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Properties

Label 1824.1019
Order \( 2^{5} \cdot 3 \cdot 19 \)
Exponent \( 2^{2} \cdot 3 \cdot 19 \)
Abelian yes
$\card{\Aut(G)}$ \( 2^{11} \cdot 3^{3} \)
Perm deg. $32$
Trans deg. $1824$
Rank $3$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SmallGroup(1824, 1019);
 
Copy content gap:G := SmallGroup(1824, 1019);
 
Copy content sage_gap:G = libgap.SmallGroup(1824, 1019)
 
Copy content comment:Define the group as a permutation group
 
Copy content sage:G = PermutationGroup(['(3,6,4,5)', '(7,10,8,9)', '(1,2)', '(3,4)(5,6)', '(7,8)(9,10)', '(11,13,12)', '(14,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15)'])
 

Group information

Description:$C_2\times C_4\times C_{228}$
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: \(228\)\(\medspace = 2^{2} \cdot 3 \cdot 19 \)
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:$C_2\times C_{18}\times C_2^6.S_4$, of order \(55296\)\(\medspace = 2^{11} \cdot 3^{3} \)
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()
 
Nilpotency class:$1$
Copy content comment:Nilpotency class of the group
 
Copy content magma:NilpotencyClass(G);
 
Copy content gap:NilpotencyClassOfGroup(G);
 
Copy content sage_gap:G.NilpotencyClassOfGroup()
 
Derived length:$1$
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 abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).

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 12 19 38 57 76 114 228
Elements 1 7 2 24 14 48 18 126 36 432 252 864 1824
Conjugacy classes   1 7 2 24 14 48 18 126 36 432 252 864 1824
Divisions 1 7 1 12 7 12 1 7 1 12 7 12 80
Autjugacy classes 1 2 1 1 2 1 1 2 1 1 2 1 16

Minimal presentations

Permutation degree:$32$
Transitive degree:$1824$
Rank: $3$
Inequivalent generating triples: $34671$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible none not computed none
Arbitrary 3 not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Presentation:Abelian group $\langle a, b, c \mid a^{2}=b^{4}=c^{228}=1 \rangle$ 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, -2, -2, -3, -19, 36, 80, 102, 166]); a,b,c := Explode([G.1, G.2, G.4]); AssignNames(~G, ["a", "b", "b2", "c", "c2", "c4", "c12"]);
 
Copy content gap:G := PcGroupCode(503848304969021487744595235,1824); a := G.1; b := G.2; c := G.4;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(503848304969021487744595235,1824)'); a = G.1; b = G.2; c = G.4;
 
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(503848304969021487744595235,1824)'); a = G.1; b = G.2; c = G.4;
 
Permutation group:Degree $32$ $\langle(3,6,4,5), (7,10,8,9), (1,2), (3,4)(5,6), (7,8)(9,10), (11,13,12), (14,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 32 | (3,6,4,5), (7,10,8,9), (1,2), (3,4)(5,6), (7,8)(9,10), (11,13,12), (14,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15) >;
 
Copy content gap:G := Group( (3,6,4,5), (7,10,8,9), (1,2), (3,4)(5,6), (7,8)(9,10), (11,13,12), (14,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15) );
 
Copy content sage:G = PermutationGroup(['(3,6,4,5)', '(7,10,8,9)', '(1,2)', '(3,4)(5,6)', '(7,8)(9,10)', '(11,13,12)', '(14,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15)'])
 
Matrix group:$\left\langle \left(\begin{array}{rr} 1 & 5 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 39 & 0 \\ 0 & 39 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 37 \end{array}\right), \left(\begin{array}{rr} 39 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 77 & 0 \\ 0 & 77 \end{array}\right), \left(\begin{array}{rr} 94 & 0 \\ 0 & 37 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/95\Z)$
Copy content comment:Define the group as a matrix group with coefficients in GLZN
 
Copy content magma:G := MatrixGroup< 2, Integers(95) | [[1, 5, 0, 1], [39, 0, 0, 39], [37, 0, 0, 37], [39, 0, 0, 1], [11, 0, 0, 11], [77, 0, 0, 77], [94, 0, 0, 37]] >;
 
