LMFDB - Abstract group 8568.c: $C

Show commands: Gap / Magma / SageMath

magma:G := CyclicGroup(8568);

gap:G := CyclicGroup(8568);

sage:G = CyclicPermutationGroup(8568)

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(1666896348260402524237996416445714783,8568)'); a = G.1;

Group information

Description:	$C_{8568}$
Order:	$8568$$\medspace = 2^{3} \cdot 3^{2} \cdot 7 \cdot 17 $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order()
Exponent:	$8568$$\medspace = 2^{3} \cdot 3^{2} \cdot 7 \cdot 17 $	comment:Exponent of the group magma:Exponent(G); gap:Exponent(G); sage:G.exponent() sage_gap:G.Exponent()
Automorphism group:	$C_2^3\times C_6\times C_{48}$, of order $2304$$\medspace = 2^{8} \cdot 3^{2} $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage_gap:G.AutomorphismGroup()
Composition factors:	$C_2$ x 3, $C_3$ x 2, $C_7$, $C_{17}$	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	9	12	14	17	18	21	24	28	34	36	42	51	56	63	68	72	84	102	119	126	136	153	168	204	238	252	306	357	408	476	504	612	714	952	1071	1224	1428	2142	2856	4284	8568
Elements	1	1	2	2	2	6	4	6	4	6	16	6	12	8	12	16	12	12	32	24	36	32	24	24	32	96	36	64	96	48	64	96	72	96	192	128	192	144	192	192	384	576	384	384	576	768	1152	2304	8568
Conjugacy classes	1	1	2	2	2	6	4	6	4	6	16	6	12	8	12	16	12	12	32	24	36	32	24	24	32	96	36	64	96	48	64	96	72	96	192	128	192	144	192	192	384	576	384	384	576	768	1152	2304	8568
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:	$41$
Transitive degree:	$8568$
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:	$\langle a \mid a^{8568}=1 \rangle$ < a \| a^8568 >
comment:Define the group with the given generators and relations magma:G := PCGroup([7, -2, -2, -2, -3, -3, -7, -17, 14, 36, 58, 108, 137, 334]); a := Explode([G.1]); AssignNames(~G, ["a", "a2", "a4", "a8", "a24", "a72", "a504"]); gap:G := PcGroupCode(1666896348260402524237996416445714783,8568); 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(1666896348260402524237996416445714783,8568)'); 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(1666896348260402524237996416445714783,8568)'); a = G.1;
Permutation group:	Degree $41$ $\langle(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16,17)(18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)\rangle$ (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)
comment:Define the group as a permutation group magma:G := PermutationGroup< 41 \| (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) >; 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) ); 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)'])
Matrix group:	$\left\langle \left(\begin{array}{rr} 276 & 11 \\ 125 & 276 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{307})$
comment:Define the group as a matrix group with coefficients in GLFp magma:G := MatrixGroup< 2, GF(307) \| [[276, 11, 125, 276]] >; gap:G := Group([[[ Z(307)^140, Z(307)^4 ], [ Z(307)^3, Z(307)^140 ]]]); sage:MS = MatrixSpace(GF(307), 2, 2) G = MatrixGroup([MS([[276, 11], [125, 276]])])
Direct product:	$C_8$ $\, \times\, $ $C_9$ $\, \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:	$C_{4284}$ . $C_2$	$C_{2856}$ . $C_3$	$C_{2142}$ . $C_4$	$C_{1428}$ . $C_6$	all 32

