LMFDB - Abstract group 4056.bb: $C_{13}^2:(C_4\times S

comment:Define the group as a permutation group

magma:G := PermutationGroup< 26 | (1,17,5,24)(2,22,4,19)(3,14)(6,16,13,25)(7,21,12,20)(8,26,11,15)(9,18,10,23), (1,17,6,24,11,18,3,25,8,19,13,26,5,20,10,14,2,21,7,15,12,22,4,16,9,23) >;

gap:G := Group( (1,17,5,24)(2,22,4,19)(3,14)(6,16,13,25)(7,21,12,20)(8,26,11,15)(9,18,10,23), (1,17,6,24,11,18,3,25,8,19,13,26,5,20,10,14,2,21,7,15,12,22,4,16,9,23) );

sage:G = PermutationGroup(['(1,17,5,24)(2,22,4,19)(3,14)(6,16,13,25)(7,21,12,20)(8,26,11,15)(9,18,10,23)', '(1,17,6,24,11,18,3,25,8,19,13,26,5,20,10,14,2,21,7,15,12,22,4,16,9,23)'])

sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(510441394630661718194494619427342127350372583007196380197151,4056)'); a = G.1; b = G.2; c = G.5; d = G.6;

Group information

Description:	$C_{13}^2:(C_4\times S_3)$
Order:	$4056$$\medspace = 2^{3} \cdot 3 \cdot 13^{2} $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order()
Exponent:	$156$$\medspace = 2^{2} \cdot 3 \cdot 13 $	comment:Exponent of the group magma:Exponent(G); gap:Exponent(G); sage:G.exponent() sage_gap:G.Exponent()
Automorphism group:	$D_{13}^2.(C_6\times S_3)$, of order $24336$$\medspace = 2^{4} \cdot 3^{2} \cdot 13^{2} $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage_gap:G.AutomorphismGroup()
Composition factors:	$C_2$ x 3, $C_3$, $C_{13}$ x 2	comment:Composition factors of the group magma:CompositionFactors(G); gap:CompositionSeries(G); sage:G.composition_series() sage_gap:G.CompositionSeries()
Derived length:	$3$	comment:Derived length of the group magma:DerivedLength(G); gap:DerivedLength(G); sage_gap:G.DerivedLength()

This group is nonabelian, monomial (hence solvable), 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	4	6	12	13	26
Elements	1	247	338	1352	338	676	168	936	4056
Conjugacy classes	1	3	1	4	1	2	10	6	28
Divisions	1	3	1	2	1	1	4	2	15
Autjugacy classes	1	2	1	3	1	2	3	1	14

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	4	12	24	36	72
Irr. complex chars.	8	4	0	12	4	0	0	28
Irr. rational chars.	4	4	1	0	1	4	1	15

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$26$
Rank:	$2$
Inequivalent generating pairs:	$168$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	24
Arbitrary	12	12	24

