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Properties

Label 1824.279
Order \( 2^{5} \cdot 3 \cdot 19 \)
Exponent \( 2^{3} \cdot 3 \cdot 19 \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{3} \cdot 19 \)
$\card{Z(G)}$ \( 2^{2} \cdot 19 \)
$\card{\Aut(G)}$ \( 2^{6} \cdot 3^{3} \)
$\card{\mathrm{Out}(G)}$ \( 2^{3} \cdot 3^{2} \)
Perm deg. $31$
Trans deg. $456$
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SmallGroup(1824, 279);
 
Copy content gap:G := SmallGroup(1824, 279);
 
Copy content sage_gap:G = libgap.SmallGroup(1824, 279)
 
Copy content comment:Define the group as a permutation group
 
Copy content sage:G = PermutationGroup(['(2,4)(5,6,7,9,8,10,11,12)', '(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31)', '(5,7,8,11)(6,9,10,12)', '(5,8)(6,10)(7,11)(9,12)', '(2,3,4)', '(1,2)(3,4)', '(1,3)(2,4)'])
 

Group information

Description:$A_4:C_{152}$
Order: \(1824\)\(\medspace = 2^{5} \cdot 3 \cdot 19 \)
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()
 
Exponent: \(456\)\(\medspace = 2^{3} \cdot 3 \cdot 19 \)
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()
 
Automorphism group:$C_2^2\times C_{18}\times S_4$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$ x 5, $C_3$, $C_{19}$
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()
 
Derived length:$3$
Copy content comment:Derived length of the group
 
Copy content magma:DerivedLength(G);
 
Copy content gap:DerivedLength(G);
 
Copy content sage_gap:G.DerivedLength()
 

This group is nonabelian and monomial (hence solvable).

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 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 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 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 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 comment:Determine if the group G is simple
 
Copy content magma:IsSimple(G);
 
Copy content gap:IsSimpleGroup(G);
 
Copy content sage_gap:G.IsSimpleGroup()
 

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))
 

Order 1 2 3 4 6 8 12 19 38 57 76 114 152 228
Elements 1 7 8 8 8 48 16 18 126 144 144 144 864 288 1824
Conjugacy classes   1 3 1 4 1 8 2 18 54 18 72 18 144 36 380
Divisions 1 3 1 2 1 2 1 1 3 1 2 1 2 1 22
Autjugacy classes 1 3 1 2 1 2 1 1 3 1 2 1 2 1 22

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:G.CharacterDegrees()
 

Dimension 1 2 3 4 6 12 18 36 54 72 108 216
Irr. complex chars.   152 76 152 0 0 0 0 0 0 0 0 0 380
Irr. rational chars. 2 3 2 2 1 1 2 3 2 2 1 1 22

Minimal presentations

Permutation degree:$31$
Transitive degree:$456$
Rank: $2$
Inequivalent generating pairs: $720$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 3 6 216
Arbitrary 3 5 25

Constructions

Show commands: Gap / Magma / SageMath


Presentation: $\langle a, b, c, d \mid a^{8}=b^{3}=c^{2}=d^{38}=[a,c]=[c,d]=1, b^{a}=b^{2}, d^{a}=cd, c^{b}=cd^{19}, d^{b}=cd^{20} \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([7, -2, -2, -2, -3, -2, 2, -19, 14, 36, 451, 4120, 3029, 5192, 124]); a,b,c,d := Explode([G.1, G.4, G.5, G.6]); AssignNames(~G, ["a", "a2", "a4", "b", "c", "d", "d2"]);
 
Copy content gap:G := PcGroupCode(199761335934849200767429266023540284505,1824); a := G.1; b := G.4; c := G.5; d := G.6;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(199761335934849200767429266023540284505,1824)'); a = G.1; b = G.4; c = G.5; d = G.6;
 
Copy content sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(199761335934849200767429266023540284505,1824)'); a = G.1; b = G.4; c = G.5; d = G.6;
 
