The Klein Configuration -- representations

Question

In the Klein configuration there are 60 points and 60 planes, with each point lying on 15 planes and each plane passing through 15 points. A $60_{15}$ configuration.

Page 34 of Abstract Configurations in Algebraic Geometry has one description, and Kummer's Quartic Surface by Hudson (which I've ordered) has another. But I find both of these descriptions murky.

The incidence graph between the planes and points would be a regular graph of 120 points and valence 15, but I can't find any references to such a graph. For the Kummer $16_6$ configuration, it's easy to find the Kummer graph.

A labeled set of planes and point would also have nice properties in combinatorial designs, but I haven't found it in the Handbook of Combinatorial Designs and other sources. I can't find it as a pseudoline arrangement.

Does anyone have a handy representation? Is there a famous graph I'm missing?

Answer 1

As now seen at Klein Configuration, the arxiv paper "Unexpected properties of the Klein configuration of 60 points in P3" has come up with a graphical realization.

Answer 2

Figured it out from the 1905 book Kummer's quartic surface by Hudson.

There are 30 lines, each with an opposing skew line indicated by reversal.

12 . 13 . 14 . 15 . 16 . 23 . 24 . 25 . 26 . 34 . 35 . 36 . 45 . 46 . 56
21 . 31 . 41 . 51 . 61 . 32 . 42 . 52 . 62 . 43 . 53 . 63 . 54 . 64 . 65

Here are the sixty points, based on odd permutations and line intersections.

12-34-65 . 12-43-56 . 21-34-56 . 21-43-65 . 12-35-46 . 12-53-64
21-35-64 . 21-53-46 . 12-36-54 . 12-63-45 . 21-36-45 . 21-63-54
13-24-56 . 13-42-65 . 31-24-65 . 31-42-56 . 13-25-64 . 13-52-46
31-25-46 . 31-52-64 . 13-26-45 . 13-62-54 . 31-26-54 . 31-62-45
14-23-65 . 14-32-56 . 41-23-56 . 41-32-65 . 14-25-36 . 14-52-63
41-25-63 . 41-52-36 . 14-26-53 . 14-62-35 . 41-26-35 . 41-62-53
15-23-46 . 15-32-64 . 51-23-64 . 51-32-46 . 15-24-63 . 15-42-36
51-24-36 . 51-42-63 . 15-26-34 . 15-62-43 . 51-26-43 . 51-62-34
16-23-54 . 16-32-45 . 61-23-45 . 61-32-54 . 16-24-35 . 16-42-53
61-24-53 . 61-42-35 . 16-25-43 . 16-52-34 . 61-25-34 . 61-52-43

For the sixty planes, swap the two ending digits for even permutations and coplanar lines making a face.

Pick a point or plane. There are 15 members in the other set containing those 3 lines.

Can anyone find a graphical representation?