LMFDB - Abstract group 1336.4: $Q_8\times C

Show commands: Gap / Magma / Oscar / SageMath

comment:Construction of abstract group

magma:G := SmallGroup(1336, 4);

gap:G := SmallGroup(1336, 4);

sage:G = PermutationGroup(['(1,6,2,5)(3,8,4,7)', '(1,4,2,3)(5,7,6,8)', '(1,2)(3,4)(5,6)(7,8)', '(9,175,174,173,172,171,170,169,168,167,166,165,164,163,162,161,160,159,158,157,156,155,154,153,152,151,150,149,148,147,146,145,144,143,142,141,140,139,138,137,136,135,134,133,132,131,130,129,128,127,126,125,124,123,122,121,120,119,118,117,116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,100,99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66,65,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10)'])

sage_gap:G = libgap.SmallGroup(1336, 4)

oscar:G = small_group(1336, 4)

Group information

Description:	$Q_8\times C_{167}$
Order:	$1336$$\medspace = 2^{3} \cdot 167 $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order() oscar:order(G)
Exponent:	$668$$\medspace = 2^{2} \cdot 167 $	comment:Exponent of the group magma:Exponent(G); gap:Exponent(G); sage:G.exponent() sage_gap:G.Exponent() oscar:exponent(G)
Automorphism group:	$C_{166}\times S_4$, of order $3984$$\medspace = 2^{4} \cdot 3 \cdot 83 $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage:libgap(G).AutomorphismGroup() sage_gap:G.AutomorphismGroup() oscar:automorphism_group(G)
Composition factors:	$C_2$ x 3, $C_{167}$	comment:Composition factors of the group magma:CompositionFactors(G); gap:CompositionSeries(G); sage:G.composition_series() sage_gap:G.CompositionSeries() oscar:composition_series(G)
Nilpotency class:	$2$	comment:Nilpotency class of the group magma:NilpotencyClass(G); gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi; sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1 sage_gap:G.NilpotencyClassOfGroup() if G.IsNilpotentGroup() else -1 oscar:if is_nilpotent(G) nilpotency_class(G) end
Derived length:	$2$	comment:Derived length of the group magma:DerivedLength(G); gap:DerivedLength(G); sage:libgap(G).DerivedLength() sage_gap:G.DerivedLength() oscar:derived_length(G)

This group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).

comment:Determine if the group G is abelian

magma:IsAbelian(G);

gap:IsAbelian(G);

sage:G.is_abelian()

sage_gap:G.IsAbelian()

oscar:is_abelian(G)

comment:Determine if the group G is cyclic

magma:IsCyclic(G);

gap:IsCyclic(G);

sage:G.is_cyclic()

sage_gap:G.IsCyclic()

oscar:is_cyclic(G)

comment:Determine if the group G is nilpotent

magma:IsNilpotent(G);

gap:IsNilpotentGroup(G);

sage:G.is_nilpotent()

sage_gap:G.IsNilpotentGroup()

oscar:is_nilpotent(G)

comment:Determine if the group G is solvable

magma:IsSolvable(G);

gap:IsSolvableGroup(G);

sage:G.is_solvable()

sage_gap:G.IsSolvableGroup()

oscar:is_solvable(G)

comment:Determine if the group G is supersolvable

gap:IsSupersolvableGroup(G);

sage:G.is_supersolvable()

sage_gap:G.IsSupersolvableGroup()

oscar:is_supersolvable(G)

comment:Determine if the group G is simple

magma:IsSimple(G);

gap:IsSimpleGroup(G);

sage:G.is_simple()

sage_gap:G.IsSimpleGroup()

oscar:is_simple(G)

