LMFDB - Abstract group 11424.d: $C

Show commands: Gap / Magma / SageMath

magma:G := CyclicGroup(11424);

gap:G := CyclicGroup(11424);

sage:G = CyclicPermutationGroup(11424)

comment:Define the group as a permutation group

sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(328522457404204952571712827029216524559712607,11424)'); a = G.1;

Group information

Description:	$C_{11424}$
Order:	$11424$$\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 17 $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order()
Exponent:	$11424$$\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 17 $	comment:Exponent of the group magma:Exponent(G); gap:Exponent(G); sage:G.exponent() sage_gap:G.Exponent()
Automorphism group:	$C_2^3\times C_8\times C_{48}$, of order $3072$$\medspace = 2^{10} \cdot 3 $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage_gap:G.AutomorphismGroup()
Composition factors:	$C_2$ x 5, $C_3$, $C_7$, $C_{17}$	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 cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,7,17$), hyperelementary, metacyclic, metabelian, a Z-group, 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	7	8	12	14	16	17	21	24	28	32	34	42	48	51	56	68	84	96	102	112	119	136	168	204	224	238	272	336	357	408	476	544	672	714	816	952	1428	1632	1904	2856	3808	5712	11424
Elements	1	1	2	2	2	6	4	4	6	8	16	12	8	12	16	16	12	16	32	24	32	24	32	32	48	96	64	48	64	96	96	128	96	192	128	192	256	192	192	256	384	384	512	768	768	1536	1536	3072	11424
Conjugacy classes	1	1	2	2	2	6	4	4	6	8	16	12	8	12	16	16	12	16	32	24	32	24	32	32	48	96	64	48	64	96	96	128	96	192	128	192	256	192	192	256	384	384	512	768	768	1536	1536	3072	11424
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	48
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	48

Minimal presentations

Permutation degree:	$59$
Transitive degree:	$11424$
Rank:	$1$
Inequivalent generators:	$1$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	1	not computed	not computed
Arbitrary	1	not computed	not computed

Constructions

Show commands: Gap / Magma / SageMath

Presentation:	$\langle a \mid a^{11424}=1 \rangle$ < a \| a^11424 >
comment:Define the group with the given generators and relations magma:G := PCGroup([8, -2, -2, -2, -2, -2, -3, -7, -17, 16, 41, 66, 91, 116, 189, 446]); a := Explode([G.1]); AssignNames(~G, ["a", "a2", "a4", "a8", "a16", "a32", "a96", "a672"]); gap:G := PcGroupCode(328522457404204952571712827029216524559712607,11424); a := G.1; sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(328522457404204952571712827029216524559712607,11424)'); a = G.1; sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(328522457404204952571712827029216524559712607,11424)'); a = G.1;
Permutation group:	Degree $59$ $\langle(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32) \!\cdots\! \rangle$ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59)
comment:Define the group as a permutation group magma:G := PermutationGroup< 59 \| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59) >; gap:G := Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59) ); sage:G = PermutationGroup(['(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59)'])
Matrix group:	$\left\langle \left(\begin{array}{rr} 216 & 99 \\ 219 & 216 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{239})$
comment:Define the group as a matrix group with coefficients in GLFp magma:G := MatrixGroup< 2, GF(239) \| [[216, 99, 219, 216]] >; gap:G := Group([[[ Z(239)^182, Z(239)^152 ], [ Z(239)^151, Z(239)^182 ]]]); sage:MS = MatrixSpace(GF(239), 2, 2) G = MatrixGroup([MS([[216, 99], [219, 216]])])
Direct product:	$C_{32}$ $\, \times\, $ $C_3$ $\, \times\, $ $C_7$ $\, \times\, $ $C_{17}$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{5712}$ . $C_2$	$C_{2856}$ . $C_4$	$C_{1904}$ . $C_6$	$C_{1428}$ . $C_8$	all 32

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

Homology

Primary decomposition:	$C_{32} \times C_{3} \times C_{7} \times C_{17}$	comment:The primary decomposition of the group magma:PrimaryInvariants(G); gap:AbelianInvariants(G); sage_gap:G.AbelianInvariants()
Schur multiplier:	$C_1$	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 48 subgroups, all normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{11424}$	$G/Z \simeq$ $C_1$	comment:Center of the group magma:Center(G); gap:Center(G); sage:G.center() sage_gap:G.Center()
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_{11424}$	comment:Commutator subgroup of the group G magma:CommutatorSubgroup(G); gap:DerivedSubgroup(G); sage:G.commutator() sage_gap:G.DerivedSubgroup()
Frattini:	$\Phi \simeq$ $C_{16}$	$G/\Phi \simeq$ $C_{714}$	comment:Frattini subgroup of the group G magma:FrattiniSubgroup(G); gap:FrattiniSubgroup(G); sage:G.frattini_subgroup() sage_gap:G.FrattiniSubgroup()
Fitting:	$\operatorname{Fit} \simeq$ $C_{11424}$	$G/\operatorname{Fit} \simeq$ $C_1$	comment:Fitting subgroup of the group G magma:FittingSubgroup(G); gap:FittingSubgroup(G); sage:G.fitting_subgroup() sage_gap:G.FittingSubgroup()
Radical:	$R \simeq$ $C_{11424}$	$G/R \simeq$ $C_1$	comment:Radical of the group G magma:Radical(G); gap:SolvableRadical(G); sage_gap:G.SolvableRadical()
Socle:	$\operatorname{soc} \simeq$ $C_{714}$	$G/\operatorname{soc} \simeq$ $C_{16}$	comment:Socle of the group G magma:Socle(G); gap:Socle(G); sage:G.socle() sage_gap:G.Socle()
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{32}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 11424: $C_{11424}$ ($C_1$)

