How to list all subgroups of symmetry group S_6?

Question

I learned from https://oeis.org/A005432 that $S_6$ has $1455$ subgroups, how can I list them all in mathematica?

As the comment says, the direct approach cannot solve the problem.

Answer 1

Purpose of this answer: mainly to check whether FiniteGroupData is outputting correct results up to isomorphisms.

TL;DR:

• The output of FiniteGroupData[{"SymmetricGroup", 4}, "Subgroups"] is missing the dihedral group.

• The distribution of group orders in FiniteGroupData[{"SymmetricGroup", 6}, "Subgroups"] seems correct although it could be a fluke as the output of FiniteGroupData[{"SymmetricGroup", 6}, "Subgroups"] seems to be removed of duplicates of isomorphisms (see text for further detail of how that makes a comparison with available data difficult)

---

I used the online magma calculator http://magma.maths.usyd.edu.au/calc/ . The magma code first calculates all subgroups up to conjugation and then identifies each subgroup with a canonical reference group up to isomorphism.

I am not an expert in magma programming, I used a combination of https://mathoverflow.net/questions/82873/using-magma-for-group-theory/82881#82881, https://math.stackexchange.com/questions/3324896/determine-subgroup-langle-12-1324-rangle-of-group-s-4, magma documentation and Phind which is a search engine that has GPT4 and where search results can be toggled on or off (sometimes I feel its better to ask its memory rather than rely on keyword searches)

Note: In the following : •=\[Bullet] and ⎵=\[UnderBracket]

Verifying FiniteGroupData for $S_4$

It does not seem like the output of FiniteGroupData is complete. The dihedral group which is present in https://groupprops.subwiki.org/wiki/Subgroup_structure_of_symmetric_group:S4 and https://math.stackexchange.com/questions/379841/how-to-enumerate-subgroups-of-each-order-of-s-4-by-hand is missing. The dihedral group is of order 8 whereas the other subgroups listed by FiniteGroupData are not as shown from the output of :

FiniteGroupData[{"SymmetricGroup", 4}, "Subgroups"] // Map[FiniteGroupData[#, "Order"] &]

(*{1, 2, 3, 4, 4, 6, 12, 24}*)

This group is present in the output of the following Magma code that can be used in the online calculator. The results are presented in table format at the end of this section.

// Group definition
G := SymmetricGroup(4);

// Find all subgroups of G
subgroups := Subgroups(G);

// identify each subgroup
for S in subgroups do
    id := IdentifyGroup(S`subgroup);
    groupName := GroupName(SmallGroup(id[1], id[2]));
    print "SubGroup:\n";
    S`subgroup;
    "\n-\n\nIsomorphic to :\n";
    groupName;
    print "\n---\n";
end for;

I used Mathematica to rewrite the string as a table:

StringCases[magma⎵output, 
   "Order = " ~~ (a : DigitCharacter ..) ~~ 
     Shortest[___] ~~ ("Isomorphic to :

" ~~ Shortest[b__] ~~ "

---") :> {b, ToExpression@a}] // SortBy[Last] // 
 TableForm[#, TableHeadings -> {None, {"SubGroup", "Order"}}] &

Output:

$$ \begin{array}{cc} \text{SubGroup} & \text{Order} \\ \hline \text{C1} & 1 \\ \text{C2} & 2 \\ \text{C2} & 2 \\ \text{C3} & 3 \\ \text{C2${}^{\wedge}$2} & 4 \\ \text{C2${}^{\wedge}$2} & 4 \\ \text{C4} & 4 \\ \text{S3} & 6 \\ \text{D4} & 8 \\ \text{A4} & 12 \\ \text{S4} & 24 \\ \end{array} $$

The list is the same up to isomorphism as that of https://groupprops.subwiki.org/wiki/Subgroup_structure_of_symmetric_group:S4 .