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))

sage_gap:# Sage code (using the GAP interface) to output the first two rows of the group statistics table element_orders = [g.Order() for g in G.Elements()] orders = sorted(list(set(element_orders))) print("Orders:", orders) print("Elements:", [element_orders.count(n) for n in orders], G.Order()) cc_orders = [cc.Representative().Order() for cc in G.ConjugacyClasses()] print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))

oscar:# Oscar code to output the first two rows of the group statistics table element_orders = [order(g) for g in elements(G)] orders = sort(unique(element_orders)) println("Orders: ", orders) element_counts = [count(==(n), element_orders) for n in orders] println("Elements: ", element_counts, " ", order(G)) ccs = conjugacy_classes(G) cc_orders = [order(representative(cc)) for cc in ccs] cc_counts = [count(==(n), cc_orders) for n in orders] println("Conjugacy classes: ", cc_counts, " ", length(ccs))

Order	1	2	4	167	334	668
Elements	1	1	6	166	166	996	1336
Conjugacy classes	1	1	3	166	166	498	835
Divisions	1	1	3	1	1	3	10
Autjugacy classes	1	1	1	1	1	1	6

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:# 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 G.CharacterDegrees()

oscar:# Outputs an MSet containing the absolutely irreducible degrees of G and their multiplicities. character_degrees(G)

Dimension	1	2	166	332
Irr. complex chars.	668	167	0	0	835
Irr. rational chars.	4	1	4	1	10

Minimal presentations

Permutation degree:	$175$
Transitive degree:	$1336$
Rank:	$2$
Inequivalent generating pairs:	$168$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	not computed	664
Arbitrary	not computed	not computed	not computed

Constructions

Show commands: Gap / Magma / Oscar / SageMath

Presentation:	$\langle a, b \mid b^{668}=1, a^{2}=b^{334}, b^{a}=b^{335} \rangle$ < a, b \| b^668, b^334a^-2, b^335a^-1b^-1a >
comment:Define the group with the given generators and relations magma:G := PCGroup([4, -2, -2, -2, -167, 2672, 5361, 21, 34]); a,b := Explode([G.1, G.2]); AssignNames(~G, ["a", "b", "b2", "b4"]); gap:G := PcGroupCode(231773169199562335834261,1336); 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(231773169199562335834261,1336)'); 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(231773169199562335834261,1336)'); a = G.1; b = G.2;
Permutation group:	Degree $175$ $\langle(1,6,2,5)(3,8,4,7), (1,4,2,3)(5,7,6,8), (1,2)(3,4)(5,6)(7,8), (9,175,174,173,172,171,170,169,168,167,166,165,164,163,162,161,160,159,158,157,156,155,154,153,152,151,150,149,148,147,146,145,144,143,142,141,140,139,138,137,136,135,134,133,132,131,130,129,128,127,126,125,124,123,122,121,120,119,118,117,116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,100,99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66,65,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10) \!\cdots\! \rangle$ (1,6,2,5)(3,8,4,7), (1,4,2,3)(5,7,6,8), (1,2)(3,4)(5,6)(7,8), (9,175,174,173,172,171,170,169,168,167,166,165,164,163,162,161,160,159,158,157,156,155,154,153,152,151,150,149,148,147,146,145,144,143,142,141,140,139,138,137,136,135,134,133,132,131,130,129,128,127,126,125,124,123,122,121,120,119,118,117,116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,100,99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66,65,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10)
comment:Define the group as a permutation group magma:G := PermutationGroup< 175 \| (1,6,2,5)(3,8,4,7), (1,4,2,3)(5,7,6,8), (1,2)(3,4)(5,6)(7,8), (9,175,174,173,172,171,170,169,168,167,166,165,164,163,162,161,160,159,158,157,156,155,154,153,152,151,150,149,148,147,146,145,144,143,142,141,140,139,138,137,136,135,134,133,132,131,130,129,128,127,126,125,124,123,122,121,120,119,118,117,116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,100,99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66,65,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10) >; gap:G := Group( (1,6,2,5)(3,8,4,7), (1,4,2,3)(5,7,6,8), (1,2)(3,4)(5,6)(7,8), (9,175,174,173,172,171,170,169,168,167,166,165,164,163,162,161,160,159,158,157,156,155,154,153,152,151,150,149,148,147,146,145,144,143,142,141,140,139,138,137,136,135,134,133,132,131,130,129,128,127,126,125,124,123,122,121,120,119,118,117,116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,100,99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66,65,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10) ); sage:G = PermutationGroup(['(1,6,2,5)(3,8,4,7)', '(1,4,2,3)(5,7,6,8)', '(1,2)(3,4)(5,6)(7,8)', '(9,175,174,173,172,171,170,169,168,167,166,165,164,163,162,161,160,159,158,157,156,155,154,153,152,151,150,149,148,147,146,145,144,143,142,141,140,139,138,137,136,135,134,133,132,131,130,129,128,127,126,125,124,123,122,121,120,119,118,117,116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,100,99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66,65,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10)']) sage_gap:G = gap.new('Group( (1,6,2,5)(3,8,4,7), (1,4,2,3)(5,7,6,8), (1,2)(3,4)(5,6)(7,8), (9,175,174,173,172,171,170,169,168,167,166,165,164,163,162,161,160,159,158,157,156,155,154,153,152,151,150,149,148,147,146,145,144,143,142,141,140,139,138,137,136,135,134,133,132,131,130,129,128,127,126,125,124,123,122,121,120,119,118,117,116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,100,99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66,65,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10) )') oscar:G = @permutation_group(175, (1,6,2,5)(3,8,4,7), (1,4,2,3)(5,7,6,8), (1,2)(3,4)(5,6)(7,8), (9,175,174,173,172,171,170,169,168,167,166,165,164,163,162,161,160,159,158,157,156,155,154,153,152,151,150,149,148,147,146,145,144,143,142,141,140,139,138,137,136,135,134,133,132,131,130,129,128,127,126,125,124,123,122,121,120,119,118,117,116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,100,99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66,65,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10))

