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Properties

Label 1610.6
Order \( 2 \cdot 5 \cdot 7 \cdot 23 \)
Exponent \( 2 \cdot 5 \cdot 7 \cdot 23 \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2 \cdot 23 \)
$\card{Z(G)}$ \( 23 \)
$\card{\Aut(G)}$ \( 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
$\card{\mathrm{Out}(G)}$ \( 2^{3} \cdot 3 \cdot 11 \)
Perm deg. $35$
Trans deg. $805$
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SmallGroup(1610, 6);
 
Copy content gap:G := SmallGroup(1610, 6);
 
Copy content sage_gap:G = libgap.SmallGroup(1610, 6)
 
Copy content comment:Define the group as a permutation group
 
Copy content sage:G = PermutationGroup(['(2,3)(4,5)(6,7)(9,10)(11,12)', '(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35)', '(8,9,11,12,10)', '(1,2,4,6,7,5,3)'])
 

Group information

Description:$C_{23}\times D_{35}$
Order: \(1610\)\(\medspace = 2 \cdot 5 \cdot 7 \cdot 23 \)
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: \(1610\)\(\medspace = 2 \cdot 5 \cdot 7 \cdot 23 \)
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:$C_{22}\times F_5\times F_7$, of order \(18480\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$, $C_5$, $C_7$, $C_{23}$
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, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.

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 5 7 23 35 46 115 161 805
Elements 1 35 4 6 22 24 770 88 132 528 1610
Conjugacy classes   1 1 2 3 22 12 22 44 66 264 437
Divisions 1 1 1 1 1 1 1 1 1 1 10
Autjugacy classes 1 1 1 1 1 1 1 1 1 1 10

Minimal presentations

Permutation degree:$35$
Transitive degree:$805$
Rank: $2$
Inequivalent generating pairs: $72$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 2 not computed not computed
Arbitrary not computed not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Presentation: $\langle a, b \mid a^{2}=b^{805}=1, b^{a}=b^{139} \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([4, -2, -5, -7, -23, 2225, 45, 16682, 94]); a,b := Explode([G.1, G.2]); AssignNames(~G, ["a", "b", "b5", "b35"]);
 
Copy content gap:G := PcGroupCode(67266284439552090639,1610); 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(67266284439552090639,1610)'); 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(67266284439552090639,1610)'); a = G.1; b = G.2;
 
Permutation group:Degree $35$ $\langle(2,3)(4,5)(6,7)(9,10)(11,12), (13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35), (8,9,11,12,10), (1,2,4,6,7,5,3)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 35 | (2,3)(4,5)(6,7)(9,10)(11,12), (13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35), (8,9,11,12,10), (1,2,4,6,7,5,3) >;
 
Copy content gap:G := Group( (2,3)(4,5)(6,7)(9,10)(11,12), (13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35), (8,9,11,12,10), (1,2,4,6,7,5,3) );
 
Copy content sage:G = PermutationGroup(['(2,3)(4,5)(6,7)(9,10)(11,12)', '(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35)', '(8,9,11,12,10)', '(1,2,4,6,7,5,3)'])
 
Matrix group:$\left\langle \left(\begin{array}{rr} 51 & 45 \\ 92 & 51 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 138 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{139})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 2, GF(139) | [[51, 45, 92, 51], [1, 0, 0, 138]] >;
 
Copy content gap:G := Group([[[ Z(139)^10, Z(139)^30 ], [ Z(139)^29, Z(139)^10 ]], [[ Z(139)^0, 0*Z(139) ], [ 0*Z(139), Z(139)^69 ]]]);
 
Copy content sage:MS = MatrixSpace(GF(139), 2, 2) G = MatrixGroup([MS([[51, 45], [92, 51]]), MS([[1, 0], [0, 138]])])
 
Direct product: $C_{23}$ $\, \times\, $ $D_{35}$
Semidirect product: $C_{161}$ $\,\rtimes\,$ $D_5$ $C_{115}$ $\,\rtimes\,$ $D_7$ $C_{805}$ $\,\rtimes\,$ $C_2$ $C_{35}$ $\,\rtimes\,$ $C_{46}$ all 6
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: $C_{46} \simeq C_{2} \times C_{23}$
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 104 subgroups in 16 conjugacy classes, 10 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_{23}$ $G/Z \simeq$ $D_{35}$
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: $G' \simeq$ $C_{35}$ $G/G' \simeq$ $C_{46}$
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: $\Phi \simeq$ $C_1$ $G/\Phi \simeq$ $C_{23}\times D_{35}$
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: $\operatorname{Fit} \simeq$ $C_{805}$ $G/\operatorname{Fit} \simeq$ $C_2$
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: $R \simeq$ $C_{23}\times D_{35}$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_{805}$ $G/\operatorname{soc} \simeq$ $C_2$
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: $P_{ 2 } \simeq$ $C_2$
5-Sylow subgroup: $P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup: $P_{ 7 } \simeq$ $C_7$
23-Sylow subgroup: $P_{ 23 } \simeq$ $C_{23}$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $C_{23}\times D_{35}$ $\rhd$ $C_{35}$ $\rhd$ $C_1$
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 $C_{23}\times D_{35}$ $\rhd$ $C_{805}$ $\rhd$ $C_{161}$ $\rhd$ $C_{23}$ $\rhd$ $C_1$
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 $C_{23}\times D_{35}$ $\rhd$ $C_{35}$
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 $C_1$ $\lhd$ $C_{23}$
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 2 larger groups in the database.

This group is a maximal quotient of 3 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

The $437 \times 437$ character table is not available for this group.

Rational character table

The $10 \times 10$ rational character table is not available for this group.