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Properties

Label 360.118
Order \( 2^{3} \cdot 3^{2} \cdot 5 \)
Exponent \( 2^{2} \cdot 3 \cdot 5 \)
Simple yes
$\card{G^{\mathrm{ab}}}$ \( 1 \)
$\card{Z(G)}$ \( 1 \)
$\card{\Aut(G)}$ \( 2^{5} \cdot 3^{2} \cdot 5 \)
$\card{\mathrm{Out}(G)}$ \( 2^{2} \)
Perm deg. $6$
Trans deg. $6$
Rank $2$

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Show commands: Gap / Magma / Oscar / SageMath

This is the only alternating group $A_n$ whose automorphism group is not $S_n$. Instead, it is a group of order $1440$ containing $S_n$ with index $2$. There is an outer automorphism that exchanges the $3$-cycles such as $(123)$ with the products of two $3$-cycles such as $(123)(456)$.

There is also an exceptional isomorphism with $\PSL_2(\mathbb{F}_9)$, and it is the third smallest nonabelian simple group.

Copy content comment:Define group as an alternating group
 
Copy content magma:G := AlternatingGroup(6);
 
Copy content gap:G := AlternatingGroup(6);
 
Copy content sage:G = AlternatingGroup(6)
 
Copy content sage_gap:G = libgap.eval('AlternatingGroup(6)')
 
Copy content oscar:G = alternating_group(6)
 

Group information

Description:$A_6$
Order: \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \)
Copy content comment:Order of the group
 
Copy content magma:Order(G);
 
Copy content gap:Order(G);
 
Copy content sage:G.order()
 
Copy content sage_gap:G.Order()
 
Copy content oscar:order(G)
 
Exponent: \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Copy content comment:Exponent of the group
 
Copy content magma:Exponent(G);
 
Copy content gap:Exponent(G);
 
Copy content sage:G.exponent()
 
Copy content sage_gap:G.Exponent()
 
Copy content oscar:exponent(G)
 
Automorphism group:$S_6:C_2$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage:libgap(G).AutomorphismGroup()
 
Copy content sage_gap:G.AutomorphismGroup()
 
Copy content oscar:automorphism_group(G)
 
Composition factors:$A_6$
Copy content comment:Composition factors of the group
 
Copy content magma:CompositionFactors(G);
 
Copy content gap:CompositionSeries(G);
 
Copy content sage:G.composition_series()
 
Copy content sage_gap:G.CompositionSeries()
 
Copy content oscar:composition_series(G)
 
Derived length:$0$
Copy content comment:Derived length of the group
 
Copy content magma:DerivedLength(G);
 
Copy content gap:DerivedLength(G);
 
Copy content sage:libgap(G).DerivedLength()
 
Copy content sage_gap:G.DerivedLength()
 
Copy content oscar:derived_length(G)
 

This group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).

Copy content comment:Determine if the group G is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content gap:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content sage_gap:G.IsAbelian()
 
Copy content oscar:is_abelian(G)
 
Copy content comment:Determine if the group G is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content gap:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content sage_gap:G.IsCyclic()
 
Copy content oscar:is_cyclic(G)
 
Copy content comment:Determine if the group G is nilpotent
 
Copy content magma:IsNilpotent(G);
 
Copy content gap:IsNilpotentGroup(G);
 
Copy content sage:G.is_nilpotent()
 
Copy content sage_gap:G.IsNilpotentGroup()
 
Copy content oscar:is_nilpotent(G)
 
Copy content comment:Determine if the group G is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content gap:IsSolvableGroup(G);
 
Copy content sage:G.is_solvable()
 
Copy content sage_gap:G.IsSolvableGroup()
 
Copy content oscar:is_solvable(G)
 
Copy content comment:Determine if the group G is supersolvable
 
Copy content gap:IsSupersolvableGroup(G);
 
Copy content sage:G.is_supersolvable()
 
Copy content sage_gap:G.IsSupersolvableGroup()
 
Copy content oscar:is_supersolvable(G)
 
Copy content comment:Determine if the group G is simple
 
Copy content magma:IsSimple(G);
 
Copy content gap:IsSimpleGroup(G);
 
Copy content sage:G.is_simple()
 
Copy content sage_gap:G.IsSimpleGroup()
 
