magma: G := SmallGroup(1440, 5848);
gap: G := SmallGroup(1440, 5848);
sage_gap: G = libgap.SmallGroup(1440, 5848)
sage: G = PermutationGroup(['(1,3)(2,4)(7,9,8)', '(1,4,5,3,2)(6,9,8,7)'])
Group information
Description: $S_4\times A_5$ Order: \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \)
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 :$S_4\times S_5$ , of order \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \)
gap: AutomorphismGroup(G);
magma: AutomorphismGroup(G);
sage_gap: G.AutomorphismGroup()
Composition factors :$C_2$ x 3 , $C_3$ , $A_5$
magma: CompositionFactors(G);
gap: CompositionSeries(G);
sage: G.composition_series()
sage_gap: G.CompositionSeries()
Derived length: $3$
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 :
$\langle(1,3)(2,4)(7,9,8), (1,4,5,3,2)(6,9,8,7)\rangle$
magma: G := PermutationGroup< 9 | (1,3)(2,4)(7,9,8), (1,4,5,3,2)(6,9,8,7) >;
gap: G := Group( (1,3)(2,4)(7,9,8), (1,4,5,3,2)(6,9,8,7) );
sage: G = PermutationGroup(['(1,3)(2,4)(7,9,8)', '(1,4,5,3,2)(6,9,8,7)'])
Transitive group :20T203 24T2963 30T258 30T262 all 10
Direct product :$S_4$ $\, \times\, $ $A_5$
Semidirect product :$(A_4\times A_5)$ $\,\rtimes\,$ $C_2$ $A_4$ $\,\rtimes\,$ $(C_2\times A_5)$ $(C_2^2\times A_5)$ $\,\rtimes\,$ $S_3$ $C_2^2$ $\,\rtimes\,$ $(S_3\times A_5)$ more information
Trans. wreath product :not isomorphic to a non-trivial transitive wreath product
Elements of the group are displayed as permutations of degree 9.
Homology
Abelianization : $C_{2} $
magma: quo< G | CommutatorSubgroup(G) >;
gap: FactorGroup(G, DerivedSubgroup(G));
sage: G.quotient(G.commutator())
Schur multiplier :$C_{2}^{2}$
gap: AbelianInvariantsMultiplier(G);
sage: G.homology(2)
sage_gap: G.AbelianInvariantsMultiplier()
Commutator length :$1$
gap: CommutatorLength(G);
sage_gap: G.CommutatorLength()
Subgroups
magma: Subgroups(G);
gap: AllSubgroups(G);
sage: G.subgroups()
sage_gap: G.AllSubgroups()
There are 4298 subgroups in 154 conjugacy classes , 8 normal , and all normal subgroups are characteristic.
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 1440: $S_4\times A_5$ Order 720: $A_4\times A_5$ Order 480: $D_4\times A_5$ Order 360: $S_3\times A_5$ Order 288: $A_4\times S_4$ Order 240: $C_2^2\times A_5$ x 2, $C_4\times A_5$ , $D_5\times S_4$ Order 180: $\GL(2,4)$ Order 144: $A_4^2$ , $S_3\times S_4$ Order 120: $C_2\times A_5$ x 2, $D_5\times A_4$ , $C_5:S_4$ , $C_5\times S_4$ Order 96: $C_2^2\times S_4$ , $D_4\times A_4$ Order 80: $D_4\times D_5$ Order 72: $S_3\times A_4$ x 2, $C_3:S_4$ , $C_3\times S_4$ Order 60: $C_5\times A_4$ , $S_3\times D_5$ , $A_5$ Order 48: $C_2^2\times A_4$ x 3, $C_2\times S_4$ x 2, $C_2^2:A_4$ , $S_3\times D_4$ , $C_4\times A_4$ Order 40: $C_5:D_4$ x 2, $C_2\times D_{10}$ x 2, $D_{20}$ , $C_4\times D_5$ , $C_5\times D_4$ Order 36: $C_3\times A_4$ x 2, $S_3^2$ Order 32: $C_2^2\times D_4$ Order 30: $D_{15}$ , $C_3\times D_5$ , $C_5\times S_3$ Order 24: $C_2\times D_6$ x 3, $C_2\times A_4$ x 3, $S_4$ x 3, $C_3:D_4$ x 2, $D_{12}$ , $C_4\times S_3$ , $C_3\times D_4$ Order 20: $D_{10}$ x 5, $C_2\times C_{10}$ x 2, $C_{20}$ , $C_5:C_4$ Order 18: $C_3\times S_3$ x 2, $C_3:S_3$ Order 16: $C_2\times D_4$ x 4, $C_2^4$ x 2, $C_2^2\times C_4$ Order 15: $C_{15}$ Order 12: $D_6$ x 7, $A_4$ x 5, $C_2\times C_6$ x 3, $C_{12}$ , $C_3:C_4$ Order 10: $D_5$ x 3, $C_{10}$ x 2 Order 9: $C_3^2$ Order 8: $C_2^3$ x 7, $D_4$ x 6, $C_2\times C_4$ x 2 Order 6: $S_3$ x 6, $C_6$ x 3 Order 5: $C_5$ Order 4: $C_2^2$ x 12, $C_4$ x 2 Order 3: $C_3$ x 3 Order 2: $C_2$ x 5 Order 1: $C_1$
Classes of subgroups up to automorphism
Order 1440: $S_4\times A_5$ Order 720: $A_4\times A_5$ Order 480: $D_4\times A_5$ Order 360: $S_3\times A_5$ Order 288: $A_4\times S_4$ Order 240: $C_2^2\times A_5$ x 2, $C_4\times A_5$ , $D_5\times S_4$ Order 180: $\GL(2,4)$ Order 144: $A_4^2$ , $S_3\times S_4$ Order 120: $C_2\times A_5$ x 2, $D_5\times A_4$ , $C_5:S_4$ , $C_5\times S_4$ Order 96: $C_2^2\times S_4$ , $D_4\times A_4$ Order 80: $D_4\times D_5$ Order 72: $S_3\times A_4$ x 2, $C_3:S_4$ , $C_3\times S_4$ Order 60: $C_5\times A_4$ , $S_3\times D_5$ , $A_5$ Order 48: $C_2^2\times A_4$ x 3, $C_2\times S_4$ x 2, $C_2^2:A_4$ , $S_3\times D_4$ , $C_4\times A_4$ Order 40: $C_5:D_4$ x 2, $C_2\times D_{10}$ x 2, $D_{20}$ , $C_4\times D_5$ , $C_5\times D_4$ Order 36: $C_3\times A_4$ x 2, $S_3^2$ Order 32: $C_2^2\times D_4$ Order 30: $D_{15}$ , $C_3\times D_5$ , $C_5\times S_3$ Order 24: $C_2\times D_6$ x 3, $C_2\times A_4$ x 3, $S_4$ x 3, $C_3:D_4$ x 2, $D_{12}$ , $C_4\times S_3$ , $C_3\times D_4$ Order 20: $D_{10}$ x 5, $C_2\times C_{10}$ x 2, $C_{20}$ , $C_5:C_4$ Order 18: $C_3\times S_3$ x 2, $C_3:S_3$ Order 16: $C_2\times D_4$ x 4, $C_2^4$ x 2, $C_2^2\times C_4$ Order 15: $C_{15}$ Order 12: $D_6$ x 7, $A_4$ x 5, $C_2\times C_6$ x 3, $C_{12}$ , $C_3:C_4$ Order 10: $D_5$ x 3, $C_{10}$ x 2 Order 9: $C_3^2$ Order 8: $C_2^3$ x 7, $D_4$ x 5, $C_2\times C_4$ x 2 Order 6: $S_3$ x 6, $C_6$ x 3 Order 5: $C_5$ Order 4: $C_2^2$ x 12, $C_4$ x 2 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
$S_4\times A_5$
$\rhd$
$A_4\times A_5$
$\rhd$
$C_2^2\times A_5$
$\rhd$
$A_5$
magma: DerivedSeries(G);
gap: DerivedSeriesOfGroup(G);
sage: G.derived_series()
sage_gap: G.DerivedSeriesOfGroup()
Chief series
$S_4\times A_5$
$\rhd$
$S_4$
$\rhd$
$A_4$
$\rhd$
$C_2^2$
$\rhd$
$C_1$
magma: ChiefSeries(G);
gap: ChiefSeries(G);
sage_gap: G.ChiefSeries()
Lower central series
$S_4\times A_5$
$\rhd$
$A_4\times A_5$
magma: LowerCentralSeries(G);
gap: LowerCentralSeriesOfGroup(G);
sage: G.lower_central_series()
sage_gap: G.LowerCentralSeriesOfGroup()
Upper central series
$C_1$
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 21 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 $25 \times 25$ character table .
Alternatively, you may search for characters of this group with desired properties.
See the $20 \times 20$ rational character table .
