magma: G := PermutationGroup< 11 | (1,3)(2,4,5,6)(7,8,9)(10,11), (1,2)(5,6)(8,9) >;
gap: G := Group( (1,3)(2,4,5,6)(7,8,9)(10,11), (1,2)(5,6)(8,9) );
sage: G = PermutationGroup(['(1,3)(2,4,5,6)(7,8,9)(10,11)', '(1,2)(5,6)(8,9)'])
Group information
Description: $D_6\times A_6$ Order: \(4320\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \)
magma: Order(G);
gap: Order(G);
sage: G.order()
sage_gap: G.Order()
Exponent: \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
magma: Exponent(G);
gap: Exponent(G);
sage: G.exponent()
sage_gap: G.Exponent()
Automorphism group :$C_2\times S_6:D_6$ , of order \(17280\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \)
gap: AutomorphismGroup(G);
magma: AutomorphismGroup(G);
sage_gap: G.AutomorphismGroup()
Composition factors :$C_2$ x 2 , $C_3$ , $A_6$
magma: CompositionFactors(G);
gap: CompositionSeries(G);
sage: G.composition_series()
sage_gap: G.CompositionSeries()
Derived length: $2$
magma: DerivedLength(G);
gap: DerivedLength(G);
sage_gap: G.DerivedLength()
This group is nonabelian and nonsolvable .
magma: IsAbelian(G);
gap: IsAbelian(G);
sage: G.is_abelian()
sage_gap: G.IsAbelian()
magma: IsCyclic(G);
gap: IsCyclic(G);
sage: G.is_cyclic()
sage_gap: G.IsCyclic()
magma: IsNilpotent(G);
gap: IsNilpotentGroup(G);
sage: G.is_nilpotent()
sage_gap: G.IsNilpotentGroup()
magma: IsSolvable(G);
gap: IsSolvableGroup(G);
sage: G.is_solvable()
sage_gap: G.IsSolvableGroup()
gap: IsSupersolvableGroup(G);
sage: G.is_supersolvable()
sage_gap: G.IsSupersolvableGroup()
magma: IsSimple(G);
gap: IsSimpleGroup(G);
sage_gap: G.IsSimpleGroup()
Group statistics
magma: // Magma code to output the first two rows of the group statistics table
element_orders := [Order(g) : g in G];
orders := Set(element_orders);
printf "Orders: %o\n", orders;
printf "Elements: %o %o\n", [#[x : x in element_orders | x eq n] : n in orders], Order(G);
cc_orders := [cc[1] : cc in ConjugacyClasses(G)];
printf "Conjugacy classes: %o %o\n", [#[x : x in cc_orders | x eq n] : n in orders], #cc_orders;
gap: # Gap code to output the first two rows of the group statistics table
element_orders := List(Elements(G), g -> Order(g));
orders := Set(element_orders);
Print("Orders: ", orders, "\n");
element_counts := List(orders, n -> Length(Filtered(element_orders, x -> x = n)));
Print("Elements: ", element_counts, " ", Size(G), "\n");
cc_orders := List(ConjugacyClasses(G), cc -> Order(Representative(cc)));
cc_counts := List(orders, n -> Length(Filtered(cc_orders, x -> x = n)));
Print("Conjugacy classes: ", cc_counts, " ", Length(ConjugacyClasses(G)), "\n");
sage: # Sage code to output the first two rows of the group statistics table