Copy content oscar:is_simple(G)
 

Group statistics

Copy content comment:Compute statistics for the group G
 
Copy content 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;
 
Copy content 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");
 
Copy content 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))
 
Copy content sage_gap:# Sage code (using the GAP interface) to output the first two rows of the group statistics table element_orders = [g.Order() for g in G.Elements()] orders = sorted(list(set(element_orders))) print("Orders:", orders) print("Elements:", [element_orders.count(n) for n in orders], G.Order()) cc_orders = [cc.Representative().Order() for cc in G.ConjugacyClasses()] print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))
 
Copy content oscar:# Oscar code to output the first two rows of the group statistics table element_orders = [order(g) for g in elements(G)] orders = sort(unique(element_orders)) println("Orders: ", orders) element_counts = [count(==(n), element_orders) for n in orders] println("Elements: ", element_counts, " ", order(G)) ccs = conjugacy_classes(G) cc_orders = [order(representative(cc)) for cc in ccs] cc_counts = [count(==(n), cc_orders) for n in orders] println("Conjugacy classes: ", cc_counts, " ", length(ccs))
 

Order 1 2 3 4 5
Elements 1 45 80 90 144 360
Conjugacy classes   1 1 2 1 2 7
Divisions 1 1 2 1 1 6
Autjugacy classes 1 1 1 1 1 5

Copy content comment:Compute statistics about the characters of G
 
Copy content 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);
 
Copy content 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);
 
Copy content 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)]
 
Copy content sage_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 G.CharacterDegrees()
 
Copy content oscar:# Outputs an MSet containing the absolutely irreducible degrees of G and their multiplicities. character_degrees(G)
 

Dimension 1 5 8 9 10 16
Irr. complex chars.   1 2 2 1 1 0 7
Irr. rational chars. 1 2 0 1 1 1 6

Minimal presentations

Permutation degree:$6$
Transitive degree:$6$
Rank: $2$
Inequivalent generating pairs: $53$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 5 5 5
Arbitrary 5 5 5

Constructions

Show commands: Gap / Magma / Oscar / SageMath


Groups of Lie type:$\PSL(2,9)$, $\PSU(2,9)$, $\Omega(3,9)$, $\OmegaMinus(4,3)$, $\PSOMinus(4,3)$, $\POmega(3,9)$, $\POmegaMinus(4,3)$
Copy content magma:G := PSL(2,9);
 
Copy content gap:G := PSL(2,9);
 
Copy content sage:G = PSL(2,9)
 
Copy content oscar:F = GF(9); al = F.0; MS = MatrixSpace(F, 2, 2) G = MatrixGroup([MS([[al^1, 0], [0, al^7]]), MS([[al^4, 1], [al^4, 0]])])
 
Copy content magma:G := PSU(2,9);
 
Copy content gap:G := PSU(2,9);
 
Copy content sage:G = PSU(2,9)
 
Copy content magma:G := Omega(3,9);
 
Copy content gap:G := Omega(3,9);
 
Copy content magma:G := OmegaMinus(4,3);
 
Copy content magma:G := PSOMinus(4,3);
 
Copy content magma:G := POmega(3,9);
 
Copy content gap:G := POmega(3,9);
 
Copy content magma:G := POmegaMinus(4,3);
 
Permutation group: $\langle(2,3,4,5,6), (1,2,3,4,5)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 6 | (2,3,4,5,6), (1,2,3,4,5) >;
 
Copy content gap:G := Group( (2,3,4,5,6), (1,2,3,4,5) );
 
Copy content sage:G = PermutationGroup(['(2,3,4,5,6)', '(1,2,3,4,5)'])
 
Copy content sage_gap:G = gap.new('Group( (2,3,4,5,6), (1,2,3,4,5) )')
 
Copy content oscar:G = @permutation_group(6, (2,3,4,5,6), (1,2,3,4,5))
 
Matrix group:$\left\langle \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ -1 & 1 & 1 & -1 & -1 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & -1 & -1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 \end{array}\right) \right\rangle \subseteq \GL_{5}(\Z)$
Copy content comment:Define the group as a matrix group with coefficients in Z
 
Copy content magma:G := MatrixGroup< 5, Integers() | [[1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 1, 1, -1, -1, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, -1, -1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0]] >;
 
