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Properties

Label 406.1
Order \( 2 \cdot 7 \cdot 29 \)
Exponent \( 2 \cdot 7 \cdot 29 \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2 \cdot 7 \)
$\card{Z(G)}$ \( 1 \)
$\card{\Aut(G)}$ \( 2^{2} \cdot 7 \cdot 29 \)
$\card{\mathrm{Out}(G)}$ \( 2 \)
Perm deg. $29$
Trans deg. $29$
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SmallGroup(406, 1);
 
Copy content gap:G := SmallGroup(406, 1);
 
Copy content sage_gap:G = libgap.SmallGroup(406, 1)
 
Copy content comment:Define the group as a permutation group
 
Copy content sage:G = PermutationGroup(['(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)', '(2,17,25,8,26,24,21)(3,4,20,15,22,18,12)(5,7,10,29,14,6,23)(9,13,19,28,27,11,16)', '(1,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2)'])
 

Group information

Description:$C_{29}:C_{14}$
Order: \(406\)\(\medspace = 2 \cdot 7 \cdot 29 \)
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: \(406\)\(\medspace = 2 \cdot 7 \cdot 29 \)
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:$F_{29}$, of order \(812\)\(\medspace = 2^{2} \cdot 7 \cdot 29 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$, $C_7$, $C_{29}$
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:$2$
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 a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).

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 7 14 29
Elements 1 29 174 174 28 406
Conjugacy classes   1 1 6 6 2 16
Divisions 1 1 1 1 1 5
Autjugacy classes 1 1 6 6 1 15

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 6 14 28
Irr. complex chars.   14 0 2 0 16
Irr. rational chars. 2 2 0 1 5

Minimal presentations

Permutation degree:$29$
Transitive degree:$29$
Rank: $2$
Inequivalent generating pairs: $144$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 14 14 28
Arbitrary 14 14 28

Constructions

Show commands: Gap / Magma / SageMath


Presentation: $\langle a, b \mid a^{14}=b^{29}=1, b^{a}=b^{13} \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([3, -2, -7, -29, 6, 1640, 1517]); a,b := Explode([G.1, G.3]); AssignNames(~G, ["a", "a2", "b"]);
 
Copy content gap:G := PcGroupCode(2666644510823,406); a := G.1; b := G.3;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(2666644510823,406)'); a = G.1; b = G.3;
 
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(2666644510823,406)'); a = G.1; b = G.3;
 
Permutation group:Degree $29$ $\langle(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 29 | (2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16), (2,17,25,8,26,24,21)(3,4,20,15,22,18,12)(5,7,10,29,14,6,23)(9,13,19,28,27,11,16), (1,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2) >;
 
Copy content gap:G := Group( (2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16), (2,17,25,8,26,24,21)(3,4,20,15,22,18,12)(5,7,10,29,14,6,23)(9,13,19,28,27,11,16), (1,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2) );
 
Copy content sage:G = PermutationGroup(['(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)', '(2,17,25,8,26,24,21)(3,4,20,15,22,18,12)(5,7,10,29,14,6,23)(9,13,19,28,27,11,16)', '(1,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2)'])
 
Matrix group:$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 20 & 0 \\ 0 & 6 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{29})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 2, GF(29) | [[1, 1, 0, 1], [20, 0, 0, 6]] >;
 
Copy content gap:G := Group([[[ Z(29)^0, Z(29)^0 ], [ 0*Z(29), Z(29)^0 ]], [[ Z(29)^24, 0*Z(29) ], [ 0*Z(29), Z(29)^6 ]]]);
 
Copy content sage:MS = MatrixSpace(GF(29), 2, 2) G = MatrixGroup([MS([[1, 1], [0, 1]]), MS([[20, 0], [0, 6]])])
 
Transitive group: 29T5 more information
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: $D_{29}$ $\,\rtimes\,$ $C_7$ $C_{29}$ $\,\rtimes\,$ $C_{14}$ $(C_{29}:C_7)$ $\,\rtimes\,$ $C_2$ more information
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization: $C_{14} \simeq C_{2} \times C_{7}$
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: $C_1$
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 92 subgroups in 8 conjugacy classes, 5 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_1$ $G/Z \simeq$ $C_{29}:C_{14}$
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: $G' \simeq$ $C_{29}$ $G/G' \simeq$ $C_{14}$
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: $\Phi \simeq$ $C_1$ $G/\Phi \simeq$ $C_{29}:C_{14}$
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: $\operatorname{Fit} \simeq$ $C_{29}$ $G/\operatorname{Fit} \simeq$ $C_{14}$
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: $R \simeq$ $C_{29}:C_{14}$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_{29}$ $G/\operatorname{soc} \simeq$ $C_{14}$
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: $P_{ 2 } \simeq$ $C_2$
7-Sylow subgroup: $P_{ 7 } \simeq$ $C_7$
29-Sylow subgroup: $P_{ 29 } \simeq$ $C_{29}$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $C_{29}:C_{14}$ $\rhd$ $C_{29}$ $\rhd$ $C_1$
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 $C_{29}:C_{14}$ $\rhd$ $C_{29}:C_7$ $\rhd$ $C_{29}$ $\rhd$ $C_1$
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 $C_{29}:C_{14}$ $\rhd$ $C_{29}$
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 $C_1$
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 7 larger groups in the database.