element_orders = [g.order() for g in G]
orders = sorted(list(set(element_orders)))
print("Orders:", orders)
print("Elements:", [element_orders.count(n) for n in orders], G.order())
cc_orders = [cc[0].order() for cc in G.conjugacy_classes()]
print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))
magma: // Outputs [<d_1,c_1>, <d_2,c_2>, ...] where c_i is the number of irr. complex chars. of G with degree d_i
CharacterDegrees(G);
gap: # Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i
CharacterDegrees(G);
sage: # Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i
character_degrees = [c[0] for c in G.character_table()]
[[n, character_degrees.count(n)] for n in set(character_degrees)]
sage_gap: G.CharacterDegrees()
Minimal presentations
Constructions
Permutation group :Degree $11$
$\langle(1,3)(2,4,5,6)(7,8,9)(10,11), (1,2)(5,6)(8,9)\rangle$
magma: G := PermutationGroup< 11 | (1,3)(2,4,5,6)(7,8,9)(10,11), (1,2)(5,6)(8,9) >;
gap: G := Group( (1,3)(2,4,5,6)(7,8,9)(10,11), (1,2)(5,6)(8,9) );
sage: G = PermutationGroup(['(1,3)(2,4,5,6)(7,8,9)(10,11)', '(1,2)(5,6)(8,9)'])
Transitive group :36T4983 more information
Direct product :$C_2$ $\, \times\, $ $S_3$ $\, \times\, $ $A_6$
Semidirect product :$(C_6\times A_6)$ $\,\rtimes\,$ $C_2$ $C_6$ $\,\rtimes\,$ $(C_2\times A_6)$ $(C_3\times A_6)$ $\,\rtimes\,$ $C_2^2$ $C_3$ $\,\rtimes\,$ $(C_2^2\times A_6)$ more information
Trans. wreath product :not isomorphic to a non-trivial transitive wreath product
Aut. group :$\Aut(C_2^9.(C_6\times S_3))$ $\Aut(C_2^5.C_2^8:C_{10})$ $\Aut(C_2^5.C_2^8:C_{10})$
Elements of the group are displayed as permutations of degree 11.
Homology
Subgroups
magma: Subgroups(G);
gap: AllSubgroups(G);
sage: G.subgroups()
sage_gap: G.AllSubgroups()
There are 25138 subgroups in 434 conjugacy classes , 14 normal (10 characteristic ).
Characteristic subgroups are shown in this color . Normal (but not characteristic) subgroups are shown in this color .
Special subgroups
Hi
diagram
profile
all subgroups
normal subgroups
up to conjugacy
up to automorphism
Normal subgroups
Normal subgroups up to automorphism
Classes of subgroups up to conjugation
Order 4320: $D_6\times A_6$ Order 2160: $S_3\times A_6$ x 2, $C_6\times A_6$ Order 1440: $C_2^2\times A_6$ Order 1080: $C_3\times A_6$ Order 720: $C_2\times A_6$ x 3, $D_6\times A_5$ x 2 Order 432: $C_3^2:C_4\times D_6$ Order 360: $S_3\times A_5$ x 4, $C_6\times A_5$ x 2, $A_6$ Order 288: $D_6\times S_4$ x 2 Order 240: $C_2^2\times A_5$ x 2 Order 216: $C_3^2:C_4\times S_3$ x 4, $C_6:S_3^2$ , $C_2\times C_3^3:C_4$ , $C_2\times C_3^2:C_{12}$ Order 180: $\GL(2,4)$ x 2 Order 144: $S_3\times S_4$ x 8, $A_4\times D_6$ x 2, $C_6:S_4$ x 2, $C_6\times S_4$ x 