Copy content gap:G := Group([[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [-1, 1, 1, -1, -1], [0, 0, 0, 1, 0]], [[0, 0, 0, 1, 0], [0, 0, 1, 0, 0], [1, -1, -1, 1, 1], [0, 0, 0, 0, 1], [1, 0, 0, 0, 0]]]);
 
Copy content sage:MS = MatrixSpace(Integers(), 5, 5) G = MatrixGroup([MS([[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [-1, 1, 1, -1, -1], [0, 0, 0, 1, 0]]), MS([[0, 0, 0, 1, 0], [0, 0, 1, 0, 0], [1, -1, -1, 1, 1], [0, 0, 0, 0, 1], [1, 0, 0, 0, 0]])])
 
Copy content sage_gap:G = gap.new('Group([[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [-1, 1, 1, -1, -1], [0, 0, 0, 1, 0]], [[0, 0, 0, 1, 0], [0, 0, 1, 0, 0], [1, -1, -1, 1, 1], [0, 0, 0, 0, 1], [1, 0, 0, 0, 0]]])')
 
Copy content oscar:G = matrix_group([matrix(ZZ, [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [-1, 1, 1, -1, -1], [0, 0, 0, 1, 0]]), matrix(ZZ, [[0, 0, 0, 1, 0], [0, 0, 1, 0, 0], [1, -1, -1, 1, 1], [0, 0, 0, 0, 1], [1, 0, 0, 0, 0]])])
 
$\left\langle \left(\begin{array}{rrrr} 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{2})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 4, GF(2) | [[1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0], [0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0]] >;
 
Copy content gap:G := Group([[[ Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0 ], [ Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ]], [[ 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ], [ Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ], [ Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2) ]]]);
 
Copy content sage:MS = MatrixSpace(GF(2), 4, 4) G = MatrixGroup([MS([[1, 0, 1, 1], [1, 1, 1, 0], [0, 0, 0, 1], [0, 0, 1, 0]]), MS([[0, 1, 1, 0], [0, 1, 1, 1], [1, 1, 1, 1], [1, 0, 1, 0]])])
 
Copy content sage_gap:G = gap.new('Group([[[ Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0 ], [ Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ]], [[ 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ], [ Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ], [ Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2) ]]])')
 
Copy content oscar:G = matrix_group([matrix(GF(2), [[1, 0, 1, 1], [1, 1, 1, 0], [0, 0, 0, 1], [0, 0, 1, 0]]), matrix(GF(2), [[0, 1, 1, 0], [0, 1, 1, 1], [1, 1, 1, 1], [1, 0, 1, 0]])])
 
Transitive group: 6T15 10T26 15T20 20T89 all 8
Copy content magma:G := TransitiveGroup(6, 15);
 
Copy content gap:G := TransitiveGroup(6, 15);
 
Copy content sage:G = TransitiveGroup(6, 15)
 
Copy content sage_gap:G = libgap.TransitiveGroup(6, 15)
 
Copy content oscar:G = transitive_group(6, 15)
 
Copy content magma:G := TransitiveGroup(10, 26);
 
Copy content gap:G := TransitiveGroup(10, 26);
 
Copy content sage:G = TransitiveGroup(10, 26)
 
Copy content sage_gap:G = libgap.TransitiveGroup(10, 26)
 
Copy content oscar:G = transitive_group(10, 26)
 
Copy content magma:G := TransitiveGroup(15, 20);
 
Copy content gap:G := TransitiveGroup(15, 20);
 
Copy content sage:G = TransitiveGroup(15, 20)
 
Copy content sage_gap:G = libgap.TransitiveGroup(15, 20)
 
Copy content oscar:G = transitive_group(15, 20)
 
Copy content magma:G := TransitiveGroup(20, 89);
 
Copy content gap:G := TransitiveGroup(20, 89);
 
Copy content sage:G = TransitiveGroup(20, 89)
 
Copy content sage_gap:G = libgap.TransitiveGroup(20, 89)
 
Copy content oscar:G = transitive_group(20, 89)
 
Copy content magma:G := TransitiveGroup(30, 88);
 
Copy content gap:G := TransitiveGroup(30, 88);
 
Copy content sage:G = TransitiveGroup(30, 88)
 
