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.

  • $\begingroup$ Is $A_6$ the symmetric group? That type of notation is usually reserved for the alternating group, i.e., the subgroup of even symmetries. $\endgroup$ – Vidit Nanda Aug 7 '13 at 21:52
  • $\begingroup$ I mean the simple group with 360 elements. Yes. Symmetric. $\endgroup$ – user13559 Aug 7 '13 at 22:25
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    $\begingroup$ You mean "Yes. Alternating." ("Symmetric" would be $S_6$, not $A_6$.) This group has a nontrivial triple cover $3.A_6$, with a faithful $3$-dimensional representation that projectivizes to an action of $A_6$ on ${\bf P}^2$. There's some information and reference in the Wikipedia page on the "Valentiner group": en.wikipedia.org/wiki/Valentiner_group $\phantom\infty$ It's known that $A_6$ and $A_7$ are the only simple alternating groups whose Schur multiplier is cyclic of order $6$; the others have Schur multiplier ${\bf Z}/2{\bf Z}$. $\endgroup$ – Noam D. Elkies Aug 7 '13 at 23:09

As noted in the comments, this is a classical object, credited to [Valentiner 1889]; the paper by Robert Crass 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.

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