2, $C_6^2:C_4$ Order 120: $C_2\times A_5$ x 6, $S_3\times D_{10}$ Order 108: $C_3:S_3^2$ x 4, $C_3^3:C_4$ x 2, $C_3^2:C_{12}$ x 2, $C_3^2:D_6$ , $C_3^2:D_6$ , $C_3^2\times D_6$ Order 96: $C_2^2\times S_4$ x 2, $D_4\times D_6$ Order 72: $C_2\times C_3^2:C_4$ x 6, $S_3\times A_4$ x 4, $C_3:S_4$ x 4, $C_3\times S_4$ x 4, $C_6\times A_4$ x 2, $S_3\times D_6$ x 2, $C_6:D_6$ Order 60: $S_3\times D_5$ x 4, $A_5$ x 2, $D_{30}$ , $S_3\times C_{10}$ , $C_3\times D_{10}$ Order 54: $C_3^2:S_3$ x 2, $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2, $C_3^2\times C_6$ Order 48: $C_2\times S_4$ x 14, $S_3\times D_4$ x 8, $C_2^2\times D_6$ x 2, $C_2^2\times A_4$ x 2, $C_6:D_4$ x 2, $C_6\times D_4$ , $C_2\times D_{12}$ , $C_4\times D_6$ Order 40: $C_2\times D_{10}$ Order 36: $C_6:S_3$ x 10, $S_3^2$ x 8, $C_3^2:C_4$ x 4, $C_6\times S_3$ x 4, $C_3\times A_4$ x 2, $C_6^2$ Order 32: $C_2^2\times D_4$ Order 30: $D_{15}$ x 2, $C_3\times D_5$ x 2, $C_5\times S_3$ x 2, $C_{30}$ Order 27: $C_3^3$ Order 24: $C_2\times D_6$ x 13, $S_4$ x 12, $C_3:D_4$ x 8, $C_2\times A_4$ x 8, $D_{12}$ x 4, $C_4\times S_3$ x 4, $C_3\times D_4$ x 4, $C_2^2\times C_6$ x 2, $C_2\times C_{12}$ , $C_6:C_4$ Order 20: $D_{10}$ x 6, $C_2\times C_{10}$ Order 18: $C_3:S_3$ x 12, $C_3\times S_3$ x 8, $C_3\times C_6$ x 7 Order 16: $C_2\times D_4$ x 12, $C_2^4$ x 2, $C_2^2\times C_4$ Order 15: $C_{15}$ Order 12: $D_6$ x 26, $C_2\times C_6$ x 7, $A_4$ x 4, $C_{12}$ x 2, $C_3:C_4$ x 2 Order 10: $D_5$ x 4, $C_{10}$ x 3 Order 9: $C_3^2$ x 5 Order 8: $D_4$ x 16, $C_2^3$ x 13, $C_2\times C_4$ x 6 Order 6: $S_3$ x 16, $C_6$ x 11 Order 5: $C_5$ Order 4: $C_2^2$ x 17, $C_4$ x 4 Order 3: $C_3$ x 5 Order 2: $C_2$ x 7 Order 1: $C_1$
Classes of subgroups up to automorphism
Order 4320: $D_6\times A_6$ Order 2160: $C_6\times A_6$ , $S_3\times A_6$ Order 1440: $C_2^2\times A_6$ Order 1080: $C_3\times A_6$ Order 720: $C_2\times A_6$ x 2, $D_6\times A_5$ Order 432: $C_3^2:C_4\times D_6$ Order 360: $C_6\times A_5$ , $S_3\times A_5$ , $A_6$ Order 288: $D_6\times S_4$ Order 240: $C_2^2\times A_5$ Order 216: $C_3^2:C_4\times S_3$ x 2, $C_6:S_3^2$ , $C_2\times C_3^3:C_4$ , $C_2\times C_3^2:C_{12}$ Order 180: $\GL(2,4)$ Order 144: $S_3\times S_4$ x 2, $C_6^2:C_4$ , $A_4\times D_6$ , $C_6:S_4$ , $C_6\times S_4$ Order 120: $C_2\times A_5$ x 2, $S_3\times D_{10}$ Order 108: $C_3:S_3^2$ x 2, $C_3^2:C_{12}$ x 2, $C_3^2:D_6$ , $C_3^2:D_6$ , $C_3^2\times D_6$ , $C_3^3:C_4$ Order 96: $C_2^2\times S_4$ , $D_4\times D_6$ Order 72: $C_2\times C_3^2:C_4$ x 4, $C_3\times S_4$ x 2, $C_6:D_6$ , $C_6\times A_4$ , $S_3\times D_6$ , $S_3\times A_4$ , $C_3:S_4$ Order 60: $S_3\times D_5$ x 2, $A_5$ , $D_{30}$ , $S_3\times C_{10}$ , $C_3\times D_{10}$ Order 54: $C_3^2:C_6$ x 2, $C_3^2\times C_6$ , $C_3^2:S_3$ , $S_3\times C_3^2$ Order 48: $C_2\times S_4$ x 5, $S_3\times D_4$ x 3, $C_2^2\times D_6$ , $C_2^2\times A_4$ , $C_6\times D_4$ , $C_6:D_4$ , $C_2\times D_{12}$ , $C_4\times D_6$ Order 40: $C_2\times D_{10}$ Order 36: $C_6:S_3$ x 6, $C_3^2:C_4$ x 3, $C_6\times S_3$ x 2, $S_3^2$ x 2, $C_6^2$ , $C_3\times A_4$ Order 32: $C_2^2\times D_4$ Order 30: $C_3\times D_5$ x 2, $C_{30}$ , $D_{15}$ , $C_5\times S_3$ Order 27: $C_3^3$ Order 24: $C_2\times D_6$ x 5, $S_4$ x 4, $C_2\times A_4$ x 3, $C_3\times D_4$ x 3, $C_3:D_4$ x 2, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_2\times C_{12}$ , $C_6:C_4$ , $C_2^2\times C_6$ Order 20: $D_{10}$ x 4, $C_2\times C_{10}$ Order 18: $C_3:S_3$ x 5, $C_3\times C_6$ x 4, $C_3\times S_3$ x 3 Order 16: $C_2\times D_4$ x 6, $C_2^4$ , $C_2^2\times C_4$ Order 15: $C_{15}$ Order 12: $D_6$ x 11, $C_2\times C_6$ x 4, $A_4$ x 2, $C_{12}$ x 2, $C_3:C_4$ Order 10: $D_5$ x 3, $C_{10}$ x 2 Order 9: $C_3^2$ x 3 Order 8: $D_4$ x 7, $C_2^3$ x 5, $C_2\times C_4$ x 4 Order 6: $C_6$ x 6, $S_3$ x 6 Order 5: $C_5$ Order 4: $C_2^2$ x 9, $C_4$ x 3 Order 3: $C_3$ x 3 Order 2: $C_2$ x 5 Order 1: $C_1$
Normal subgroups (quotient in parentheses)
Normal subgroups up to automorphism (quotient in parentheses)
Series
Derived series
$D_6\times A_6$
$\rhd$
$C_3\times A_6$
$\rhd$
$A_6$
magma: DerivedSeries(G);
gap: DerivedSeriesOfGroup(G);
sage: G.derived_series()
sage_gap: G.DerivedSeriesOfGroup()
Chief series
$D_6\times A_6$
$\rhd$
$D_6$
$\rhd$
$C_6$
$\rhd$
$C_3$
$\rhd$
$C_1$
magma: ChiefSeries(G);
gap: ChiefSeries(G);
sage_gap: G.ChiefSeries()
Lower central series
$D_6\times A_6$
$\rhd$
$C_3\times A_6$
magma: LowerCentralSeries(G);
gap: LowerCentralSeriesOfGroup(G);
sage: G.lower_central_series()
sage_gap: G.LowerCentralSeriesOfGroup()
Upper central series
$C_1$
$\lhd$
$C_2$
magma: UpperCentralSeries(G);
gap: UpperCentralSeriesOfGroup(G);
sage: G.upper_central_series()
sage_gap: G.UpperCentralSeriesOfGroup()
Supergroups
This group is a maximal subgroup of 23 larger groups in the database.
This group is a maximal quotient of 12 larger groups in the database.
Character theory
magma: CharacterTable(G); // Output not guaranteed to exactly match the LMFDB table
gap: CharacterTable(G); # Output not guaranteed to exactly match the LMFDB table
sage: G.character_table() # Output not guaranteed to exactly match the LMFDB table
sage_gap: G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table
See the $42 \times 42$ character table .
Alternatively, you may search for characters of this group with desired properties.
See the $36 \times 36$ rational character table .
