from the Conway-Hulpke-Mckay table :

**9T27** - sharply triple transitive, the natural projective line with 9 points

and

**9T32** - triple transitive, but also a projective group PgamaL(2,

.

The extra-transitivity (more than three) of a projective line is explained by some twists of the line. Let' s say that the line is coordinated with some elements of a field; then, the automorphisms of that field will permute also the points on the line.

http://en.wikipedia.org/wiki/CollineationThe nice case is p prime, when

**ProjectiveLine(p prime)'''= Lin** (stick three stuck all)

for p^k +1 points on the projective line,

**ProjectiveLine(p^k)''' = Cyc[k](Lin)**. (stick three and reach the twisting)

that Gal(K/k) is cyclic of order k

An affine line should be also twisted by field automorphisms,

for example

**9T15** - the affine line on 9 points, sharply 2-transitive (order 72)and

**6T19** - the affine line with twists (order 144)