Order 5712: $C_{5712}$ ($C_2$)

Order 3808: $C_{3808}$ ($C_3$)

Order 2856: $C_{2856}$ ($C_4$)

Order 1904: $C_{1904}$ ($C_6$)

Order 1632: $C_{1632}$ ($C_7$)

Order 1428: $C_{1428}$ ($C_8$)

Order 952: $C_{952}$ ($C_{12}$)

Order 816: $C_{816}$ ($C_{14}$)

Order 714: $C_{714}$ ($C_{16}$)

Order 672: $C_{672}$ ($C_{17}$)

Order 544: $C_{544}$ ($C_{21}$)

Order 476: $C_{476}$ ($C_{24}$)

Order 408: $C_{408}$ ($C_{28}$)

Order 357: $C_{357}$ ($C_{32}$)

Order 336: $C_{336}$ ($C_{34}$)

Order 272: $C_{272}$ ($C_{42}$)

Order 238: $C_{238}$ ($C_{48}$)

Order 224: $C_{224}$ ($C_{51}$)

Order 204: $C_{204}$ ($C_{56}$)

Order 168: $C_{168}$ ($C_{68}$)

Order 136: $C_{136}$ ($C_{84}$)

Order 119: $C_{119}$ ($C_{96}$)

Order 112: $C_{112}$ ($C_{102}$)

Order 102: $C_{102}$ ($C_{112}$)

Order 96: $C_{96}$ ($C_{119}$)

Order 84: $C_{84}$ ($C_{136}$)

Order 68: $C_{68}$ ($C_{168}$)

Order 56: $C_{56}$ ($C_{204}$)

Order 51: $C_{51}$ ($C_{224}$)

Order 48: $C_{48}$ ($C_{238}$)

Order 42: $C_{42}$ ($C_{272}$)

Order 34: $C_{34}$ ($C_{336}$)

Order 32: $C_{32}$ ($C_{357}$)

Order 28: $C_{28}$ ($C_{408}$)

Order 24: $C_{24}$ ($C_{476}$)

Order 21: $C_{21}$ ($C_{544}$)

Order 17: $C_{17}$ ($C_{672}$)

Order 16: $C_{16}$ ($C_{714}$)

Order 14: $C_{14}$ ($C_{816}$)

Order 12: $C_{12}$ ($C_{952}$)

Order 8: $C_8$ ($C_{1428}$)

Order 7: $C_7$ ($C_{1632}$)

Order 6: $C_6$ ($C_{1904}$)

Order 4: $C_4$ ($C_{2856}$)

Order 3: $C_3$ ($C_{3808}$)

Order 2: $C_2$ ($C_{5712}$)

Order 1: $C_1$ ($C_{11424}$)

Series

Derived series

$C_{11424}$

$\rhd$

$C_1$

comment:Derived series of the group GF

magma:DerivedSeries(G);

gap:DerivedSeriesOfGroup(G);

sage:G.derived_series()

sage_gap:G.DerivedSeriesOfGroup()

Chief series

$C_{11424}$

$\rhd$

$C_{5712}$

$\rhd$

$C_{2856}$

$\rhd$

$C_{1428}$

$\rhd$

$C_{714}$

$\rhd$

$C_{357}$

$\rhd$

$C_{119}$

$\rhd$

$C_{17}$

$\rhd$

$C_1$

comment:Chief series of the group G

magma:ChiefSeries(G);

gap:ChiefSeries(G);

sage_gap:G.ChiefSeries()

Lower central series

$C_{11424}$

$\rhd$

$C_1$

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

$C_1$

$\lhd$

$C_{11424}$

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 2 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

The $11424 \times 11424$ character table is not available for this group.

Rational character table

The $48 \times 48$ rational character table is not available for this group.

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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.

Label	11424.d
Order	\( 2^{5} \cdot 3 \cdot 7 \cdot 17 \)
Exponent	\( 2^{5} \cdot 3 \cdot 7 \cdot 17 \)
Abelian	yes
$\card{\Aut(G)}$	\( 2^{10} \cdot 3 \)
Perm deg.	$59$
Trans deg.	$11424$
Rank	$1$