LMFDB - Abstract group 1218.5: $C_{203}:C

Show commands: Gap / Magma / SageMath

magma:G := SmallGroup(1218, 5);

gap:G := SmallGroup(1218, 5);

sage_gap:G = libgap.SmallGroup(1218, 5)

comment:Define the group as a permutation group

sage:G = PermutationGroup(['(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)', '(31,32,33)(34,36,35)', '(30,31,33,35,32,34,36)', '(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,29,27,25,23,21,19,17,15,13,11,9,7,5,3)'])

Group information

Description:	$C_{203}:C_6$
Order:	$1218$$\medspace = 2 \cdot 3 \cdot 7 \cdot 29 $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order()
Exponent:	$1218$$\medspace = 2 \cdot 3 \cdot 7 \cdot 29 $	comment:Exponent of the group magma:Exponent(G); gap:Exponent(G); sage:G.exponent() sage_gap:G.Exponent()
Automorphism group:	$C_{29}:(C_7^2:(C_2\times C_{12}))$, of order $34104$$\medspace = 2^{3} \cdot 3 \cdot 7^{2} \cdot 29 $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage_gap:G.AutomorphismGroup()
Composition factors:	$C_2$, $C_3$, $C_7$, $C_{29}$	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 and a Z-group (hence solvable, supersolvable, monomial, metacyclic, 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	6	7	14	29	87	203
Elements	1	29	14	406	6	174	28	392	168	1218
Conjugacy classes	1	1	2	2	2	2	14	28	28	80
Divisions	1	1	1	1	1	1	1	1	1	9
Autjugacy classes	1	1	2	2	1	1	1	2	1	12

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	6	28	56	168
Irr. complex chars.	6	42	4	28	0	0	0	80
Irr. rational chars.	2	2	0	2	1	1	1	9

Minimal presentations

Permutation degree:	$36$
Transitive degree:	$203$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	168
Arbitrary	5	8	34

Constructions

Show commands: Gap / Magma / SageMath

Presentation:	$\langle a, b \mid a^{6}=b^{203}=1, b^{a}=b^{144} \rangle$ < a, b \| a^6,b^203, b^144a^-1b^-1*a >
comment:Define the group with the given generators and relations magma:G := PCGroup([4, -2, -3, -7, -29, 8, 10370, 1086, 94, 18819]); a,b := Explode([G.1, G.3]); AssignNames(~G, ["a", "a2", "b", "b7"]); gap:G := PcGroupCode(22871664083634580314295,1218); a := G.1; b := G.3; sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(22871664083634580314295,1218)'); a = G.1; b = 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(22871664083634580314295,1218)'); a = G.1; b = G.3;
Permutation group:	Degree $36$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25) \!\cdots\! \rangle$ (2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29), (31,32,33)(34,36,35), (30,31,33,35,32,34,36), (1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,29,27,25,23,21,19,17,15,13,11,9,7,5,3)
comment:Define the group as a permutation group magma:G := PermutationGroup< 36 \| (2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29), (31,32,33)(34,36,35), (30,31,33,35,32,34,36), (1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,29,27,25,23,21,19,17,15,13,11,9,7,5,3) >; gap:G := Group( (2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29), (31,32,33)(34,36,35), (30,31,33,35,32,34,36), (1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,29,27,25,23,21,19,17,15,13,11,9,7,5,3) ); sage:G = PermutationGroup(['(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)', '(31,32,33)(34,36,35)', '(30,31,33,35,32,34,36)', '(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,29,27,25,23,21,19,17,15,13,11,9,7,5,3)'])
Direct product:	$(C_7:C_3)$ $\, \times\, $ $D_{29}$
Semidirect product:	$C_{203}$ $\,\rtimes\,$ $C_6$	$(C_7\times D_{29})$ $\,\rtimes\,$ $C_3$	$C_7$ $\,\rtimes\,$ $(C_3\times D_{29})$	$(C_7:C_{87})$ $\,\rtimes\,$ $C_2$	all 5
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:	$C_{6} \simeq C_{2} \times C_{3}$	comment:The abelianization of the group magma:quo< G \| CommutatorSubgroup(G) >; gap:FactorGroup(G, DerivedSubgroup(G)); sage:G.quotient(G.commutator())
Schur multiplier:	$C_1$	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 320 subgroups in 16 conjugacy classes, 9 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_{203}:C_6$	comment:Center of the group magma:Center(G); gap:Center(G); sage:G.center() sage_gap:G.Center()
Commutator:	$G' \simeq$ $C_{203}$	$G/G' \simeq$ $C_6$	comment:Commutator subgroup of the group G magma:CommutatorSubgroup(G); gap:DerivedSubgroup(G); sage:G.commutator() sage_gap:G.DerivedSubgroup()
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{203}:C_6$	comment:Frattini subgroup of the group G magma:FrattiniSubgroup(G); gap:FrattiniSubgroup(G); sage:G.frattini_subgroup() sage_gap:G.FrattiniSubgroup()
Fitting:	$\operatorname{Fit} \simeq$ $C_{203}$	$G/\operatorname{Fit} \simeq$ $C_6$	comment:Fitting subgroup of the group G magma:FittingSubgroup(G); gap:FittingSubgroup(G); sage:G.fitting_subgroup() sage_gap:G.FittingSubgroup()
Radical:	$R \simeq$ $C_{203}:C_6$	$G/R \simeq$ $C_1$	comment:Radical of the group G magma:Radical(G); gap:SolvableRadical(G); sage_gap:G.SolvableRadical()
Socle:	$\operatorname{soc} \simeq$ $C_{203}$	$G/\operatorname{soc} \simeq$ $C_6$	comment:Socle of the group G magma:Socle(G); gap:Socle(G); sage:G.socle() sage_gap:G.Socle()
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
29-Sylow subgroup:	$P_{ 29 } \simeq$ $C_{29}$

