LMFDB - Abstract group 12.5: $C_2\times C

Show commands: Gap / Magma / SageMath

magma:G := SmallGroup(12, 5);

gap:G := SmallGroup(12, 5);

sage_gap:G = libgap.SmallGroup(12, 5)

comment:Define the group as a permutation group

sage:G = PermutationGroup(['(1,2)', '(3,4)', '(5,7,6)'])

Group information

Description:	$C_2\times C_6$
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:	$D_6$, of order $12$$\medspace = 2^{2} \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()
Nilpotency class:	$1$	comment:Nilpotency class of the group magma:NilpotencyClass(G); gap:NilpotencyClassOfGroup(G); sage_gap:G.NilpotencyClassOfGroup()
Derived length:	$1$	comment:Derived length of the group magma:DerivedLength(G); gap:DerivedLength(G); 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.

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	6
Elements	1	3	2	6	12
Conjugacy classes	1	3	2	6	12
Divisions	1	3	1	3	8
Autjugacy classes	1	1	1	1	4

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
Irr. complex chars.	12	0	12
Irr. rational chars.	4	4	8

Minimal presentations

Permutation degree:	$7$
Transitive degree:	$12$
Rank:	$2$
Inequivalent generating pairs:	$4$

Minimal degrees of faithful linear representations

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

Constructions

Show commands: Gap / Magma / SageMath

Presentation:	Abelian group $\langle a, b \mid a^{2}=b^{6}=1 \rangle$ < a, b \| a^2,b^6 >
comment:Define the group with the given generators and relations magma:G := PCGroup([3, -2, -2, -3, 16]); a,b := Explode([G.1, G.2]); AssignNames(~G, ["a", "b", "b2"]); gap:G := PcGroupCode(273,12); a := G.1; b := G.2; sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(273,12)'); a = G.1; b = G.2; sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(273,12)'); a = G.1; b = G.2;
Permutation group:	$\langle(1,2), (3,4), (5,7,6)\rangle$ (1,2), (3,4), (5,7,6)
comment:Define the group as a permutation group magma:G := PermutationGroup< 7 \| (1,2), (3,4), (5,7,6) >; gap:G := Group( (1,2), (3,4), (5,7,6) ); sage:G = PermutationGroup(['(1,2)', '(3,4)', '(5,7,6)'])
Matrix group:	$\left\langle \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrr} 0 & -1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{3}(\Z)$
comment:Define the group as a matrix group with coefficients in Z magma:G := MatrixGroup< 3, Integers() \| [[1, 0, 0, 0, 1, 0, 0, 0, -1], [0, -1, 0, 1, 1, 0, 0, 0, 1]] >; gap:G := Group([[[1, 0, 0], [0, 1, 0], [0, 0, -1]], [[0, -1, 0], [1, 1, 0], [0, 0, 1]]]); sage:MS = MatrixSpace(Integers(), 3, 3) G = MatrixGroup([MS([[1, 0, 0], [0, 1, 0], [0, 0, -1]]), MS([[0, -1, 0], [1, 1, 0], [0, 0, 1]])])
	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 6 & 0 \\ 0 & 6 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{7})$
comment:Define the group as a matrix group with coefficients in GLFp magma:G := MatrixGroup< 2, GF(7) \| [[1, 0, 0, 3], [6, 0, 0, 6]] >; gap:G := Group([[[ Z(7)^0, 0Z(7) ], [ 0Z(7), Z(7) ]], [[ Z(7)^3, 0Z(7) ], [ 0Z(7), Z(7)^3 ]]]); sage:MS = MatrixSpace(GF(7), 2, 2) G = MatrixGroup([MS([[1, 0], [0, 3]]), MS([[6, 0], [0, 6]])])
Transitive group:	12T2				more information
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_3$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(C_{21})$	$\Aut(C_{28})$	$\Aut(C_{36})$	$\Aut(C_{42})$

