LMFDB - Abstract group 524160.a: $\SOMinus(4,8)$

magma:G := SOMinus(4, 8);

comment:Define the group as a permutation group

gap:G := Group( (1,3,10)(2,6,18)(4,13,12)(5,15,19)(7,21,26)(8,25,20)(9,27,43)(11,29,47)(14,31,50)(16,35,23)(22,28,46)(24,38,39)(30,48,42)(33,37,52)(34,44,53)(36,54,62)(40,41,56)(45,61,59)(49,51,57)(55,64,63)(58,65,60), (1,2,5,14,30,20,40,56,63,10,22,36,33,53,59,64,48,54,11,28,15,25,43,60,35,46,62,13,8,24,39,55,19,21,42,51,4,7,16,32,49,41,57,65,52,3,9,26,44,27,45,58,50,61,12,23,34,18,38,6,17,37,31), (1,4)(2,8)(3,12)(5,15)(6,20)(7,23)(10,13)(14,33)(16,26)(18,25)(21,35)(22,28)(24,34)(27,43)(29,47)(31,52)(37,50)(38,53)(39,44)(40,58)(41,60)(42,48)(45,59)(49,55)(51,63)(54,62)(56,65)(57,64), (1,4,10,3,11,12,13)(2,7,22,9,28,23,8)(5,16,36,26,15,34,24)(6,19,20,41,59,45,60)(14,32,33,44,25,18,39)(17,21,40,57,64,58,35)(27,43,38,55,30,49,53)(31,51,63,52,54,61,62)(37,42,56,65,48,50,46) );

sage:G = PermutationGroup(['(1,3,10)(2,6,18)(4,13,12)(5,15,19)(7,21,26)(8,25,20)(9,27,43)(11,29,47)(14,31,50)(16,35,23)(22,28,46)(24,38,39)(30,48,42)(33,37,52)(34,44,53)(36,54,62)(40,41,56)(45,61,59)(49,51,57)(55,64,63)(58,65,60)', '(1,2,5,14,30,20,40,56,63,10,22,36,33,53,59,64,48,54,11,28,15,25,43,60,35,46,62,13,8,24,39,55,19,21,42,51,4,7,16,32,49,41,57,65,52,3,9,26,44,27,45,58,50,61,12,23,34,18,38,6,17,37,31)', '(1,4)(2,8)(3,12)(5,15)(6,20)(7,23)(10,13)(14,33)(16,26)(18,25)(21,35)(22,28)(24,34)(27,43)(29,47)(31,52)(37,50)(38,53)(39,44)(40,58)(41,60)(42,48)(45,59)(49,55)(51,63)(54,62)(56,65)(57,64)', '(1,4,10,3,11,12,13)(2,7,22,9,28,23,8)(5,16,36,26,15,34,24)(6,19,20,41,59,45,60)(14,32,33,44,25,18,39)(17,21,40,57,64,58,35)(27,43,38,55,30,49,53)(31,51,63,52,54,61,62)(37,42,56,65,48,50,46)'])

Group information

Description:	$\SOMinus(4,8)$
Order:	$524160$$\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order()
Exponent:	$16380$$\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 $	comment:Exponent of the group magma:Exponent(G); gap:Exponent(G); sage:G.exponent() sage_gap:G.Exponent()
Automorphism group:	$\SL(2,64).C_6$, of order $1572480$$\medspace = 2^{7} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 13 $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage_gap:G.AutomorphismGroup()
Composition factors:	$C_2$, $\SL(2,64)$	comment:Composition factors of the group magma:CompositionFactors(G); gap:CompositionSeries(G); sage:G.composition_series() sage_gap:G.CompositionSeries()
Derived length:	$1$	comment:Derived length of the group magma:DerivedLength(G); gap:DerivedLength(G); sage_gap:G.DerivedLength()

This group is nonabelian, almost simple, and nonsolvable.

