Properties

Label 504.156
Order \( 2^{3} \cdot 3^{2} \cdot 7 \)
Exponent \( 2 \cdot 3^{2} \cdot 7 \)
Simple yes
$\card{G^{\mathrm{ab}}}$ \( 1 \)
$\card{Z(G)}$ \( 1 \)
$\card{\Aut(G)}$ \( 2^{3} \cdot 3^{3} \cdot 7 \)
$\card{\mathrm{Out}(G)}$ \( 3 \)
Perm deg. $9$
Trans deg. $9$
Rank $2$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Construction of abstract group
 
Copy content magma:G := SL(2,8);
 
Copy content gap:G := SL(2,8);
 
Copy content sage:G = SL(2,8)
 
Copy content sage_gap:G = libgap.SmallGroup(504, 156)
 
Copy content oscar:G = SL(2,8)
 

Group information

Description:$\SL(2,8)$
Order: \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \)
Copy content comment:Order of the group
 
Copy content magma:Order(G);
 
Copy content gap:Order(G);
 
Copy content sage:G.order()
 
Copy content sage_gap:G.Order()
 
Copy content oscar:order(G)
 
Exponent: \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \)
Copy content comment:Exponent of the group
 
Copy content magma:Exponent(G);
 
Copy content gap:Exponent(G);
 
Copy content sage:G.exponent()
 
Copy content sage_gap:G.Exponent()
 
Copy content oscar:exponent(G)
 
Automorphism group:${}^2G(2,3)$, of order \(1512\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 7 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage:libgap(G).AutomorphismGroup()
 
Copy content sage_gap:G.AutomorphismGroup()
 
Copy content oscar:automorphism_group(G)
 
Composition factors:$\SL(2,8)$
Copy content comment:Composition factors of the group
 
Copy content magma:CompositionFactors(G);
 
Copy content gap:CompositionSeries(G);
 
Copy content sage:G.composition_series()
 
Copy content sage_gap:G.CompositionSeries()
 
Copy content oscar:composition_series(G)
 
Derived length:$0$
Copy content comment:Derived length of the group
 
Copy content magma:DerivedLength(G);
 
Copy content gap:DerivedLength(G);
 
Copy content sage:libgap(G).DerivedLength()
 
Copy content sage_gap:G.DerivedLength()
 
Copy content oscar:derived_length(G)
 

This group is nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and an A-group.

Copy content comment:Determine if the group G is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content gap:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content sage_gap:G.IsAbelian()
 
Copy content oscar:is_abelian(G)
 
Copy content comment:Determine if the group G is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content gap:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content sage_gap:G.IsCyclic()
 
Copy content oscar:is_cyclic(G)
 
Copy content comment:Determine if the group G is nilpotent
 
Copy content magma:IsNilpotent(G);
 
Copy content gap:IsNilpotentGroup(G);
 
Copy content sage:G.is_nilpotent()
 
Copy content sage_gap:G.IsNilpotentGroup()
 
Copy content oscar:is_nilpotent(G)
 
Copy content comment:Determine if the group G is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content gap:IsSolvableGroup(G);
 
Copy content sage:G.is_solvable()
 
Copy content sage_gap:G.IsSolvableGroup()
 
Copy content oscar:is_solvable(G)
 
Copy content comment:Determine if the group G is supersolvable
 
Copy content gap:IsSupersolvableGroup(G);
 
Copy content sage:G.is_supersolvable()
 
Copy content sage_gap:G.IsSupersolvableGroup()
 
Copy content oscar:is_supersolvable(G)
 
Copy content comment:Determine if the group G is simple
 
Copy content magma:IsSimple(G);
 
Copy content gap:IsSimpleGroup(G);
 
Copy content sage:G.is_simple()
 
Copy content sage_gap:G.IsSimpleGroup()
 
Copy content oscar:is_simple(G)
 

Group statistics

Copy content comment:Compute statistics for the group G
 
Copy content 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;
 
Copy content 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");
 
Copy content 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))
 
Copy content 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))
 
Copy content 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 3 7 9
Elements 1 63 56 216 168 504
Conjugacy classes   1 1 1 3 3 9
Divisions 1 1 1 1 1 5
Autjugacy classes 1 1 1 1 1 5

Copy content comment:Compute statistics about the characters of G
 
Copy content 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);
 
Copy content 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);
 
Copy content 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)]
 
Copy content 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()
 
Copy content oscar:# Outputs an MSet containing the absolutely irreducible degrees of G and their multiplicities. character_degrees(G)
 

Dimension 1 7 8 9 21 27
Irr. complex chars.   1 4 1 3 0 0 9
Irr. rational chars. 1 1 1 0 1 1 5

Minimal presentations

Permutation degree:$9$
Transitive degree:$9$
Rank: $2$
Inequivalent generating pairs: $142$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 7 7 7
Arbitrary 7 7 7

Constructions

Show commands: Gap / Magma / Oscar / SageMath


Groups of Lie type:$\SL(2,8)$, $\PSL(2,8)$, $\PGL(2,8)$, $\SO(3,8)$, $\SU(2,8)$, $\PSO(3,8)$, $\PSU(2,8)$, $\Orth(3,8)$, $\Omega(3,8)$, $\PO(3,8)$, $\PU(2,8)$, $\POmega(3,8)$, $\GSO(3,8)$, $\GSU(2,8)$
Copy content magma:G := SL(2,8);
 
Copy content gap:G := SL(2,8);
 
Copy content sage:G = SL(2,8)
 
Copy content oscar:G = SL(2,8)
 
Copy content magma:G := PSL(2,8);
 
Copy content gap:G := PSL(2,8);
 
Copy content sage:G = PSL(2,8)
 
Copy content magma:G := PGL(2,8);
 