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

Homology

Primary decomposition:	$C_{8} \times C_{9} \times C_{7} \times C_{17}$	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:	$Z \simeq$ $C_{8568}$	$G/Z \simeq$ $C_1$	comment:Center of the group magma:Center(G); gap:Center(G); sage:G.center() sage_gap:G.Center()
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_{8568}$	comment:Commutator subgroup of the group G magma:CommutatorSubgroup(G); gap:DerivedSubgroup(G); sage:G.commutator() sage_gap:G.DerivedSubgroup()
Frattini:	$\Phi \simeq$ $C_{12}$	$G/\Phi \simeq$ $C_{714}$	comment:Frattini subgroup of the group G magma:FrattiniSubgroup(G); gap:FrattiniSubgroup(G); sage:G.frattini_subgroup() sage_gap:G.FrattiniSubgroup()
Fitting:	$\operatorname{Fit} \simeq$ $C_{8568}$	$G/\operatorname{Fit} \simeq$ $C_1$	comment:Fitting subgroup of the group G magma:FittingSubgroup(G); gap:FittingSubgroup(G); sage:G.fitting_subgroup() sage_gap:G.FittingSubgroup()
Radical:	$R \simeq$ $C_{8568}$	$G/R \simeq$ $C_1$	comment:Radical of the group G magma:Radical(G); gap:SolvableRadical(G); sage_gap:G.SolvableRadical()
Socle:	$\operatorname{soc} \simeq$ $C_{714}$	$G/\operatorname{soc} \simeq$ $C_{12}$	comment:Socle of the group G magma:Socle(G); gap:Socle(G); sage:G.socle() sage_gap:G.Socle()
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$
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 (quotient in parentheses)

Order 8568: $C_{8568}$ ($C_1$)

Order 4284: $C_{4284}$ ($C_2$)

Order 2856: $C_{2856}$ ($C_3$)

Order 2142: $C_{2142}$ ($C_4$)

Order 1428: $C_{1428}$ ($C_6$)

Order 1224: $C_{1224}$ ($C_7$)

Order 1071: $C_{1071}$ ($C_8$)

Order 952: $C_{952}$ ($C_9$)

Order 714: $C_{714}$ ($C_{12}$)

Order 612: $C_{612}$ ($C_{14}$)

Order 504: $C_{504}$ ($C_{17}$)

Order 476: $C_{476}$ ($C_{18}$)

Order 408: $C_{408}$ ($C_{21}$)

Order 357: $C_{357}$ ($C_{24}$)

Order 306: $C_{306}$ ($C_{28}$)

Order 252: $C_{252}$ ($C_{34}$)

Order 238: $C_{238}$ ($C_{36}$)

Order 204: $C_{204}$ ($C_{42}$)

Order 168: $C_{168}$ ($C_{51}$)

Order 153: $C_{153}$ ($C_{56}$)

Order 136: $C_{136}$ ($C_{63}$)

Order 126: $C_{126}$ ($C_{68}$)

Order 119: $C_{119}$ ($C_{72}$)

Order 102: $C_{102}$ ($C_{84}$)

Order 84: $C_{84}$ ($C_{102}$)

Order 72: $C_{72}$ ($C_{119}$)

Order 68: $C_{68}$ ($C_{126}$)

Order 63: $C_{63}$ ($C_{136}$)

Order 56: $C_{56}$ ($C_{153}$)

Order 51: $C_{51}$ ($C_{168}$)

Order 42: $C_{42}$ ($C_{204}$)

Order 36: $C_{36}$ ($C_{238}$)

Order 34: $C_{34}$ ($C_{252}$)

Order 28: $C_{28}$ ($C_{306}$)

Order 24: $C_{24}$ ($C_{357}$)

Order 21: $C_{21}$ ($C_{408}$)

Order 18: $C_{18}$ ($C_{476}$)

Order 17: $C_{17}$ ($C_{504}$)

Order 14: $C_{14}$ ($C_{612}$)

Order 12: $C_{12}$ ($C_{714}$)

Order 9: $C_9$ ($C_{952}$)

Order 8: $C_8$ ($C_{1071}$)

Order 7: $C_7$ ($C_{1224}$)

Order 6: $C_6$ ($C_{1428}$)

Order 4: $C_4$ ($C_{2142}$)

Order 3: $C_3$ ($C_{2856}$)

Order 2: $C_2$ ($C_{4284}$)

Order 1: $C_1$ ($C_{8568}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 8568: $C_{8568}$ ($C_1$)

Order 4284: $C_{4284}$ ($C_2$)