Copy content sage_gap:G = libgap.TransitiveGroup(30, 88)
 
Copy content oscar:G = transitive_group(30, 88)
 
Copy content magma:G := TransitiveGroup(36, 555);
 
Copy content gap:G := TransitiveGroup(36, 555);
 
Copy content sage:G = TransitiveGroup(36, 555)
 
Copy content sage_gap:G = libgap.TransitiveGroup(36, 555)
 
Copy content oscar:G = transitive_group(36, 555)
 
Copy content magma:G := TransitiveGroup(40, 304);
 
Copy content gap:G := TransitiveGroup(40, 304);
 
Copy content sage:G = TransitiveGroup(40, 304)
 
Copy content sage_gap:G = libgap.TransitiveGroup(40, 304)
 
Copy content oscar:G = transitive_group(40, 304)
 
Copy content magma:G := TransitiveGroup(45, 49);
 
Copy content gap:G := TransitiveGroup(45, 49);
 
Copy content sage:G = TransitiveGroup(45, 49)
 
Copy content sage_gap:G = libgap.TransitiveGroup(45, 49)
 
Copy content oscar:G = transitive_group(45, 49)
 
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: not isomorphic to a non-trivial semidirect product
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 6.

Homology

Abelianization: $C_1 $
Copy content comment:The abelianization of the group
 
Copy content magma:quo< G | CommutatorSubgroup(G) >;
 
Copy content gap:FactorGroup(G, DerivedSubgroup(G));
 
Copy content sage:G.quotient(G.commutator())
 
Copy content sage_gap:G.FactorGroup(G.DerivedSubgroup())
 
Copy content oscar:quo(G, derived_subgroup(G)[1])
 
Schur multiplier: $C_{6}$
Copy content comment:The Schur multiplier of the group
 
Copy content gap:AbelianInvariantsMultiplier(G);
 
Copy content sage:G.homology(2)
 
Copy content sage_gap:G.AbelianInvariantsMultiplier()
 
Commutator length: $1$
Copy content comment:The commutator length of the group
 
Copy content gap:CommutatorLength(G);
 
Copy content sage_gap:G.CommutatorLength()
 

Subgroups

Copy content comment:List of subgroups of the group
 
Copy content magma:Subgroups(G);
 
Copy content gap:AllSubgroups(G);
 
Copy content sage:G.subgroups()
 
Copy content sage_gap:G.AllSubgroups()
 
Copy content oscar:subgroups(G)
 

There are 501 subgroups in 22 conjugacy classes, 2 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_1$ $G/Z \simeq$ $A_6$
Copy content comment:Center of the group
 
Copy content magma:Center(G);
 
Copy content gap:Center(G);
 
Copy content sage:G.center()
 
Copy content sage_gap:G.Center()
 
Copy content oscar:center(G)
 
Commutator: $G' \simeq$ $A_6$ $G/G' \simeq$ $C_1$
Copy content comment:Commutator subgroup of the group G
 
Copy content magma:CommutatorSubgroup(G);
 
Copy content gap:DerivedSubgroup(G);
 
Copy content sage:G.commutator()
 
Copy content sage_gap:G.DerivedSubgroup()
 
Copy content oscar:derived_subgroup(G)
 
Frattini: $\Phi \simeq$ $C_1$ $G/\Phi \simeq$ $A_6$
Copy content comment:Frattini subgroup of the group G
 
Copy content magma:FrattiniSubgroup(G);
 
Copy content gap:FrattiniSubgroup(G);
 
Copy content sage:G.frattini_subgroup()
 
Copy content sage_gap:G.FrattiniSubgroup()
 
Copy content oscar:frattini_subgroup(G)
 
Fitting: $\operatorname{Fit} \simeq$ $C_1$ $G/\operatorname{Fit} \simeq$ $A_6$
Copy content comment:Fitting subgroup of the group G
 
Copy content magma:FittingSubgroup(G);
 
Copy content gap:FittingSubgroup(G);
 
Copy content sage:G.fitting_subgroup()
 
Copy content sage_gap:G.FittingSubgroup()
 
Copy content oscar:fitting_subgroup(G)
 
Radical: $R \simeq$ $C_1$ $G/R \simeq$ $A_6$
Copy content comment:Radical of the group G
 
