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Properties

Label 84.15
Order \( 2^{2} \cdot 3 \cdot 7 \)
Exponent \( 2 \cdot 3 \cdot 7 \)
Abelian yes
$\card{\Aut(G)}$ \( 2^{3} \cdot 3^{2} \)
Perm deg. $14$
Trans deg. $84$
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SmallGroup(84, 15);
 
Copy content gap:G := SmallGroup(84, 15);
 
Copy content sage_gap:G = libgap.SmallGroup(84, 15)
 
Copy content comment:Define the group as a permutation group
 
Copy content sage:G = PermutationGroup(['(1,2)', '(3,4)', '(5,7,6)', '(8,14,13,12,11,10,9)'])
 

Group information

Description:$C_2\times C_{42}$
Order: \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
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: \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \)
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:$C_6\times D_6$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
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 2, $C_3$, $C_7$
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), elementary for $p = 2$ (hence hyperelementary), and metacyclic.

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 6 7 14 21 42
Elements 1 3 2 6 6 18 12 36 84
Conjugacy classes   1 3 2 6 6 18 12 36 84
Divisions 1 3 1 3 1 3 1 3 16
Autjugacy classes 1 1 1 1 1 1 1 1 8

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 6 12
Irr. complex chars.   84 0 0 0 84
Irr. rational chars. 4 4 4 4 16

Minimal presentations

Permutation degree:$14$
Transitive degree:$84$
Rank: $2$
Inequivalent generating pairs: $32$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible none none none
Arbitrary 2 3 8

Constructions

Show commands: Gap / Magma / SageMath


Presentation:Abelian group $\langle a, b \mid a^{2}=b^{42}=1 \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([4, -2, -2, -3, -7, 21, 46]); a,b := Explode([G.1, G.2]); AssignNames(~G, ["a", "b", "b2", "b6"]);
 
Copy content gap:G := PcGroupCode(60391019,84); a := G.1; b := G.2;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(60391019,84)'); a = G.1; b = G.2;
 
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(60391019,84)'); a = G.1; b = G.2;
 
Permutation group:Degree $14$ $\langle(1,2), (3,4), (5,7,6), (8,14,13,12,11,10,9)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 14 | (1,2), (3,4), (5,7,6), (8,14,13,12,11,10,9) >;
 
Copy content gap:G := Group( (1,2), (3,4), (5,7,6), (8,14,13,12,11,10,9) );
 
Copy content sage:G = PermutationGroup(['(1,2)', '(3,4)', '(5,7,6)', '(8,14,13,12,11,10,9)'])
 
Matrix group:$\left\langle \left(\begin{array}{rr} 30 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 42 & 0 \\ 0 & 42 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{43})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 2, GF(43) | [[30, 0, 0, 13], [42, 0, 0, 42]] >;
 
Copy content gap:G := Group([[[ Z(43)^11, 0*Z(43) ], [ 0*Z(43), Z(43)^32 ]], [[ Z(43)^21, 0*Z(43) ], [ 0*Z(43), Z(43)^21 ]]]);
 
Copy content sage:MS = MatrixSpace(GF(43), 2, 2) G = MatrixGroup([MS([[30, 0], [0, 13]]), MS([[42, 0], [0, 42]])])
 
Direct product: $C_2$ ${}^2$ $\, \times\, $ $C_3$ $\, \times\, $ $C_7$
Semidirect product: not isomorphic to a non-trivial semidirect product
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Aut. group: $\Aut(C_{129})$ $\Aut(C_{147})$ $\Aut(C_{172})$ $\Aut(C_{196})$ all 6

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

Homology

Primary decomposition: $C_{2}^{2} \times C_{3} \times C_{7}$
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: $C_{2}$
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 20 subgroups, all normal (8 characteristic).

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

Special subgroups

Center: $Z \simeq$ $C_2\times C_{42}$ $G/Z \simeq$ $C_1$
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: $G' \simeq$ $C_1$ $G/G' \simeq$ $C_2\times C_{42}$
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: $\Phi \simeq$ $C_1$ $G/\Phi \simeq$ $C_2\times C_{42}$
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: $\operatorname{Fit} \simeq$ $C_2\times C_{42}$ $G/\operatorname{Fit} \simeq$ $C_1$
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: $R \simeq$ $C_2\times C_{42}$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_2\times C_{42}$ $G/\operatorname{soc} \simeq$ $C_1$
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()
 
2-Sylow subgroup: $P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup: $P_{ 7 } \simeq$ $C_7$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $C_2\times C_{42}$ $\rhd$ $C_1$
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 $C_2\times C_{42}$ $\rhd$ $C_{42}$ $\rhd$ $C_{21}$ $\rhd$ $C_7$ $\rhd$ $C_1$
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 $C_2\times C_{42}$ $\rhd$ $C_1$
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 $C_1$ $\lhd$ $C_2\times C_{42}$
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()
 

Supergroups

This group is a maximal subgroup of 77 larger groups in the database.

This group is a maximal quotient of 62 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
 

Complex character table

See the $84 \times 84$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

1A 2A 2B 2C 3A 6A 6B 6C 7A 14A 14B 14C 21A 42A 42B 42C
Size 1 1 1 1 2 2 2 2 6 6 6 6 12 12 12 12
2 P 1A 1A 1A 1A 3A 3A 3A 3A 7A 7A 7A 7A 21A 21A 21A 21A
3 P 1A 2A 2B 2C 1A 2A 2B 2C 7A 14A 14B 14C 7A 14A 14B 14C
7 P 1A 2A 2B 2C 3A 6A 6B 6C 1A 2A 2B 2C 3A 6A 6B 6C
84.15.1a 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
84.15.1b 1 −1 −1 1 1 1 −1 −1 1 1 −1 1 1 −1 −1 −1
84.15.1c 1 −1 1 −1 1 −1 −1 1 1 −1 −1 −1 1 1 1 1
84.15.1d 1 1 −1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 −1 −1
84.15.1e 2 2 2 2 −1 −1 −1 −1 2 2 2 2 −1 −1 −1 −1
84.15.1f 2 −2 −2 2 −1 −1 1 1 2 2 −2 2 −1 1 1 1
84.15.1g 2 −2 2 −2 −1 1 1 −1 2 −2 −2 −2 −1 −1 −1 −1
84.15.1h 2 2 −2 −2 −1 1 −1 1 2 −2 2 −2 −1 1 1 1
84.15.1i 6 6 6 6 6 6 6 6 −1 −1 −1 −1 −1 −1 −1 −1
84.15.1j 6 −6 −6 6 6 6 −6 −6 −1 −1 1 −1 −1 1 1 1
84.15.1k 6 −6 6 −6 6 −6 −6 6 −1 1 1 1 −1 −1 −1 −1
84.15.1l 6 6 −6 −6 6 −6 6 −6 −1 1 −1 1 −1 1 1 1
84.15.1m 12 12 12 12 −6 −6 −6 −6 −2 −2 −2 −2 1 1 1 1
84.15.1n 12 −12 −12 12 −6 −6 6 6 −2 −2 2 −2 1 −1 −1 −1
84.15.1o 12 −12 12 −12 −6 6 6 −6 −2 2 2 2 1 1 1 1
84.15.1p 12 12 −12 −12 −6 6 −6 6 −2 2 −2 2 1 −1 −1 −1