TITLE: I don't understand symmetries of the Fano plane
QUESTION [5 upvotes]: Hello I got a picture of the Fano plane, but there are 5 points on every line. Why aren't there 3?
And I cannot see what it's symmetry group is. I have been told it's $PSL_2(7)$ but that doesn't help.


How can I see the symmetries of the Fano plane by this or any other diagram?


I have now computed the line-point incidence relationship from the matrix group $GL_3(\mathbb F_2)$ and got the usual diagram  but I cannot see the symmetries of this at all.. so I found a nicer way to lay out the diagram but I stil can't see them

I want to be able to read the symmetry group off of the diagram, I know a priori that it is $GL_3(\mathbb F_2)$. It would be even cooler if we could see that it's $PSL_2(\mathbb F_7)$ just by looking.


From suggestions in the comments I suppose that I have constructed the second fano plane diagram using $PSL_3(2)$, and shown the two diagrams equivalent.
I put the same labeling and coloring on both diagrams. The first has $C_7$ symmetry and the second has $D_{2 \cdot 3}$, which together generate:
gap> g:=Group((1,2,3,4,5,6,7),(1,2,5)(3,6,4),(1,5)(3,4));;
gap> StructureDescription(g); 
"PSL(3,2)"

I don't know how to get that without GAP. I guess these are all the symmetries of the fano plane, but I don't know how to show that either.
I just spotted another type of symmetry in the second fano plane diagram: $(6\;0)(1\;4)$ and so I can see the entirety of $PSL_2(7) = \langle(1\;2\;5)(3\;6\;4),(1\;5)(3\;4),(6\;0)(1\;4)\rangle$ from that one diagram.

how can we see that together the $C_7$ and $D_{2 \cdot 3}$ make $GL_3(2)$ or even better $PSL_2(7)$? or how can we construct the first diagram from $PSL_2(7)$?


I was looking at the wikipedia page for Klein quartic and there are dual pictures: tiling of hyperbolic plane by heptagons and triangles.. Can we use this geometry of the diagrams and the 3/7-duality to explain the isomorphism of the matrix groups?

REPLY [2 votes]: Thanks very much to all the comments which progressively got me further to understanding the fano plane. To finally resolve this OldJohn suggested just count all the symmetries using the fact that every two points determine a unique line!
In the second diagram: Take one point anywhere 7 choices, then another point anywhere 6 choices, that determines a line - without loss of generality we can take that line to be the base of the triangle: Now there are 2x2 symmetries of the top part of the triangle. -- hardest part for me was realizing all the lines of the fano plane are actually circles. Therefore there are exactly 168 symmetries!