Constructions

Show commands: Gap / Magma / SageMath

Presentation:	${\langle a, b, c, d \mid b^{12}=c^{13}=d^{13}=[c,d]=1, a^{2}=b^{6}c^{4}d^{11}, b^{a}= \!\cdots\! \rangle}$ < a, b, c, d \| b^12,c^13,d^13,[c,d], b^6c^4d^11a^-2, b^5c^10d^8a^-1b^-1a, c^2d^12a^-1c^-1a, c^5d^11a^-1d^-1a, c^6db^-1c^-1b, d^2b^-1d^-1*b >
comment:Define the group with the given generators and relations magma:G := PCGroup([6, -2, -2, -2, -3, -13, 13, 21240, 32953, 31, 41402, 50, 49347, 113764, 6850, 20536, 10192, 127877, 11243, 11249, 4235]); a,b,c,d := Explode([G.1, G.2, G.5, G.6]); AssignNames(~G, ["a", "b", "b2", "b4", "c", "d"]); gap:G := PcGroupCode(510441394630661718194494619427342127350372583007196380197151,4056); a := G.1; b := G.2; c := G.5; d := G.6; sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(510441394630661718194494619427342127350372583007196380197151,4056)'); a = G.1; b = G.2; c = G.5; d = G.6; sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(510441394630661718194494619427342127350372583007196380197151,4056)'); a = G.1; b = G.2; c = G.5; d = G.6;
Permutation group:	Degree $26$ $\langle(1,17,5,24)(2,22,4,19)(3,14)(6,16,13,25)(7,21,12,20)(8,26,11,15)(9,18,10,23), (1,17,6,24,11,18,3,25,8,19,13,26,5,20,10,14,2,21,7,15,12,22,4,16,9,23)\rangle$ (1,17,5,24)(2,22,4,19)(3,14)(6,16,13,25)(7,21,12,20)(8,26,11,15)(9,18,10,23), (1,17,6,24,11,18,3,25,8,19,13,26,5,20,10,14,2,21,7,15,12,22,4,16,9,23)
comment:Define the group as a permutation group magma:G := PermutationGroup< 26 \| (1,17,5,24)(2,22,4,19)(3,14)(6,16,13,25)(7,21,12,20)(8,26,11,15)(9,18,10,23), (1,17,6,24,11,18,3,25,8,19,13,26,5,20,10,14,2,21,7,15,12,22,4,16,9,23) >; gap:G := Group( (1,17,5,24)(2,22,4,19)(3,14)(6,16,13,25)(7,21,12,20)(8,26,11,15)(9,18,10,23), (1,17,6,24,11,18,3,25,8,19,13,26,5,20,10,14,2,21,7,15,12,22,4,16,9,23) ); sage:G = PermutationGroup(['(1,17,5,24)(2,22,4,19)(3,14)(6,16,13,25)(7,21,12,20)(8,26,11,15)(9,18,10,23)', '(1,17,6,24,11,18,3,25,8,19,13,26,5,20,10,14,2,21,7,15,12,22,4,16,9,23)'])
Transitive group:	26T30	39T42			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{13}^2:S_3)$ $\,\rtimes\,$ $C_4$ (2)	$(C_{13}^2:C_4)$ $\,\rtimes\,$ $S_3$	$C_{13}^2$ $\,\rtimes\,$ $(C_4\times S_3)$	$(C_{13}^2:C_3)$ $\,\rtimes\,$ $(C_2\times C_4)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{13}:D_{13})$ . $D_6$	$(C_{13}^2:D_6)$ . $C_2$	$(C_{13}^2:C_6)$ . $C_2^2$		more information

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

Homology

Abelianization:	$C_{2} \times C_{4} $	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 4090 subgroups in 57 conjugacy classes, 12 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_{13}^2:(C_4\times S_3)$	comment:Center of the group magma:Center(G); gap:Center(G); sage:G.center() sage_gap:G.Center()
Commutator:	$G' \simeq$ $C_{13}^2:C_3$	$G/G' \simeq$ $C_2\times C_4$	comment:Commutator subgroup of the group G magma:CommutatorSubgroup(G); gap:DerivedSubgroup(G); sage:G.commutator() sage_gap:G.DerivedSubgroup()
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{13}^2:(C_4\times S_3)$	comment:Frattini subgroup of the group G magma:FrattiniSubgroup(G); gap:FrattiniSubgroup(G); sage:G.frattini_subgroup() sage_gap:G.FrattiniSubgroup()
Fitting:	$\operatorname{Fit} \simeq$ $C_{13}^2$	$G/\operatorname{Fit} \simeq$ $C_4\times S_3$	comment:Fitting subgroup of the group G magma:FittingSubgroup(G); gap:FittingSubgroup(G); sage:G.fitting_subgroup() sage_gap:G.FittingSubgroup()
Radical:	$R \simeq$ $C_{13}^2:(C_4\times S_3)$	$G/R \simeq$ $C_1$	comment:Radical of the group G magma:Radical(G); gap:SolvableRadical(G); sage_gap:G.SolvableRadical()
Socle:	$\operatorname{soc} \simeq$ $C_{13}^2$	$G/\operatorname{soc} \simeq$ $C_4\times S_3$	comment:Socle of the group G magma:Socle(G); gap:Socle(G); sage:G.socle() sage_gap:G.Socle()
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}^2$

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.