Compare this result with that of FiniteGroupData :

FiniteGroupData[{"SymmetricGroup", 4}, "Subgroups"] // 
   Map[{#, FiniteGroupData[#, "Order"]} &] // 
  TableForm[#, 
    TableHeadings -> {None, {"Subgroup", "Order"}}] & // TeXForm

$$ \begin{array}{cc} \text{Subgroup} & \text{Order} \\ \hline \text{Trivial} & 1 \\ \text{$\{$CyclicGroup, 2$\}$} & 2 \\ \text{$\{$CyclicGroup, 3$\}$} & 3 \\ \text{$\{$CyclicGroup, 4$\}$} & 4 \\ \text{Vierergruppe} & 4 \\ \text{$\{$SymmetricGroup, 3$\}$} & 6 \\ \text{$\{$AlternatingGroup, 4$\}$} & 12 \\ \text{$\{$SymmetricGroup, 4$\}$} & 24 \\ \end{array} $$

Notice that:

• FiniteGroupData does not show duplicates of isomorphism groups. I am not sure but this might make it difficult to reconstruct the permutation groups even with FiniteGroupData[#, "PermutationGroupRepresentation"] & as naively information seems to be removed by creating isomorphisms and deleting duplicates.

• $D_4$ which is a group of order 8 is missing in the output from FiniteGroupData. It is however present in FiniteGroupData[{"SymmetricGroup", 4}, "SylowSubgroups"] as mentioned in the comments.

Verifying FiniteGroupData for $S_6$

The outputs are given at the end of this section but they are quite long. As a quick check, I removed the duplicate subgroups (notice that this is potentially a risk as some of the subgroups in the output of FiniteGroupData for $S_4$ were isomorphic to each other). I sorted the orders in both lists and I checked that the sorted lists of orders are the same

I got the following outputs :

Magma (with duplicates)

I used the same code with $S_6$. Changing the group in the magma code to G := SymmetricGroup(6);

$$ \begin{array}{cc} \text{SubGroup} & \text{Order} \\ \hline \text{C1} & 1 \\ \text{C2} & 2 \\ \text{C2} & 2 \\ \text{C2} & 2 \\ \text{C3} & 3 \\ \text{C3} & 3 \\ \text{C2${}^{\wedge}$2} & 4 \\ \text{C2${}^{\wedge}$2} & 4 \\ \text{C2${}^{\wedge}$2} & 4 \\ \text{C2${}^{\wedge}$2} & 4 \\ \text{C2${}^{\wedge}$2} & 4 \\ \text{C4} & 4 \\ \text{C4} & 4 \\ \text{C5} & 5 \\ \text{C6} & 6 \\ \text{C6} & 6 \\ \text{S3} & 6 \\ \text{S3} & 6 \\ \text{S3} & 6 \\ \text{S3} & 6 \\ \text{C2${}^{\wedge}$3} & 8 \\ \text{C2${}^{\wedge}$3} & 8 \\ \text{C2*C4} & 8 \\ \text{D4} & 8 \\ \text{D4} & 8 \\ \text{D4} & 8 \\ \text{D4} & 8 \\ \text{C3${}^{\wedge}$2} & 9 \\ \text{D5} & 10 \\ \text{A4} & 12 \\ \text{A4} & 12 \\ \text{D6} & 12 \\ \text{D6} & 12 \\ \text{C2*D4} & 16 \\ \text{C3*S3} & 18 \\ \text{C3*S3} & 18 \\ \text{C3:S3} & 18 \\ \text{F5} & 20 \\ \text{C2*A4} & 24 \\ \text{C2*A4} & 24 \\ \text{S4} & 24 \\ \text{S4} & 24 \\ \text{S4} & 24 \\ \text{S4} & 24 \\ \text{C3:S3.C2} & 36 \\ \text{S3${}^{\wedge}$2} & 36 \\ \text{S3${}^{\wedge}$2} & 36 \\ \text{C2*S4} & 48 \\ \text{C2*S4} & 48 \\ \text{A5} & 60 \\ \text{A5} & 60 \\ \text{S3wrC2} & 72 \\ \text{S5} & 120 \\ \text{S5} & 120 \\ \text{A6} & 360 \\ \text{S6} & 720 \\ \end{array} $$

