Show commands: Gap / Magma / SageMath

magma:G := AlternatingGroup(4);

gap:G := AlternatingGroup(4);

sage:G = AlternatingGroup(4)

sage_gap:G = libgap.SmallGroup(12, 3)

comment:Define the group as a permutation group

Group information

Description:	$A_4$
Order:	$12$$\medspace = 2^{2} \cdot 3 $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order()
Exponent:	$6$$\medspace = 2 \cdot 3 $	comment:Exponent of the group magma:Exponent(G); gap:Exponent(G); sage:G.exponent() sage_gap:G.Exponent()
Automorphism group:	$S_4$, of order $24$$\medspace = 2^{3} \cdot 3 $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage_gap:G.AutomorphismGroup()
Composition factors:	$C_2$ x 2, $C_3$	comment:Composition factors of the group magma:CompositionFactors(G); gap:CompositionSeries(G); sage:G.composition_series() sage_gap:G.CompositionSeries()
Derived length:	$2$	comment:Derived length of the group magma:DerivedLength(G); gap:DerivedLength(G); sage_gap:G.DerivedLength()

This group is nonabelian, monomial (hence solvable), metabelian, and an A-group.

comment:Determine if the group G is abelian

magma:IsAbelian(G);

gap:IsAbelian(G);

sage:G.is_abelian()

sage_gap:G.IsAbelian()

comment:Determine if the group G is cyclic

magma:IsCyclic(G);

gap:IsCyclic(G);

sage:G.is_cyclic()

sage_gap:G.IsCyclic()

comment:Determine if the group G is nilpotent

magma:IsNilpotent(G);

gap:IsNilpotentGroup(G);

sage:G.is_nilpotent()

sage_gap:G.IsNilpotentGroup()

comment:Determine if the group G is solvable

magma:IsSolvable(G);

gap:IsSolvableGroup(G);

sage:G.is_solvable()

sage_gap:G.IsSolvableGroup()

comment:Determine if the group G is supersolvable

gap:IsSupersolvableGroup(G);

sage:G.is_supersolvable()

sage_gap:G.IsSupersolvableGroup()

comment:Determine if the group G is simple

magma:IsSimple(G);

gap:IsSimpleGroup(G);

sage_gap:G.IsSimpleGroup()

Group statistics

comment:Compute statistics for the group G

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;

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");

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	8	12
Conjugacy classes	1	1	2	4
Divisions	1	1	1	3
Autjugacy classes	1	1	1	3

comment:Compute statistics about the characters of G

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);

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);

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)]

sage_gap:G.CharacterDegrees()

Dimension	1	2	3
Irr. complex chars.	3	0	1	4
Irr. rational chars.	1	1	1	3

Minimal presentations

Permutation degree:	$4$
Transitive degree:	$4$
Rank:	$2$
Inequivalent generating pairs:	$4$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	3	3	3
Arbitrary	3	3	3