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	5	6	7	9	13	14	18	21	63	65
Elements	1	4615	4160	32760	8064	29120	12480	12480	24192	112320	87360	24960	74880	96768	524160
Conjugacy classes	1	2	1	1	1	1	3	3	3	3	3	3	9	12	46
Divisions	1	2	1	1	1	1	1	1	1	1	1	1	1	1	15
Autjugacy classes	1	2	1	1	1	1	1	1	1	1	1	1	3	4	20

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	64	65	126	130	195	378	390	1170	1512
Irr. complex chars.	2	2	14	16	12	0	0	0	0	0	46
Irr. rational chars.	2	2	2	1	0	4	1	1	1	1	15

Minimal presentations

Permutation degree:	$65$
Transitive degree:	$65$
Rank:	$2$
Inequivalent generating pairs:	$128697$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	64	64	64
Arbitrary	64	64	64

Constructions

Show commands: Gap / Magma / SageMath

Groups of Lie type:	$\SOMinus(4,8)$, $\GOMinus(4,8)$, $\PSOMinus(4,8)$, $\PGOMinus(4,8)$, $\CSOMinus(4,8)$
Permutation group:	Degree $65$ $\langle(1,3,10)(2,6,18)(4,13,12)(5,15,19)(7,21,26)(8,25,20)(9,27,43)(11,29,47) \!\cdots\! \rangle$ (1,3,10)(2,6,18)(4,13,12)(5,15,19)(7,21,26)(8,25,20)(9,27,43)(11,29,47)(14,31,50)(16,35,23)(22,28,46)(24,38,39)(30,48,42)(33,37,52)(34,44,53)(36,54,62)(40,41,56)(45,61,59)(49,51,57)(55,64,63)(58,65,60), (1,2,5,14,30,20,40,56,63,10,22,36,33,53,59,64,48,54,11,28,15,25,43,60,35,46,62,13,8,24,39,55,19,21,42,51,4,7,16,32,49,41,57,65,52,3,9,26,44,27,45,58,50,61,12,23,34,18,38,6,17,37,31), (1,4)(2,8)(3,12)(5,15)(6,20)(7,23)(10,13)(14,33)(16,26)(18,25)(21,35)(22,28)(24,34)(27,43)(29,47)(31,52)(37,50)(38,53)(39,44)(40,58)(41,60)(42,48)(45,59)(49,55)(51,63)(54,62)(56,65)(57,64), (1,4,10,3,11,12,13)(2,7,22,9,28,23,8)(5,16,36,26,15,34,24)(6,19,20,41,59,45,60)(14,32,33,44,25,18,39)(17,21,40,57,64,58,35)(27,43,38,55,30,49,53)(31,51,63,52,54,61,62)(37,42,56,65,48,50,46)
comment:Define the group as a permutation group magma:G := PermutationGroup< 65 \| (1,3,10)(2,6,18)(4,13,12)(5,15,19)(7,21,26)(8,25,20)(9,27,43)(11,29,47)(14,31,50)(16,35,23)(22,28,46)(24,38,39)(30,48,42)(33,37,52)(34,44,53)(36,54,62)(40,41,56)(45,61,59)(49,51,57)(55,64,63)(58,65,60), (1,2,5,14,30,20,40,56,63,10,22,36,33,53,59,64,48,54,11,28,15,25,43,60,35,46,62,13,8,24,39,55,19,21,42,51,4,7,16,32,49,41,57,65,52,3,9,26,44,27,45,58,50,61,12,23,34,18,38,6,17,37,31), (1,4)(2,8)(3,12)(5,15)(6,20)(7,23)(10,13)(14,33)(16,26)(18,25)(21,35)(22,28)(24,34)(27,43)(29,47)(31,52)(37,50)(38,53)(39,44)(40,58)(41,60)(42,48)(45,59)(49,55)(51,63)(54,62)(56,65)(57,64), (1,4,10,3,11,12,13)(2,7,22,9,28,23,8)(5,16,36,26,15,34,24)(6,19,20,41,59,45,60)(14,32,33,44,25,18,39)(17,21,40,57,64,58,35)(27,43,38,55,30,49,53)(31,51,63,52,54,61,62)(37,42,56,65,48,50,46) >; gap:G := Group( (1,3,10)(2,6,18)(4,13,12)(5,15,19)(7,21,26)(8,25,20)(9,27,43)(11,29,47)(14,31,50)(16,35,23)(22,28,46)(24,38,39)(30,48,42)(33,37,52)(34,44,53)(36,54,62)(40,41,56)(45,61,59)(49,51,57)(55,64,63)(58,65,60), (1,2,5,14,30,20,40,56,63,10,22,36,33,53,59,64,48,54,11,28,15,25,43,60,35,46,62,13,8,24,39,55,19,21,42,51,4,7,16,32,49,41,57,65,52,3,9,26,44,27,45,58,50,61,12,23,34,18,38,6,17,37,31), (1,4)(2,8)(3,12)(5,15)(6,20)(7,23)(10,13)(14,33)(16,26)(18,25)(21,35)(22,28)(24,34)(27,43)(29,47)(31,52)(37,50)(38,53)(39,44)(40,58)(41,60)(42,48)(45,59)(49,55)(51,63)(54,62)(56,65)(57,64), (1,4,10,3,11,12,13)(2,7,22,9,28,23,8)(5,16,36,26,15,34,24)(6,19,20,41,59,45,60)(14,32,33,44,25,18,39)(17,21,40,57,64,58,35)(27,43,38,55,30,49,53)(31,51,63,52,54,61,62)(37,42,56,65,48,50,46) ); sage:G = PermutationGroup(['(1,3,10)(2,6,18)(4,13,12)(5,15,19)(7,21,26)(8,25,20)(9,27,43)(11,29,47)(14,31,50)(16,35,23)(22,28,46)(24,38,39)(30,48,42)(33,37,52)(34,44,53)(36,54,62)(40,41,56)(45,61,59)(49,51,57)(55,64,63)(58,65,60)', '(1,2,5,14,30,20,40,56,63,10,22,36,33,53,59,64,48,54,11,28,15,25,43,60,35,46,62,13,8,24,39,55,19,21,42,51,4,7,16,32,49,41,57,65,52,3,9,26,44,27,45,58,50,61,12,23,34,18,38,6,17,37,31)', '(1,4)(2,8)(3,12)(5,15)(6,20)(7,23)(10,13)(14,33)(16,26)(18,25)(21,35)(22,28)(24,34)(27,43)(29,47)(31,52)(37,50)(38,53)(39,44)(40,58)(41,60)(42,48)(45,59)(49,55)(51,63)(54,62)(56,65)(57,64)', '(1,4,10,3,11,12,13)(2,7,22,9,28,23,8)(5,16,36,26,15,34,24)(6,19,20,41,59,45,60)(14,32,33,44,25,18,39)(17,21,40,57,64,58,35)(27,43,38,55,30,49,53)(31,51,63,52,54,61,62)(37,42,56,65,48,50,46)'])
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$\SL(2,64)$ $\,\rtimes\,$ $C_2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\SOMinus(4,8)$.