Direct product:	$C_{167}$ $\, \times\, $ $Q_8$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{668}$ . $C_2$ (3)	$C_4$ . $C_{334}$ (3)	$C_{334}$ . $C_2^2$	$C_2$ . $(C_2\times C_{334})$	more information

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

Homology

Abelianization:	$C_{2} \times C_{334} \simeq C_{2}^{2} \times C_{167}$	comment:The abelianization of the group magma:quo< G \| CommutatorSubgroup(G) >; gap:FactorGroup(G, DerivedSubgroup(G)); sage:G.quotient(G.commutator()) sage_gap:G.FactorGroup(G.DerivedSubgroup()) oscar:quo(G, derived_subgroup(G)[1])
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()

oscar:subgroups(G)

There are 12 subgroups, all normal (6 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{334}$	$G/Z \simeq$ $C_2^2$	comment:Center of the group magma:Center(G); gap:Center(G); sage:G.center() sage_gap:G.Center() oscar:center(G)
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_2\times C_{334}$	comment:Commutator subgroup of the group G magma:CommutatorSubgroup(G); gap:DerivedSubgroup(G); sage:G.commutator() sage_gap:G.DerivedSubgroup() oscar:derived_subgroup(G)
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times C_{334}$	comment:Frattini subgroup of the group G magma:FrattiniSubgroup(G); gap:FrattiniSubgroup(G); sage:G.frattini_subgroup() sage_gap:G.FrattiniSubgroup() oscar:frattini_subgroup(G)
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_{167}$	$G/\operatorname{Fit} \simeq$ $C_1$	comment:Fitting subgroup of the group G magma:FittingSubgroup(G); gap:FittingSubgroup(G); sage:G.fitting_subgroup() sage_gap:G.FittingSubgroup() oscar:fitting_subgroup(G)
Radical:	$R \simeq$ $Q_8\times C_{167}$	$G/R \simeq$ $C_1$	comment:Radical of the group G magma:Radical(G); gap:SolvableRadical(G); sage_gap:G.SolvableRadical() oscar:solvable_radical(G)
Socle:	$\operatorname{soc} \simeq$ $C_{334}$	$G/\operatorname{soc} \simeq$ $C_2^2$	comment:Socle of the group G magma:Socle(G); gap:Socle(G); sage:G.socle() sage_gap:G.Socle() oscar:socle(G)
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_8$
167-Sylow subgroup:	$P_{ 167 } \simeq$ $C_{167}$