Permutation group:Degree $31$ $\langle(2,4)(5,6,7,9,8,10,11,12), (13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 31 | (2,4)(5,6,7,9,8,10,11,12), (13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31), (5,7,8,11)(6,9,10,12), (5,8)(6,10)(7,11)(9,12), (2,3,4), (1,2)(3,4), (1,3)(2,4) >;
 
Copy content gap:G := Group( (2,4)(5,6,7,9,8,10,11,12), (13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31), (5,7,8,11)(6,9,10,12), (5,8)(6,10)(7,11)(9,12), (2,3,4), (1,2)(3,4), (1,3)(2,4) );
 
Copy content sage:G = PermutationGroup(['(2,4)(5,6,7,9,8,10,11,12)', '(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31)', '(5,7,8,11)(6,9,10,12)', '(5,8)(6,10)(7,11)(9,12)', '(2,3,4)', '(1,2)(3,4)', '(1,3)(2,4)'])
 
Direct product: $C_{19}$ $\, \times\, $ $(A_4:C_8)$
Semidirect product: $A_4$ $\,\rtimes\,$ $C_{152}$ $(A_4\times C_{19})$ $\,\rtimes\,$ $C_8$ $C_2^2$ $\,\rtimes\,$ $(C_3:C_{152})$ $(C_2\times C_{38})$ $\,\rtimes\,$ $(C_3:C_8)$ more information
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $C_{76}$ . $S_4$ $C_4$ . $(C_{19}\times S_4)$ $(A_4\times C_{76})$ . $C_2$ $(A_4\times C_{38})$ . $C_4$ all 12

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization: $C_{152} \simeq C_{8} \times C_{19}$
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())
 
Schur multiplier: $C_{2}$
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()
 

There are 136 subgroups in 52 conjugacy classes, 20 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_{76}$ $G/Z \simeq$ $S_4$
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()
 
Commutator: $G' \simeq$ $A_4$ $G/G' \simeq$ $C_{152}$
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()
 
Frattini: $\Phi \simeq$ $C_4$ $G/\Phi \simeq$ $C_{19}\times S_4$
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()
 
Fitting: $\operatorname{Fit} \simeq$ $C_2^2\times C_{76}$ $G/\operatorname{Fit} \simeq$ $S_3$
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()
 
Radical: $R \simeq$ $A_4:C_{152}$ $G/R \simeq$ $C_1$
Copy content comment:Radical of the group G
 
Copy content magma:Radical(G);
 
Copy content gap:SolvableRadical(G);
 
Copy content sage_gap:G.SolvableRadical()
 
Socle: $\operatorname{soc} \simeq$ $C_2^2\times C_{38}$ $G/\operatorname{soc} \simeq$ $C_3:C_4$
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()
 
2-Sylow subgroup: $P_{ 2 } \simeq$ $C_2^2:C_8$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3$
19-Sylow subgroup: $P_{ 19 } \simeq$ $C_{19}$

Subgroup diagram and profile

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
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Subgroup information

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Series

Derived series $A_4:C_{152}$ $\rhd$ $A_4$ $\rhd$ $C_2^2$ $\rhd$ $C_1$
Copy content comment:Derived series of the group GF
 
Copy content magma:DerivedSeries(G);
 
Copy content gap:DerivedSeriesOfGroup(G);
 
Copy content sage:G.derived_series()
 
Copy content sage_gap:G.DerivedSeriesOfGroup()
 
Chief series $A_4:C_{152}$ $\rhd$ $A_4\times C_{76}$ $\rhd$ $A_4\times C_{38}$ $\rhd$ $A_4\times C_{19}$ $\rhd$ $C_2\times C_{38}$ $\rhd$ $C_{19}$ $\rhd$ $C_1$
Copy content comment:Chief series of the group G
 
Copy content magma:ChiefSeries(G);
 
Copy content gap:ChiefSeries(G);
 
Copy content sage_gap:G.ChiefSeries()
 
Lower central series $A_4:C_{152}$ $\rhd$ $A_4$
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()
 
Upper central series $C_1$ $\lhd$ $C_{76}$
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()
 

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
 

Complex character table

See the $380 \times 380$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $22 \times 22$ rational character table.