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

$C_{203}:C_6$

$\rhd$

$C_{203}$

$\rhd$

$C_1$

comment:Derived series of the group GF

magma:DerivedSeries(G);

gap:DerivedSeriesOfGroup(G);

sage:G.derived_series()

sage_gap:G.DerivedSeriesOfGroup()

Chief series

$C_{203}:C_6$

$\rhd$

$C_7:C_{87}$

$\rhd$

$C_{203}$

$\rhd$

$C_{29}$

$\rhd$

$C_1$

comment:Chief series of the group G

magma:ChiefSeries(G);

gap:ChiefSeries(G);

sage_gap:G.ChiefSeries()

Lower central series

$C_{203}:C_6$

$\rhd$

$C_{203}$

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()

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 $80 \times 80$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	3A	6A	7A	14A	29A	87A	203A
	Size	1	29	14	406	6	174	28	392	168
	2 P	1A	1A	3A	3A	7A	7A	29A	87A	203A
	3 P	1A	2A	1A	2A	7A	14A	29A	29A	203A
	7 P	1A	2A	3A	6A	1A	2A	29A	87A	29A
	29 P	1A	2A	3A	6A	7A	14A	1A	3A	7A
1218.5.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1218.5.1b		$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$
1218.5.1c		$2$	$2$	$- 1$	$- 1$	$2$	$2$	$2$	$- 1$	$2$
1218.5.1d		$2$	$- 2$	$- 1$	$1$	$2$	$- 2$	$2$	$- 1$	$2$
1218.5.2a		$28$	$0$	$28$	$0$	$28$	$0$	$- 1$	$- 1$	$- 1$
1218.5.2b		$56$	$0$	$- 28$	$0$	$56$	$0$	$- 2$	$1$	$- 2$
1218.5.3a		$6$	$6$	$0$	$0$	$- 1$	$- 1$	$6$	$0$	$- 1$
1218.5.3b		$6$	$- 6$	$0$	$0$	$- 1$	$1$	$6$	$0$	$- 1$
1218.5.6a		$168$	$0$	$0$	$0$	$- 28$	$0$	$- 6$	$0$	$1$

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	1218.5
Order	\( 2 \cdot 3 \cdot 7 \cdot 29 \)
Exponent	\( 2 \cdot 3 \cdot 7 \cdot 29 \)

Nilpotent	no
Solvable	yes
$\card{G^{\mathrm{ab}}}$	\( 2 \cdot 3 \)
$\card{Z(G)}$	\( 1 \)
$\card{\Aut(G)}$	\( 2^{3} \cdot 3 \cdot 7^{2} \cdot 29 \)
$\card{\mathrm{Out}(G)}$	\( 2^{2} \cdot 7 \)
Perm deg.	$36$
Trans deg.	$203$
Rank	$2$

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Fields

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Database

Properties

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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

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.