FiniteGroupData+ Magma

In the code below, the magma⎵output is the string output from the magma calculator :

FiniteGroupData•SubGroups = 
  FiniteGroupData[{"SymmetricGroup", 6}, "Subgroups"];

orders = 
  FiniteGroupData[{"SymmetricGroup", 6}, "Subgroups"] // 
    Map[FiniteGroupData[#, "Order"] &] // Sort;

output = 
  StringCases[magma⎵output, 
    "Order = " ~~ (a : DigitCharacter ..) ~~ 
      Shortest[___] ~~ ("Isomorphic to :

" ~~ Shortest[b__] ~~ "

---") :> {b, ToExpression@a}] // SortBy[Last];

magma⎵subgroups = 
  output // DeleteDuplicatesBy[First] // Map[First];
{FiniteGroupData\[Bullet]SubGroups, magma\[UnderBracket]subgroups, 
   orders} // Transpose // TableForm

The output is wide the reader might need to use the horizontal scroll at the bottom.

NOTE that the table is only sorted by group order, the subgroups between the output of Magma and that of Mathematica do not necessarily correspond on each line.

$$ \begin{array}{ccc} \text{FiniteGroupData} & \text{Magma} & \text{Order} \\ \hline \text{Trivial} & \text{C1} & 1 \\ \text{$\{$CyclicGroup, 2$\}$} & \text{C2} & 2 \\ \text{$\{$CyclicGroup, 3$\}$} & \text{C3} & 3 \\ \text{$\{$CyclicGroup, 4$\}$} & \text{C2${}^{\wedge}$2} & 4 \\ \text{Vierergruppe} & \text{C4} & 4 \\ \text{$\{$CyclicGroup, 5$\}$} & \text{C5} & 5 \\ \text{$\{$CyclicGroup, 6$\}$} & \text{C6} & 6 \\ \text{$\{$AbelianGroup, $\{$2, 2, 2$\}\}$} & \text{S3} & 6 \\ \text{$\{$AbelianGroup, $\{$4, 2$\}\}$} & \text{C2${}^{\wedge}$3} & 8 \\ \text{$\{$SymmetricGroup, 3$\}$} & \text{C2*C4} & 8 \\ \text{$\{$DihedralGroup, 4$\}$} & \text{D4} & 8 \\ \text{$\{$AbelianGroup, $\{$3, 3$\}\}$} & \text{C3${}^{\wedge}$2} & 9 \\ \text{$\{$DihedralGroup, 5$\}$} & \text{D5} & 10 \\ \text{$\{$DihedralGroup, 6$\}$} & \text{A4} & 12 \\ \text{$\{$AlternatingGroup, 4$\}$} & \text{D6} & 12 \\ \text{$\{$DirectProduct, $\{\{$CyclicGroup, 2$\}$, $\{$DihedralGroup, 4$\}\}\}$} & \text{C2*D4} & 16 \\ \text{$\{$SemidirectProduct, $\{\{$CyclicGroup, 2$\}$, $\{$AbelianGroup, $\{$3, 3$\}\}\}\}$} & \text{C3*S3} & 18 \\ \text{$\{$DirectProduct, $\{\{$CyclicGroup, 3$\}$, $\{$SymmetricGroup, 3$\}\}\}$} & \text{C3:S3} & 18 \\ \text{$\{$SemidirectProduct, $\{\{$CyclicGroup, 4$\}$, $\{$CyclicGroup, 5$\}\}\}$} & \text{F5} & 20 \\ \text{$\{$DirectProduct, $\{\{$CyclicGroup, 2$\}$, $\{$AlternatingGroup, 4$\}\}\}$} & \text{C2*A4} & 24 \\ \text{$\{$SymmetricGroup, 4$\}$} & \text{S4} & 24 \\ \text{$\{$DirectProduct, $\{\{$SymmetricGroup, 3$\}$, $\{$SymmetricGroup, 3$\}\}\}$} & \text{C3:S3.C2} & 36 \\ \text{$\{$SemidirectProduct, $\{\{$CyclicGroup, 4$\}$, $\{$AbelianGroup, $\{$3, 3$\}\}\}\}$} & \text{S3${}^{\wedge}$2} & 36 \\ \text{$\{$DirectProduct, $\{\{$CyclicGroup, 2$\}$, $\{$SymmetricGroup, 4$\}\}\}$} & \text{C2*S4} & 48 \\ \text{$\{$AlternatingGroup, 5$\}$} & \text{A5} & 60 \\ \text{$\{$SemidirectProduct, $\{\{$CyclicGroup, 2$\}$, $\{$DirectProduct, $\{\{$SymmetricGroup, 3$\}$, $\{$SymmetricGroup, 3$\}\}\}\}\}$} & \text{S3wrC2} & 72 \\ \text{$\{$SymmetricGroup, 5$\}$} & \text{S5} & 120 \\ \text{$\{$AlternatingGroup, 6$\}$} & \text{A6} & 360 \\ \text{$\{$SymmetricGroup, 6$\}$} & \text{S6} & 720 \\ \end{array} $$