Constructions

Show commands: Gap / Magma / SageMath

Groups of Lie type:	$\PSL(2,3)$, $\PSU(2,3)$, $\Omega(3,3)$, $\POmega(3,3)$, $\PSigmaL(2,3)$, $\AGL(1,4)$
Presentation:	$\langle a, b, c \mid a^{3}=b^{2}=c^{2}=[b,c]=1, b^{a}=bc, c^{a}=b \rangle$ < a, b, c \| a^3,b^2,c^2,[b,c], bca^-1b^-1a, ba^-1c^-1*a >
comment:Define the group with the given generators and relations magma:G := PCGroup([3, -3, -2, 2, 55, 29]); a,b,c := Explode([G.1, G.2, G.3]); AssignNames(~G, ["a", "b", "c"]); gap:G := PcGroupCode(9572,12); a := G.1; b := G.2; c := G.3; sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(9572,12)'); a = G.1; b = G.2; c = G.3; sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(9572,12)'); a = G.1; b = G.2; c = G.3;
Permutation group:	$\langle(2,4,3), (1,3)(2,4), (1,2)(3,4)\rangle$ (2,4,3), (1,3)(2,4), (1,2)(3,4)
comment:Define the group as a permutation group magma:G := PermutationGroup< 4 \| (2,4,3), (1,3)(2,4), (1,2)(3,4) >; gap:G := Group( (2,4,3), (1,3)(2,4), (1,2)(3,4) ); sage:G = PermutationGroup(['(2,4,3)', '(1,3)(2,4)', '(1,2)(3,4)'])
Matrix group:	$\left\langle \left(\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 1 & -1 \end{array}\right), \left(\begin{array}{rrr} 0 & -1 & 0 \\ -1 & 0 & 0 \\ -1 & 1 & -1 \end{array}\right) \right\rangle \subseteq \GL_{3}(\Z)$
comment:Define the group as a matrix group with coefficients in Z magma:G := MatrixGroup< 3, Integers() \| [[0, 0, 1, 0, 1, 0, -1, 1, -1], [0, -1, 0, -1, 0, 0, -1, 1, -1]] >; gap:G := Group([[[0, 0, 1], [0, 1, 0], [-1, 1, -1]], [[0, -1, 0], [-1, 0, 0], [-1, 1, -1]]]); sage:MS = MatrixSpace(Integers(), 3, 3) G = MatrixGroup([MS([[0, 0, 1], [0, 1, 0], [-1, 1, -1]]), MS([[0, -1, 0], [-1, 0, 0], [-1, 1, -1]])])
	$\left\langle \left(\begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 1 \end{array}\right), \left(\begin{array}{rrr} 0 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 0 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 0 & 0 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{2})$
comment:Define the group as a matrix group with coefficients in GLFp magma:G := MatrixGroup< 3, GF(2) \| [[0, 1, 0, 1, 0, 0, 1, 1, 1], [0, 1, 0, 1, 1, 1, 0, 0, 1], [0, 0, 1, 1, 1, 1, 1, 0, 0]] >; gap:G := Group([[[ 0Z(2), Z(2)^0, 0Z(2) ], [ Z(2)^0, 0Z(2), 0Z(2) ], [ Z(2)^0, Z(2)^0, Z(2)^0 ]], [[ 0Z(2), Z(2)^0, 0Z(2) ], [ Z(2)^0, Z(2)^0, Z(2)^0 ], [ 0Z(2), 0Z(2), Z(2)^0 ]], [[ 0Z(2), 0Z(2), Z(2)^0 ], [ Z(2)^0, Z(2)^0, Z(2)^0 ], [ Z(2)^0, 0Z(2), 0Z(2) ]]]); sage:MS = MatrixSpace(GF(2), 3, 3) G = MatrixGroup([MS([[0, 1, 0], [1, 0, 0], [1, 1, 1]]), MS([[0, 1, 0], [1, 1, 1], [0, 0, 1]]), MS([[0, 0, 1], [1, 1, 1], [1, 0, 0]])])
Transitive group:	4T4	6T4	12T4	more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_2^2$ $\,\rtimes\,$ $C_3$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 4.

Homology

Abelianization:	$C_{3} $	comment:The abelianization of the group magma:quo< G \| CommutatorSubgroup(G) >; gap:FactorGroup(G, DerivedSubgroup(G)); sage:G.quotient(G.commutator())
Schur multiplier:	$C_{2}$	comment:The Schur multiplier of the group gap:AbelianInvariantsMultiplier(G); sage:G.homology(2) sage_gap:G.AbelianInvariantsMultiplier()
Commutator length:	$1$	comment:The commutator length of the group gap:CommutatorLength(G); sage_gap:G.CommutatorLength()

Subgroups

comment:List of subgroups of the group

magma:Subgroups(G);

gap:AllSubgroups(G);

sage:G.subgroups()

sage_gap:G.AllSubgroups()