See a full page version of the diagram

Subgroup information

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

Series

Derived series

$C_{13}^2:(C_4\times S_3)$

$\rhd$

$C_{13}^2:C_3$

$\rhd$

$C_{13}^2$

$\rhd$

$C_1$

comment:Derived series of the group GF

magma:DerivedSeries(G);

gap:DerivedSeriesOfGroup(G);

sage:G.derived_series()

sage_gap:G.DerivedSeriesOfGroup()

Chief series

$C_{13}^2:(C_4\times S_3)$

$\rhd$

$C_{13}^2:(C_3:C_4)$

$\rhd$

$C_{13}^2:C_6$

$\rhd$

$C_{13}^2:C_3$

$\rhd$

$C_{13}^2$

$\rhd$

$C_1$

comment:Chief series of the group G

magma:ChiefSeries(G);

gap:ChiefSeries(G);

sage_gap:G.ChiefSeries()

Lower central series

$C_{13}^2:(C_4\times S_3)$

$\rhd$

$C_{13}^2:C_3$

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 3 larger groups in the database.

This group is a maximal quotient of 1 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

See the $28 \times 28$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	3A	4A	4B	6A	12A	13A	13B	13C	13D	26A	26B
	Size	1	39	39	169	338	338	1014	338	676	36	36	24	72	468	468
	2 P	1A	1A	1A	1A	3A	2C	2C	3A	6A	13A	13B	13C	13D	13A	13B
	3 P	1A	2A	2B	2C	1A	4A	4B	2C	4A	13A	13B	13C	13D	26A	26B
	13 P	1A	2A	2B	2C	3A	4A	4B	6A	12A	1A	1A	1A	1A	2A	2B
4056.bb.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
4056.bb.1b		$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$
4056.bb.1c		$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$
4056.bb.1d		$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$
4056.bb.1e		$2$	$- 2$	$2$	$- 2$	$2$	$0$	$0$	$- 2$	$0$	$2$	$2$	$2$	$2$	$- 2$	$2$
4056.bb.1f		$2$	$2$	$- 2$	$- 2$	$2$	$0$	$0$	$- 2$	$0$	$2$	$2$	$2$	$2$	$2$	$- 2$
4056.bb.2a		$2$	$0$	$0$	$2$	$- 1$	$2$	$0$	$- 1$	$- 1$	$2$	$2$	$2$	$2$	$0$	$0$
4056.bb.2b		$2$	$0$	$0$	$2$	$- 1$	$- 2$	$0$	$- 1$	$1$	$2$	$2$	$2$	$2$	$0$	$0$
4056.bb.2c		$4$	$0$	$0$	$- 4$	$- 2$	$0$	$0$	$2$	$0$	$4$	$4$	$4$	$4$	$0$	$0$
4056.bb.12a		$36$	$0$	$12$	$0$	$0$	$0$	$0$	$0$	$0$	$10$	$10$	$- 3$	$- 3$	$0$	$- 1$
4056.bb.12b		$36$	$12$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 3$	$- 3$	$- 3$	$- 3$	$- 1$	$0$
4056.bb.12c		$36$	$- 12$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 3$	$- 3$	$- 3$	$- 3$	$1$	$0$
4056.bb.12d		$36$	$0$	$- 12$	$0$	$0$	$0$	$0$	$0$	$0$	$10$	$10$	$- 3$	$- 3$	$0$	$1$
4056.bb.24a		$24$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$- 2$	$11$	$- 2$	$0$	$0$
4056.bb.24b		$72$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 6$	$- 6$	$- 6$	$7$	$0$	$0$

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	4056.bb
Order	\( 2^{3} \cdot 3 \cdot 13^{2} \)
Exponent	\( 2^{2} \cdot 3 \cdot 13 \)

Nilpotent	no
Solvable	yes
$\card{G^{\mathrm{ab}}}$	\( 2^{3} \)
$\card{Z(G)}$	\( 1 \)
$\card{\Aut(G)}$	\( 2^{4} \cdot 3^{2} \cdot 13^{2} \)
$\card{\mathrm{Out}(G)}$	\( 2 \cdot 3 \)
Perm deg.	$26$
Trans deg.	$26$
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

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