Copy content magma:Radical(G);
 
Copy content gap:SolvableRadical(G);
 
Copy content sage_gap:G.SolvableRadical()
 
Copy content oscar:solvable_radical(G)
 
Socle: $\operatorname{soc} \simeq$ $A_6$ $G/\operatorname{soc} \simeq$ $C_1$
Copy content comment:Socle of the group G
 
Copy content magma:Socle(G);
 
Copy content gap:Socle(G);
 
Copy content sage:G.socle()
 
Copy content sage_gap:G.Socle()
 
Copy content oscar:socle(G)
 
2-Sylow subgroup: $P_{ 2 } \simeq$ $D_4$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup: $P_{ 5 } \simeq$ $C_5$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $A_6$
Copy content comment:Derived series of the group G
 
Copy content magma:DerivedSeries(G);
 
Copy content gap:DerivedSeriesOfGroup(G);
 
Copy content sage:G.derived_series()
 
Copy content sage_gap:G.DerivedSeriesOfGroup()
 
Copy content oscar:derived_series(G)
 
Chief series $A_6$ $\rhd$ $C_1$
Copy content comment:Chief series of the group G
 
Copy content magma:ChiefSeries(G);
 
Copy content gap:ChiefSeries(G);
 
Copy content sage:libgap(G).ChiefSeries()
 
Copy content sage_gap:G.ChiefSeries()
 
Copy content oscar:chief_series(G)
 
Lower central series $A_6$
Copy content comment:The lower central series of the group G
 
Copy content magma:LowerCentralSeries(G);
 
Copy content gap:LowerCentralSeriesOfGroup(G);
 
Copy content sage:G.lower_central_series()
 
Copy content sage_gap:G.LowerCentralSeriesOfGroup()
 
Copy content oscar:lower_central_series(G)
 
Upper central series $C_1$
Copy content comment:The upper central series of the group G
 
Copy content magma:UpperCentralSeries(G);
 
Copy content gap:UpperCentralSeriesOfGroup(G);
 
Copy content sage:G.upper_central_series()
 
Copy content sage_gap:G.UpperCentralSeriesOfGroup()
 
Copy content oscar:upper_central_series(G)
 

Supergroups

This group is a maximal subgroup of 21 larger groups in the database.

This group is a maximal quotient of 22 larger groups in the database.

Character theory

Copy content comment:Character table
 
Copy content magma:CharacterTable(G); // Output not guaranteed to exactly match the LMFDB table
 
Copy content gap:CharacterTable(G); # Output not guaranteed to exactly match the LMFDB table
 
Copy content sage:G.character_table() # Output not guaranteed to exactly match the LMFDB table
 
Copy content sage_gap:G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table
 
Copy content oscar:character_table(G) # Output not guaranteed to exactly match the LMFDB table
 

Complex character table

1A 2A 3A 3B 4A 5A1 5A2
Size 1 45 40 40 90 72 72
2 P 1A 1A 3A 3B 2A 5A2 5A1
3 P 1A 2A 1A 1A 4A 5A2 5A1
5 P 1A 2A 3A 3B 4A 1A 1A
Type
360.118.1a R 1 1 1 1 1 1 1
360.118.5a R 5 1 −1 2 −1 0 0
360.118.5b R 5 1 2 −1 −1 0 0
360.118.8a1 R 8 0 −1 −1 0 −ζ5−1−ζ5 −ζ5−2−ζ52
360.118.8a2 R 8 0 −1 −1 0 −ζ5−2−ζ52 −ζ5−1−ζ5
360.118.9a R 9 1 0 0 1 −1 −1
360.118.10a R 10 −2 1 1 0 0 0

Rational character table

1A 2A 3A 3B 4A 5A
Size 1 45 40 40 90 144
2 P 1A 1A 3A 3B 2A 5A
3 P 1A 2A 1A 1A 4A 5A
5 P 1A 2A 3A 3B 4A 1A
360.118.1a 1 1 1 1 1 1
360.118.5a 5 1 −1 2 −1 0
360.118.5b 5 1 2 −1 −1 0
360.118.8a 16 0 −2 −2 0 1
360.118.9a 9 1 0 0 1 −1
360.118.10a 10 −2 1 1 0 0