By a paper of Scott Crass <http://xxx.lanl.gov/pdf/math/9903111v1.pdf> 
we know that $A\_6$ (Permutation on 6 elements) is an automorphism group of $\mathbb{P}\_2$ which fix a sextic. What is the geometry of this action i.e., what are the orbits of this action explicitly.



Orbits of the action of $A_6$ on $\mathbb{P}_2$
As noted in the comments, this is a classical object, credited to
[Valentiner 1889]; the [paper
by Robert Crass](http://arxiv.org/pdf/math/9903111v1.pdf) that you cite refers to that paper and also to work of
Wiman [1895] and Fricke [1926] that should contain the answer to your
question, and Crass himself collects those answers starting on page 13.


The generic point of course has trivial stabilizer. For each of the $45$
involutions $i \in A\_6$ there's a fixed line $l\_i$, and most points on each
$l\_i$ are stabilized by just $i$, and thus have an orbit of size $180$.
[The normalizer $N\_i$ is an $8$-element dihedral group, and $N\_i/\{1,i\}$
is a Klein four-group acting on $l\_i$; the orbit of the generic point on $l\_i$
consists of its four images under that four-group, and four further images
on each of the other $44$ lines, for a total of $4 \cdot 45 = 180$.]
There are two special orbits of size $60$, and one each of size $45$ and $36$,
with dihedral stabilizers of size $6$, $8$, $10$ [NB there are two
conjugacy classes of $S\_3$'s in $A\_6$, related by an outer automorphism
of $S\_6$]; when the stabilizer is dihedral of size $2n$, the point
is at the intersection of $n$ lines $l\_i$ corresponding to the $n$
non-central involutions in the group. There is also a $90$-point orbit
with cyclic stabilizer of order $4$.


You asked for "the orbits of this action explicitly"; if you want
explicit coordinates, you must specify coordinates for ${\bf P}^2$
and matrices for (generators of) the projective action of $A\_6$.
Fortunately Crass gives these coordinates and generators on pages 4ff. $-$
actually two choices ("octahedral cordinates" and "icosahedral coordinates",
citing and slightly modifying Fricke [1926]).


Alternatively you could go to the ATLAS [1985] entry for $A\_6$,
and go to the "$2 \times 3 . A\_6$" paragraph of the Constructions section.
This gives coordinates for the $45$-point orbit on the
lift of the Valentiner group to a complex reflection group of
$3 \times 3$ matrices (#27 in the list of Shephard and Todd [1954]).
Regarding them as dual coordinates for $45$ lines $l\_i$, you should be
able to compute all the intersections that give orbits shorter than $180$,
and also to find explicit matrices for the involutions $i$ that
generate the group (and thus in particular the $4$-cycles that
stabilize the $90$-point orbit).


**References** (mostly from Crass):


[ATLAS 1985] John H[orton]. Conway et al.:
*ATLAS of Finite Groups: 
Maximal Subgroups and Ordinary Characters for Simple Groups.*
Oxford: Clarendon Press, 1985.


[Fricke 1926] Robert Fricke: *Lehrbuch der Algebra* 2. Vieweg, 1926.


[Shephard and Todd 1954] G.C. Shephard and T.A. Todd:
Finite Unitary Reflection Groups,
*Canadian Journal of Mathematics* **6** (1954), 274$-$304.


[Valentiner 1889] H. Valentiner:
De endelige Transformations-Gruppers Theori,
*Videnskabernes Selskabs Skrifter, Sjette Raekke, Naturvidenskabelig og
Mathematisk Afdeling 2.* Bianco Lunos, 1889.


[Wiman 1895] Anders Wiman:
Ueber eine einfache Gruppe von 360 ebenen Collineationen,
*Mathematische Annalen* **45** (1895), 531$-$556.


