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Properties

Label 6.1
Order \( 2 \cdot 3 \)
Exponent \( 2 \cdot 3 \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2 \)
$\card{Z(G)}$ \( 1 \)
$\card{\Aut(G)}$ \( 2 \cdot 3 \)
$\card{\mathrm{Out}(G)}$ \( 1 \)
Perm deg. $3$
Trans deg. $3$
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SymmetricGroup(3);
 
Copy content gap:G := SymmetricGroup(3);
 
Copy content sage:G = SymmetricGroup(3)
 
Copy content sage_gap:G = libgap.SmallGroup(6, 1)
 
Copy content comment:Define the group as a permutation group
 

Group information

Description:$S_3$
Order: \(6\)\(\medspace = 2 \cdot 3 \)
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: \(6\)\(\medspace = 2 \cdot 3 \)
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:$S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$, $C_3$
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), hyperelementary for $p = 2$, and rational.

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
Elements 1 3 2 6
Conjugacy classes   1 1 1 3
Divisions 1 1 1 3
Autjugacy classes 1 1 1 3

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
Irr. complex chars.   2 1 3
Irr. rational chars. 2 1 3

Minimal presentations

Permutation degree:$3$
Transitive degree:$3$
Rank: $2$
Inequivalent generating pairs: $3$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 2 2 2
Arbitrary 2 2 2

Constructions

Show commands: Gap / Magma / SageMath


Groups of Lie type:$\GL(2,2)$, $\SL(2,2)$, $\PSL(2,2)$, $\PGL(2,2)$, $\SO(3,2)$, $\SU(2,2)$, $\PSO(3,2)$, $\PSU(2,2)$, $\GO(3,2)$, $\Omega(3,2)$, $\PGO(3,2)$, $\PGU(2,2)$, $\POmega(3,2)$, $\CSO(3,2)$, $\CSOPlus(2,4)$, $\CSOMinus(2,2)$, $\CSU(2,2)$, $\CO(3,2)$, $\COMinus(2,2)$, $\PGammaL(2,2)$, $\PSigmaL(2,2)$, $\PGammaU(2,2)$, $\AGL(1,3)$, $\AGammaL(1,3)$
Presentation: $\langle a, b \mid a^{2}=b^{3}=1, b^{a}=b^{2} \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([2, -2, -3, 17]); a,b := Explode([G.1, G.2]); AssignNames(~G, ["a", "b"]);
 
Copy content gap:G := PcGroupCode(25,6); 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(25,6)'); 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(25,6)'); a = G.1; b = G.2;
 
Permutation group: $\langle(2,3), (1,3,2)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 3 | (2,3), (1,3,2) >;
 
Copy content gap:G := Group( (2,3), (1,3,2) );
 
Copy content sage:G = PermutationGroup(['(2,3)', '(1,3,2)'])
 
Matrix group:$\left\langle \left(\begin{array}{rr} 0 & -1 \\ -1 & 0 \end{array}\right), \left(\begin{array}{rr} -1 & -1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z)$
Copy content comment:Define the group as a matrix group with coefficients in Z
 
Copy content magma:G := MatrixGroup< 2, Integers() | [[0, -1, -1, 0], [-1, -1, 1, 0]] >;
 
Copy content gap:G := Group([[[0, -1], [-1, 0]], [[-1, -1], [1, 0]]]);
 
Copy content sage:MS = MatrixSpace(Integers(), 2, 2) G = MatrixGroup([MS([[0, -1], [-1, 0]]), MS([[-1, -1], [1, 0]])])
 
$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{2})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 2, GF(2) | [[1, 1, 0, 1], [0, 1, 1, 0]] >;
 
Copy content gap:G := Group([[[ Z(2)^0, Z(2)^0 ], [ 0*Z(2), Z(2)^0 ]], [[ 0*Z(2), Z(2)^0 ], [ Z(2)^0, 0*Z(2) ]]]);
 
Copy content sage:MS = MatrixSpace(GF(2), 2, 2) G = MatrixGroup([MS([[1, 1], [0, 1]]), MS([[0, 1], [1, 0]])])
 
Transitive group: 3T2 6T2 more information
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: $C_3$ $\,\rtimes\,$ $C_2$ more information
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Aut. group: $\Aut(C_2^2)$

Elements of the group are displayed as permutations of degree 3.

Homology

Abelianization: $C_{2} $
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 6 subgroups in 4 conjugacy classes, 3 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_1$ $G/Z \simeq$ $S_3$
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: $G' \simeq$ $C_3$ $G/G' \simeq$ $C_2$
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: $\Phi \simeq$ $C_1$ $G/\Phi \simeq$ $S_3$
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: $\operatorname{Fit} \simeq$ $C_3$ $G/\operatorname{Fit} \simeq$ $C_2$
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: $R \simeq$ $S_3$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_3$ $G/\operatorname{soc} \simeq$ $C_2$
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: $P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $S_3$ $\rhd$ $C_3$ $\rhd$ $C_1$
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 $S_3$ $\rhd$ $C_3$ $\rhd$ $C_1$
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 $S_3$ $\rhd$ $C_3$
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 141 larger groups in the database.

This group is a maximal quotient of 212 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

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

1A 2A 3A
Size 1 3 2
2 P 1A 1A 3A
3 P 1A 2A 1A
6.1.1a 1 1 1
6.1.1b 1 −1 1
6.1.2a 2 0 −1

Additional information

This is the smallest nonabelian group, the smallest group that is not nilpotent, and the largest group with the property that any two elements of the same order are conjugate.