This group is a maximal quotient of 5 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

1A 2A 7A1 7A-1 7A2 7A-2 7A3 7A-3 14A1 14A-1 14A3 14A-3 14A5 14A-5 29A1 29A2
Size 1 29 29 29 29 29 29 29 29 29 29 29 29 29 14 14
2 P 1A 1A 7A2 7A-2 7A-3 7A3 7A-1 7A1 7A1 7A-1 7A3 7A-3 7A-2 7A2 29A2 29A1
7 P 1A 2A 7A3 7A-3 7A-1 7A1 7A2 7A-2 14A3 14A-3 14A-5 14A5 14A1 14A-1 29A2 29A1
29 P 1A 2A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 29A1 29A2
Type
406.1.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
406.1.1b R 1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1
406.1.1c1 C 1 1 ζ7−3 ζ7−1 ζ73 ζ7−2 ζ72 ζ7 ζ72 ζ7−2 ζ7−3 ζ7−1 ζ73 ζ7 1 1
406.1.1c2 C 1 1 ζ73 ζ7 ζ7−3 ζ72 ζ7−2 ζ7−1 ζ7−2 ζ72 ζ73 ζ7 ζ7−3 ζ7−1 1 1
406.1.1c3 C 1 1 ζ7−2 ζ7−3 ζ72 ζ7 ζ7−1 ζ73 ζ7−1 ζ7 ζ7−2 ζ7−3 ζ72 ζ73 1 1
406.1.1c4 C 1 1 ζ72 ζ73 ζ7−2 ζ7−1 ζ7 ζ7−3 ζ7 ζ7−1 ζ72 ζ73 ζ7−2 ζ7−3 1 1
406.1.1c5 C 1 1 ζ7−1 ζ72 ζ7 ζ7−3 ζ73 ζ7−2 ζ73 ζ7−3 ζ7−1 ζ72 ζ7 ζ7−2 1 1
406.1.1c6 C 1 1 ζ7 ζ7−2 ζ7−1 ζ73 ζ7−3 ζ72 ζ7−3 ζ73 ζ7 ζ7−2 ζ7−1 ζ72 1 1
406.1.1d1 C 1 −1 ζ7−3 ζ7−1 ζ73 ζ7−2 ζ72 ζ7 −ζ72 −ζ7−2 −ζ7−3 −ζ7−1 −ζ73 −ζ7 1 1
406.1.1d2 C 1 −1 ζ73 ζ7 ζ7−3 ζ72 ζ7−2 ζ7−1 −ζ7−2 −ζ72 −ζ73 −ζ7 −ζ7−3 −ζ7−1 1 1
406.1.1d3 C 1 −1 ζ7−2 ζ7−3 ζ72 ζ7 ζ7−1 ζ73 −ζ7−1 −ζ7 −ζ7−2 −ζ7−3 −ζ72 −ζ73 1 1
406.1.1d4 C 1 −1 ζ72 ζ73 ζ7−2 ζ7−1 ζ7 ζ7−3 −ζ7 −ζ7−1 −ζ72 −ζ73 −ζ7−2 −ζ7−3 1 1
406.1.1d5 C 1 −1 ζ7−1 ζ72 ζ7 ζ7−3 ζ73 ζ7−2 −ζ73 −ζ7−3 −ζ7−1 −ζ72 −ζ7 −ζ7−2 1 1
406.1.1d6 C 1 −1 ζ7 ζ7−2 ζ7−1 ζ73 ζ7−3 ζ72 −ζ7−3 −ζ73 −ζ7 −ζ7−2 −ζ7−1 −ζ72 1 1
406.1.14a1 R 14 0 0 0 0 0 0 0 0 0 0 0 0 0 ζ29−14+ζ29−12+ζ29−11+ζ29−10+ζ29−8+ζ29−3+ζ29−2+ζ292+ζ293+ζ298+ζ2910+ζ2911+ζ2912+ζ2914 −ζ29−14−ζ29−12−ζ29−11−ζ29−10−ζ29−8−ζ29−3−ζ29−2−1−ζ292−ζ293−ζ298−ζ2910−ζ2911−ζ2912−ζ2914
406.1.14a2 R 14 0 0 0 0 0 0 0 0 0 0 0 0 0 −ζ29−14−ζ29−12−ζ29−11−ζ29−10−ζ29−8−ζ29−3−ζ29−2−1−ζ292−ζ293−ζ298−ζ2910−ζ2911−ζ2912−ζ2914 ζ29−14+ζ29−12+ζ29−11+ζ29−10+ζ29−8+ζ29−3+ζ29−2+ζ292+ζ293+ζ298+ζ2910+ζ2911+ζ2912+ζ2914

Rational character table

1A 2A 7A 14A 29A
Size 1 29 174 174 28
2 P 1A 1A 7A 7A 29A
7 P 1A 2A 7A 14A 29A
29 P 1A 2A 1A 2A 29A
406.1.1a 1 1 1 1 1
406.1.1b 1 −1 1 −1 1
406.1.1c 6 6 −1 −1 6
406.1.1d 6 −6 −1 1 6
406.1.14a 28 0 0 0 −1