Graphic Representation of Classical Unitals on 28 Points

Question

I would like to understand the geometry of the classical unitals.
They are block designs containing $q^3+1$ points and whose blocks have cardinality $q+1$, where $q$ is a prime power. For $q=2$ (if I understand properly), the classical unital is isomorphic to the affine plane over the $3$-element field, so its geometric structure is clear. What is the geometric structure of the next unital, namely that of cardinality $3^3+1=28$ containing lines of cardinality $3+1=4$? Is there any (nice symmetric) graphical representation of this $28$-element unital that can be helpful for understanding the geometric structure of this unital, its symmetries, etc. It has only 28 points, 63 lines, so those can be drawn somehow?

Answer 1

Using the answer of Taras Banakh, I have implemented an interactive example using HTML and Javascript. It shows all lines of the unital with random colors. Every line is clickable. After clicking on a line it shows the line together with points that belong to this line. To reset — click outside any line. I don't guarantee that this example will last forever (or even for a long time), so anybody interested in it can save the source of this page and host it where he wants. If somebody wants to improve or change this example, PM me for the source and how-to-launch (I'm too lazy to write documentation, shortly it's compiled from Java source using TeaVM).