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

$Q_8\times C_{167}$

$\rhd$

$C_2$

$\rhd$

$C_1$

comment:Derived series of the group G

magma:DerivedSeries(G);

gap:DerivedSeriesOfGroup(G);

sage:G.derived_series()

sage_gap:G.DerivedSeriesOfGroup()

oscar:derived_series(G)

Chief series

$Q_8\times C_{167}$

$\rhd$

$C_{668}$

$\rhd$

$C_{334}$

$\rhd$

$C_{167}$

$\rhd$

$C_1$

comment:Chief series of the group G

magma:ChiefSeries(G);

gap:ChiefSeries(G);

sage:libgap(G).ChiefSeries()

sage_gap:G.ChiefSeries()

oscar:chief_series(G)

Lower central series

$Q_8\times C_{167}$

$\rhd$

$C_2$

$\rhd$

$C_1$

comment:The lower central series of the group G

magma:LowerCentralSeries(G);

gap:LowerCentralSeriesOfGroup(G);

sage:G.lower_central_series()

sage_gap:G.LowerCentralSeriesOfGroup()

oscar:lower_central_series(G)

Upper central series

$C_1$

$\lhd$

$C_{334}$

$\lhd$

$Q_8\times C_{167}$

comment:The upper central series of the group G

magma:UpperCentralSeries(G);

gap:UpperCentralSeriesOfGroup(G);

sage:G.upper_central_series()

sage_gap:G.UpperCentralSeriesOfGroup()

oscar:upper_central_series(G)

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

oscar:character_table(G) # Output not guaranteed to exactly match the LMFDB table

Complex character table

The $835 \times 835$ character table is not available for this group.

Rational character table

		1A	2A	4A	4B	4C	167A	334A	668A	668B	668C
	Size	1	1	2	2	2	166	166	332	332	332
	2 P	1A	1A	2A	2A	2A	167A	167A	334A	334A	334A
	167 P	1A	2A	4A	4B	4C	1A	2A	4A	4B	4C
	Schur
1336.4.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1336.4.1b		$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$
1336.4.1c		$1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$1$	$- 1$
1336.4.1d		$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$
1336.4.1e		$166$	$166$	$166$	$166$	$166$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
1336.4.1f		$166$	$166$	$- 166$	$- 166$	$166$	$- 1$	$- 1$	$1$	$1$	$- 1$
1336.4.1g		$166$	$166$	$- 166$	$166$	$- 166$	$- 1$	$- 1$	$1$	$- 1$	$1$
1336.4.1h		$166$	$166$	$166$	$- 166$	$- 166$	$- 1$	$- 1$	$- 1$	$1$	$1$
1336.4.2a	2	$2$	$- 2$	$0$	$0$	$0$	$2$	$- 2$	$0$	$0$	$0$
1336.4.2b	2	$332$	$- 332$	$0$	$0$	$0$	$- 2$	$2$	$0$	$0$	$0$

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	1336.4
Order	\( 2^{3} \cdot 167 \)
Exponent	\( 2^{2} \cdot 167 \)

Nilpotent	yes
Solvable	yes
$\card{G^{\mathrm{ab}}}$	\( 2^{2} \cdot 167 \)
$\card{Z(G)}$	\( 2 \cdot 167 \)
$\card{\Aut(G)}$	\( 2^{4} \cdot 3 \cdot 83 \)
$\card{\mathrm{Out}(G)}$	\( 2^{2} \cdot 3 \cdot 83 \)
Perm deg.	$175$
Trans deg.	$1336$
Rank	$2$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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.