Order 2856: $C_{2856}$ ($C_3$)

Order 2142: $C_{2142}$ ($C_4$)

Order 1428: $C_{1428}$ ($C_6$)

Order 1224: $C_{1224}$ ($C_7$)

Order 1071: $C_{1071}$ ($C_8$)

Order 952: $C_{952}$ ($C_9$)

Order 714: $C_{714}$ ($C_{12}$)

Order 612: $C_{612}$ ($C_{14}$)

Order 504: $C_{504}$ ($C_{17}$)

Order 476: $C_{476}$ ($C_{18}$)

Order 408: $C_{408}$ ($C_{21}$)

Order 357: $C_{357}$ ($C_{24}$)

Order 306: $C_{306}$ ($C_{28}$)

Order 252: $C_{252}$ ($C_{34}$)

Order 238: $C_{238}$ ($C_{36}$)

Order 204: $C_{204}$ ($C_{42}$)

Order 168: $C_{168}$ ($C_{51}$)

Order 153: $C_{153}$ ($C_{56}$)

Order 136: $C_{136}$ ($C_{63}$)

Order 126: $C_{126}$ ($C_{68}$)

Order 119: $C_{119}$ ($C_{72}$)

Order 102: $C_{102}$ ($C_{84}$)

Order 84: $C_{84}$ ($C_{102}$)

Order 72: $C_{72}$ ($C_{119}$)

Order 68: $C_{68}$ ($C_{126}$)

Order 63: $C_{63}$ ($C_{136}$)

Order 56: $C_{56}$ ($C_{153}$)

Order 51: $C_{51}$ ($C_{168}$)

Order 42: $C_{42}$ ($C_{204}$)

Order 36: $C_{36}$ ($C_{238}$)

Order 34: $C_{34}$ ($C_{252}$)

Order 28: $C_{28}$ ($C_{306}$)

Order 24: $C_{24}$ ($C_{357}$)

Order 21: $C_{21}$ ($C_{408}$)

Order 18: $C_{18}$ ($C_{476}$)

Order 17: $C_{17}$ ($C_{504}$)

Order 14: $C_{14}$ ($C_{612}$)

Order 12: $C_{12}$ ($C_{714}$)

Order 9: $C_9$ ($C_{952}$)

Order 8: $C_8$ ($C_{1071}$)

Order 7: $C_7$ ($C_{1224}$)

Order 6: $C_6$ ($C_{1428}$)

Order 4: $C_4$ ($C_{2142}$)

Order 3: $C_3$ ($C_{2856}$)

Order 2: $C_2$ ($C_{4284}$)

Order 1: $C_1$ ($C_{8568}$)

Series

Derived series

$C_{8568}$

$\rhd$

$C_1$

comment:Derived series of the group GF

magma:DerivedSeries(G);

gap:DerivedSeriesOfGroup(G);

sage:G.derived_series()

sage_gap:G.DerivedSeriesOfGroup()

Chief series

$C_{8568}$

$\rhd$

$C_{4284}$

$\rhd$

$C_{2142}$

$\rhd$

$C_{1071}$

$\rhd$

$C_{357}$

$\rhd$

$C_{119}$

$\rhd$

$C_{17}$

$\rhd$

$C_1$

comment:Chief series of the group G

magma:ChiefSeries(G);

gap:ChiefSeries(G);

sage_gap:G.ChiefSeries()

Lower central series

$C_{8568}$

$\rhd$

$C_1$

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

$C_1$

$\lhd$

$C_{8568}$

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 $8568 \times 8568$ character table is not available for this group.

Rational character table

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

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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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	8568.c
Order	\( 2^{3} \cdot 3^{2} \cdot 7 \cdot 17 \)
Exponent	\( 2^{3} \cdot 3^{2} \cdot 7 \cdot 17 \)
Abelian	yes
$\card{\Aut(G)}$	\( 2^{8} \cdot 3^{2} \)
Perm deg.	$41$
Trans deg.	$8568$
Rank	$1$