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

Homology

Primary decomposition:	$C_{2}^{2} \times C_{3}$	comment:The primary decomposition of the group magma:PrimaryInvariants(G); gap:AbelianInvariants(G); sage_gap:G.AbelianInvariants()
Schur multiplier:	$C_{2}$	comment:The Schur multiplier of the group gap:AbelianInvariantsMultiplier(G); sage:G.homology(2) sage_gap:G.AbelianInvariantsMultiplier()
Commutator length:	$0$	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, all normal (4 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_6$	$G/Z \simeq$ $C_1$	comment:Center of the group magma:Center(G); gap:Center(G); sage:G.center() sage_gap:G.Center()
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_2\times C_6$	comment:Commutator subgroup of the group G magma:CommutatorSubgroup(G); gap:DerivedSubgroup(G); sage:G.commutator() sage_gap:G.DerivedSubgroup()
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2\times C_6$	comment:Frattini subgroup of the group G magma:FrattiniSubgroup(G); gap:FrattiniSubgroup(G); sage:G.frattini_subgroup() sage_gap:G.FrattiniSubgroup()
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6$	$G/\operatorname{Fit} \simeq$ $C_1$	comment:Fitting subgroup of the group G magma:FittingSubgroup(G); gap:FittingSubgroup(G); sage:G.fitting_subgroup() sage_gap:G.FittingSubgroup()
Radical:	$R \simeq$ $C_2\times C_6$	$G/R \simeq$ $C_1$	comment:Radical of the group G magma:Radical(G); gap:SolvableRadical(G); sage_gap:G.SolvableRadical()
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_1$	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

$C_2\times C_6$

$\rhd$

$C_1$

comment:Derived series of the group GF

magma:DerivedSeries(G);

gap:DerivedSeriesOfGroup(G);

sage:G.derived_series()

sage_gap:G.DerivedSeriesOfGroup()

Chief series

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

comment:Chief series of the group G

magma:ChiefSeries(G);

gap:ChiefSeries(G);

sage_gap:G.ChiefSeries()

Lower central series

$C_2\times C_6$

$\rhd$

$C_1$

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

$C_1$

$\lhd$

$C_2\times C_6$

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 180 larger groups in the database.

This group is a maximal quotient of 205 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	2B	2C	3A1	3A-1	6A1	6A-1	6B1	6B-1	6C1	6C-1
	Size	1	1	1	1	1	1	1	1	1	1	1	1
	2 P	1A	1A	1A	1A	3A-1	3A1	3A1	3A-1	3A-1	3A1	3A1	3A-1
	3 P	1A	2A	2B	2C	1A	1A	2A	2A	2B	2B	2C	2C
	Type
12.5.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
12.5.1b	R	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$
12.5.1c	R	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$1$
12.5.1d	R	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$
12.5.1e1	C	$1$	$1$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$
12.5.1e2	C	$1$	$1$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$
12.5.1f1	C	$1$	$- 1$	$- 1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$ζ_{3}$	$- ζ_{3}$	$ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$
12.5.1f2	C	$1$	$- 1$	$- 1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$ζ_{3}$	$- ζ_{3}$
12.5.1g1	C	$1$	$- 1$	$1$	$- 1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- ζ_{3}^{- 1}$	$ζ_{3}$	$- ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$
12.5.1g2	C	$1$	$- 1$	$1$	$- 1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- ζ_{3}$	$ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$ζ_{3}$
12.5.1h1	C	$1$	$1$	$- 1$	$- 1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}$	$ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$
12.5.1h2	C	$1$	$1$	$- 1$	$- 1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}$

Rational character table

		1A	2A	2B	2C	3A	6A	6B	6C
	Size	1	1	1	1	2	2	2	2
	2 P	1A	1A	1A	1A	3A	3A	3A	3A
	3 P	1A	2A	2B	2C	1A	2A	2B	2C
12.5.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
12.5.1b		$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$1$	$1$
12.5.1c		$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$
12.5.1d		$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$
12.5.1e		$2$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$- 1$
12.5.1f		$2$	$- 2$	$- 2$	$2$	$- 1$	$1$	$- 1$	$- 1$
12.5.1g		$2$	$- 2$	$2$	$- 2$	$- 1$	$1$	$1$	$1$
12.5.1h		$2$	$2$	$- 2$	$- 2$	$- 1$	$- 1$	$1$	$1$

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Group information

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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

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

Label	12.5
Order	\( 2^{2} \cdot 3 \)
Exponent	\( 2 \cdot 3 \)

Abelian	yes
$\card{\Aut(G)}$	\( 2^{2} \cdot 3 \)
Perm deg.	$7$
Trans deg.	$12$
Rank	$2$