Copy content gap:G := Group([[[ZmodnZObj(1,95), ZmodnZObj(5,95)], [ZmodnZObj(0,95), ZmodnZObj(1,95)]],[[ZmodnZObj(39,95), ZmodnZObj(0,95)], [ZmodnZObj(0,95), ZmodnZObj(39,95)]],[[ZmodnZObj(37,95), ZmodnZObj(0,95)], [ZmodnZObj(0,95), ZmodnZObj(37,95)]],[[ZmodnZObj(39,95), ZmodnZObj(0,95)], [ZmodnZObj(0,95), ZmodnZObj(1,95)]],[[ZmodnZObj(11,95), ZmodnZObj(0,95)], [ZmodnZObj(0,95), ZmodnZObj(11,95)]],[[ZmodnZObj(77,95), ZmodnZObj(0,95)], [ZmodnZObj(0,95), ZmodnZObj(77,95)]],[[ZmodnZObj(94,95), ZmodnZObj(0,95)], [ZmodnZObj(0,95), ZmodnZObj(37,95)]]]);
 
Copy content sage:MS = MatrixSpace(Integers(95), 2, 2) G = MatrixGroup([MS([[1, 5], [0, 1]]), MS([[39, 0], [0, 39]]), MS([[37, 0], [0, 37]]), MS([[39, 0], [0, 1]]), MS([[11, 0], [0, 11]]), MS([[77, 0], [0, 77]]), MS([[94, 0], [0, 37]])])
 
Direct product: $C_2$ $\, \times\, $ $C_4$ ${}^2$ $\, \times\, $ $C_3$ $\, \times\, $ $C_{19}$
Semidirect product: not isomorphic to a non-trivial semidirect product
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $(C_2^2\times C_{76})$ . $C_6$ $C_6$ . $(C_2^2\times C_{76})$ $(C_2^2\times C_{228})$ . $C_2$ $(C_2\times C_{228})$ . $C_2^2$ all 24
Aut. group: $\Aut(C_{3664})$

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

Homology

Primary decomposition: $C_{2} \times C_{4}^{2} \times C_{3} \times C_{19}$
Copy content comment:The primary decomposition of the group
 
Copy content magma:PrimaryInvariants(G);
 
Copy content gap:AbelianInvariants(G);
 
Copy content sage_gap:G.AbelianInvariants()
 
Schur multiplier: $C_{2}^{2} \times C_{4}$
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: $0$
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 216 subgroups, all normal (16 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_2\times C_4\times C_{228}$ $G/Z \simeq$ $C_1$
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: $G' \simeq$ $C_1$ $G/G' \simeq$ $C_2\times C_4\times C_{228}$
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: $\Phi \simeq$ $C_2^2$ $G/\Phi \simeq$ $C_2^2\times C_{114}$
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: $\operatorname{Fit} \simeq$ $C_2\times C_4\times C_{228}$ $G/\operatorname{Fit} \simeq$ $C_1$
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: $R \simeq$ $C_2\times C_4\times C_{228}$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_2^2\times C_{114}$ $G/\operatorname{soc} \simeq$ $C_2^2$
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: $P_{ 2 } \simeq$ $C_2\times C_4^2$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3$
19-Sylow subgroup: $P_{ 19 } \simeq$ $C_{19}$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $C_2\times C_4\times C_{228}$ $\rhd$ $C_1$
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 $C_2\times C_4\times C_{228}$ $\rhd$ $C_2^2\times C_{228}$ $\rhd$ $C_2^2\times C_{114}$ $\rhd$ $C_2\times C_{114}$ $\rhd$ $C_{114}$ $\rhd$ $C_{57}$ $\rhd$ $C_{19}$ $\rhd$ $C_1$
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 $C_2\times C_4\times C_{228}$ $\rhd$ $C_1$
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 $C_1$ $\lhd$ $C_2\times C_4\times C_{228}$
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

The $1824 \times 1824$ character table is not available for this group.

Rational character table

The $80 \times 80$ rational character table is not available for this group.