Answer 2

You can install GAP and parse the output:

gapsubgroups = RunProcess[
    {"/Applications/gap-4.12.2/gap", "-q"}, (* change loc as needed *)
    "StandardOutput",
    "AllSubgroups(SymmetricGroup(6));"];

subgroups = gapsubgroups // (* ad hoc parsing -- sorry *)
       StringReplace[#, "(())" -> "{Cycles@{}}"] & //
      StringReplace[#, {"[" -> "{", "]" -> "}", 
         "(" ~~ c : ("," | DigitCharacter) ... ~~ ")" :> 
          "cyc[" <> c <> "]"}] & //
     StringReplace[#, "Group" -> "PermutationGroup@"] & //
    ToExpression //
   ReplaceAll[{c : Verbatim[Times][__cyc] :> 
      Cycles[List @@ c /. cyc -> List], c_cyc :> Cycles[{List @@ c}]}];

Checks:

Length[subgroups]
(*  1455  *)

GroupOrder /@ subgroups //
  Counts //
 BarChart[#, ChartLabels -> Automatic, ImageSize -> 500] &

Answer 3

In 13.2.1 on Windows 10

 FiniteGroupData[{"SymmetricGroup", 6}, "Subgroups"]

{Trivial,{CyclicGroup,2},{CyclicGroup,3},{CyclicGroup,4},Vierergruppe,{CyclicGroup,5},{CyclicGroup,6},{AbelianGroup,{2,2,2}},{AbelianGroup,{4,2}},{SymmetricGroup,3},{DihedralGroup,4},{AbelianGroup,{3,3}},{DihedralGroup,5},{DihedralGroup,6},{AlternatingGroup,4},{DirectProduct,{{CyclicGroup,2},{DihedralGroup,4}}},{SemidirectProduct,{{CyclicGroup,2},{AbelianGroup,{3,3}}}},{DirectProduct,{{CyclicGroup,3},{SymmetricGroup,3}}},{SemidirectProduct,{{CyclicGroup,4},{CyclicGroup,5}}},{DirectProduct,{{CyclicGroup,2},{AlternatingGroup,4}}},{SymmetricGroup,4},{DirectProduct,{{SymmetricGroup,3},{SymmetricGroup,3}}},{SemidirectProduct,{{CyclicGroup,4},{AbelianGroup,{3,3}}}},{DirectProduct,{{CyclicGroup,2},{SymmetricGroup,4}}},{AlternatingGroup,5},{SemidirectProduct,{{CyclicGroup,2},{DirectProduct,{{SymmetricGroup,3},{SymmetricGroup,3}}}}},{SymmetricGroup,5},{AlternatingGroup,6},{SymmetricGroup,6}}

and then, for example,

FiniteGroupData[{"DihedralGroup", 6}, "Subgroups"]

{"Trivial", {"CyclicGroup", 2}, {"CyclicGroup", 3}, {"CyclicGroup", 6}, {"DihedralGroup", 2}, {"DihedralGroup", 3}, {"DihedralGroup", 6}}

and so on.