There are 10 subgroups in 5 conjugacy classes, 3 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $A_4$	comment:Center of the group magma:Center(G); gap:Center(G); sage:G.center() sage_gap:G.Center()
Commutator:	$G' \simeq$ $C_2^2$	$G/G' \simeq$ $C_3$	comment:Commutator subgroup of the group G magma:CommutatorSubgroup(G); gap:DerivedSubgroup(G); sage:G.commutator() sage_gap:G.DerivedSubgroup()
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $A_4$	comment:Frattini subgroup of the group G magma:FrattiniSubgroup(G); gap:FrattiniSubgroup(G); sage:G.frattini_subgroup() sage_gap:G.FrattiniSubgroup()
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2$	$G/\operatorname{Fit} \simeq$ $C_3$	comment:Fitting subgroup of the group G magma:FittingSubgroup(G); gap:FittingSubgroup(G); sage:G.fitting_subgroup() sage_gap:G.FittingSubgroup()
Radical:	$R \simeq$ $A_4$	$G/R \simeq$ $C_1$	comment:Radical of the group G magma:Radical(G); gap:SolvableRadical(G); sage_gap:G.SolvableRadical()
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_3$	comment:Socle of the group G magma:Socle(G); gap:Socle(G); sage:G.socle() sage_gap:G.Socle()
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Subgroup diagram and profile

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Subgroup information

Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

Series

Derived series

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

comment:Derived series of the group GF

magma:DerivedSeries(G);

gap:DerivedSeriesOfGroup(G);

sage:G.derived_series()

sage_gap:G.DerivedSeriesOfGroup()

Chief series

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

comment:Chief series of the group G

magma:ChiefSeries(G);

gap:ChiefSeries(G);

sage_gap:G.ChiefSeries()

Lower central series

$A_4$

$\rhd$

$C_2^2$

comment:The lower central series of the group G

magma:LowerCentralSeries(G);

gap:LowerCentralSeriesOfGroup(G);

sage:G.lower_central_series()

sage_gap:G.LowerCentralSeriesOfGroup()

Upper central series

$C_1$

comment:The upper central series of the group G

magma:UpperCentralSeries(G);

gap:UpperCentralSeriesOfGroup(G);

sage:G.upper_central_series()

sage_gap:G.UpperCentralSeriesOfGroup()

Supergroups

This group is a maximal subgroup of 145 larger groups in the database.

This group is a maximal quotient of 119 larger groups in the database.

Character theory

comment:Character table

magma:CharacterTable(G); // Output not guaranteed to exactly match the LMFDB table

gap:CharacterTable(G); # Output not guaranteed to exactly match the LMFDB table

sage:G.character_table() # Output not guaranteed to exactly match the LMFDB table

sage_gap:G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table

Complex character table

		1A	2A	3A1	3A-1
	Size	1	3	4	4
	2 P	1A	1A	3A-1	3A1
	3 P	1A	2A	1A	1A
	Type
12.3.1a	R	$1$	$1$	$1$	$1$
12.3.1b1	C	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$
12.3.1b2	C	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$
12.3.3a	R	$3$	$- 1$	$0$	$0$

Rational character table

		1A	2A	3A
	Size	1	3	8
	2 P	1A	1A	3A
	3 P	1A	2A	1A
12.3.1a		$1$	$1$	$1$
12.3.1b		$2$	$2$	$- 1$
12.3.3a		$3$	$- 1$	$0$

Additional information

This group arises in several ways: as the rotations of a regular tetrahedron, as the alternating group on four letters, as $\PSL_2(\mathbb{F}_3)$, as $\AGL_1(\mathbb{F}_4)$ and as the von Dyck group $D(2, 3, 3)$. As such, it is part of several families of simple groups, but it itself is not simple (the products of two $2$-cycles usually generate all of $A_n$, but this fails when $n=4$). It is also the smallest group where the converse of Lagrange's theorem does not hold, since it has no subgroup of order $6$.

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Label	12.3
Order	\( 2^{2} \cdot 3 \)
Exponent	\( 2 \cdot 3 \)

Nilpotent	no
Solvable	yes
$\card{G^{\mathrm{ab}}}$	\( 3 \)
$\card{Z(G)}$	\( 1 \)
$\card{\Aut(G)}$	\( 2^{3} \cdot 3 \)
$\card{\mathrm{Out}(G)}$	\( 2 \)
Perm deg.	$4$
Trans deg.	$4$
Rank	$2$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Group information

Group statistics

Minimal presentations

Minimal degrees of faithful linear representations

Constructions

Homology

Subgroups

Special subgroups

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

Subgroup information

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups (quotient in parentheses)

Normal subgroups up to automorphism (quotient in parentheses)

Series

Supergroups

Character theory

Complex character table

Rational character table

Additional information

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.