Homology

Abelianization:	$C_{2} $	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 885500 subgroups in 127 conjugacy classes, 3 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $\SOMinus(4,8)$	comment:Center of the group magma:Center(G); gap:Center(G); sage:G.center() sage_gap:G.Center()
Commutator:	$G' \simeq$ $\SL(2,64)$	$G/G' \simeq$ $C_2$	comment:Commutator subgroup of the group G magma:CommutatorSubgroup(G); gap:DerivedSubgroup(G); sage:G.commutator() sage_gap:G.DerivedSubgroup()
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $\SOMinus(4,8)$	comment:Frattini subgroup of the group G magma:FrattiniSubgroup(G); gap:FrattiniSubgroup(G); sage:G.frattini_subgroup() sage_gap:G.FrattiniSubgroup()
Fitting:	$\operatorname{Fit} \simeq$ $C_1$	$G/\operatorname{Fit} \simeq$ $\SOMinus(4,8)$	comment:Fitting subgroup of the group G magma:FittingSubgroup(G); gap:FittingSubgroup(G); sage:G.fitting_subgroup() sage_gap:G.FittingSubgroup()
Radical:	$R \simeq$ $C_1$	$G/R \simeq$ $\SOMinus(4,8)$	comment:Radical of the group G magma:Radical(G); gap:SolvableRadical(G); sage_gap:G.SolvableRadical()
Socle:	$\operatorname{soc} \simeq$ $\SL(2,64)$	$G/\operatorname{soc} \simeq$ $C_2$	comment:Socle of the group G magma:Socle(G); gap:Socle(G); sage:G.socle() sage_gap:G.Socle()
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3\wr C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$

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

$\SOMinus(4,8)$

$\rhd$

$\SL(2,64)$

comment:Derived series of the group GF

magma:DerivedSeries(G);

gap:DerivedSeriesOfGroup(G);

sage:G.derived_series()

sage_gap:G.DerivedSeriesOfGroup()

Chief series

$\SOMinus(4,8)$

$\rhd$

$\SL(2,64)$

$\rhd$

$C_1$

comment:Chief series of the group G

magma:ChiefSeries(G);

gap:ChiefSeries(G);

sage_gap:G.ChiefSeries()

Lower central series

$\SOMinus(4,8)$

$\rhd$

$\SL(2,64)$

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 $46 \times 46$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	3A	4A	5A	6A	7A	9A	13A	14A	18A	21A	63A	65A
	Size	1	520	4095	4160	32760	8064	29120	12480	12480	24192	112320	87360	24960	74880	96768
	2 P	1A	1A	1A	3A	2B	5A	3A	7A	9A	13A	7A	9A	21A	63A	65A
	3 P	1A	2A	2B	1A	4A	5A	2A	7A	3A	13A	14A	6A	7A	21A	65A
	5 P	1A	2A	2B	3A	4A	1A	6A	7A	9A	13A	14A	18A	21A	63A	13A
	7 P	1A	2A	2B	3A	4A	5A	6A	1A	9A	13A	2A	18A	3A	9A	65A
	13 P	1A	2A	2B	3A	4A	5A	6A	7A	9A	1A	14A	18A	21A	63A	5A
524160.a.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
524160.a.1b		$1$	$- 1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$
524160.a.64a		$64$	$8$	$0$	$1$	$0$	$- 1$	$- 1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$
524160.a.64b		$64$	$- 8$	$0$	$1$	$0$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$
524160.a.65a		$65$	$- 7$	$1$	$2$	$1$	$0$	$2$	$2$	$- 1$	$0$	$0$	$- 1$	$2$	$- 1$	$0$
524160.a.65b		$65$	$7$	$1$	$2$	$- 1$	$0$	$- 2$	$2$	$- 1$	$0$	$0$	$1$	$2$	$- 1$	$0$
524160.a.65c		$195$	$27$	$3$	$6$	$3$	$0$	$0$	$- 1$	$6$	$0$	$- 1$	$0$	$- 1$	$- 1$	$0$
524160.a.65d		$195$	$21$	$3$	$- 3$	$- 3$	$0$	$3$	$6$	$0$	$0$	$0$	$0$	$- 3$	$0$	$0$
524160.a.65e		$195$	$- 21$	$3$	$- 3$	$3$	$0$	$- 3$	$6$	$0$	$0$	$0$	$0$	$- 3$	$0$	$0$
524160.a.65f		$195$	$- 27$	$3$	$6$	$- 3$	$0$	$0$	$- 1$	$6$	$0$	$1$	$0$	$- 1$	$- 1$	$0$
524160.a.126a		$126$	$0$	$- 2$	$0$	$0$	$1$	$0$	$0$	$0$	$- 4$	$0$	$0$	$0$	$0$	$1$
524160.a.126b		$378$	$0$	$- 6$	$0$	$0$	$- 12$	$0$	$0$	$0$	$1$	$0$	$0$	$0$	$0$	$1$
524160.a.126c		$1512$	$0$	$- 24$	$0$	$0$	$12$	$0$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$- 1$
524160.a.130a		$390$	$0$	$6$	$12$	$0$	$0$	$0$	$- 2$	$- 6$	$0$	$0$	$0$	$- 2$	$1$	$0$
524160.a.130b		$1170$	$0$	$18$	$- 18$	$0$	$0$	$0$	$- 6$	$0$	$0$	$0$	$0$	$3$	$0$	$0$

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Classical	Maass
Hilbert	Bianchi

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Label	524160.a
Order	\( 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
Exponent	\( 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)

Nilpotent	no
Solvable	no
$\card{G^{\mathrm{ab}}}$	\( 2 \)
$\card{Z(G)}$	\( 1 \)
$\card{\Aut(G)}$	\( 2^{7} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 13 \)
$\card{\mathrm{Out}(G)}$	\( 3 \)
Perm deg.	$65$
Trans deg.	$65$
Rank	$2$

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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.