Copy content gap:G := PGL(2,8);
 
Copy content sage:G = PGL(2,8)
 
Copy content magma:G := SO(3,8);
 
Copy content gap:G := SO(3,8);
 
Copy content sage:G = SO(3,8)
 
Copy content oscar:G = SO(3,8)
 
Copy content magma:G := SU(2,8);
 
Copy content gap:G := SU(2,8);
 
Copy content sage:G = SU(2,8)
 
Copy content oscar:G = SU(2,8)
 
Copy content magma:G := PSO(3,8);
 
Copy content gap:G := PSO(3,8);
 
Copy content magma:G := PSU(2,8);
 
Copy content gap:G := PSU(2,8);
 
Copy content sage:G = PSU(2,8)
 
Copy content magma:G := GO(3,8);
 
Copy content gap:G := GO(3,8);
 
Copy content sage:G = GO(3,8)
 
Copy content oscar:G = GO(3,8)
 
Copy content magma:G := Omega(3,8);
 
Copy content gap:G := Omega(3,8);
 
Copy content magma:G := PGO(3,8);
 
Copy content gap:G := PGO(3,8);
 
Copy content magma:G := PGU(2,8);
 
Copy content gap:G := PGU(2,8);
 
Copy content sage:G = PGU(2,8)
 
Copy content magma:G := POmega(3,8);
 
Copy content gap:G := POmega(3,8);
 
Copy content magma:G := CSO(3,8);
 
Copy content magma:G := CSU(2,8);
 
Permutation group: $\langle(1,2,8,9,7,5,3), (1,9,8,6,5,4,2)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 9 | (1,2,8,9,7,5,3), (1,9,8,6,5,4,2) >;
 
Copy content gap:G := Group( (1,2,8,9,7,5,3), (1,9,8,6,5,4,2) );
 
Copy content sage:G = PermutationGroup(['(1,2,8,9,7,5,3)', '(1,9,8,6,5,4,2)'])
 
Copy content sage_gap:G = gap.new('Group( (1,2,8,9,7,5,3), (1,9,8,6,5,4,2) )')
 
Copy content oscar:G = @permutation_group(9, (1,2,8,9,7,5,3), (1,9,8,6,5,4,2))
 
Matrix group:$\left\langle \left(\begin{array}{ll}1 & \alpha \\ \alpha^{6} & 0 \\ \end{array}\right), \left(\begin{array}{ll}1 & 1 \\ 1 & 0 \\ \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{8}) = \GL_{2}(\F_{2}[\alpha]/(\alpha^{3} + \alpha + 1))$
Copy content comment:Define the group as a matrix group with coefficients in GLFq
 
Copy content magma:F:=GF(8); al:=F.1; G := MatrixGroup< 2, F | [[1, al^1], [al^6, 0]],[[1, 1], [1, 0]] >;
 
Copy content gap:G := Group([[[Z(8)^0, Z(8)^1], [Z(8)^6, 0*Z(8)]],[[Z(8)^0, Z(8)^0], [Z(8)^0, 0*Z(8)]]]);
 
Copy content sage:F = GF(8); al = F.0; MS = MatrixSpace(F, 2, 2) G = MatrixGroup([MS([[1, al^1], [al^6, 0]]), MS([[1, 1], [1, 0]])])
 
Copy content sage_gap:G = gap.new('Group([[[Z(8)^0, Z(8)^1], [Z(8)^6, 0*Z(8)]],[[Z(8)^0, Z(8)^0], [Z(8)^0, 0*Z(8)]]])')
 
Copy content oscar:F = GF(8); al = gen(F); G = matrix_group([matrix(GF(8), [[1, al^1], [al^6, 0]]), matrix(GF(8), [[1, 1], [1, 0]])])
 
Transitive group: 9T27 28T70 36T712 more information
Copy content magma:G := TransitiveGroup(9, 27);
 
Copy content gap:G := TransitiveGroup(9, 27);
 
Copy content sage:G = TransitiveGroup(9, 27)
 
Copy content sage_gap:G = libgap.TransitiveGroup(9, 27)
 
Copy content oscar:G = transitive_group(9, 27)
 
Copy content magma:G := TransitiveGroup(28, 70);
 
Copy content gap:G := TransitiveGroup(28, 70);
 
Copy content sage:G = TransitiveGroup(28, 70)
 
Copy content sage_gap:G = libgap.TransitiveGroup(28, 70)
 
Copy content oscar:G = transitive_group(28, 70)
 
Copy content magma:G := TransitiveGroup(36, 712);
 
Copy content gap:G := TransitiveGroup(36, 712);
 
Copy content sage:G = TransitiveGroup(36, 712)
 
Copy content sage_gap:G = libgap.TransitiveGroup(36, 712)
 
Copy content oscar:G = transitive_group(36, 712)
 
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: not isomorphic to a non-trivial semidirect product
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 9.

Homology

Abelianization: $C_1 $
Copy content comment:The abelianization of the group
 
Copy content magma:quo< G | CommutatorSubgroup(G) >;
 
Copy content gap:FactorGroup(G, DerivedSubgroup(G));
 
Copy content sage:G.quotient(G.commutator())
 
Copy content sage_gap:G.FactorGroup(G.DerivedSubgroup())
 
Copy content oscar:quo(G, derived_subgroup(G)[1])
 
Schur multiplier: $C_1$
Copy content comment:The Schur multiplier of the group
 
Copy content gap:AbelianInvariantsMultiplier(G);
 
Copy content sage:G.homology(2)
 
Copy content sage_gap:G.AbelianInvariantsMultiplier()
 
Commutator length: $1$
Copy content comment:The commutator length of the group
 
Copy content gap:CommutatorLength(G);
 
Copy content sage_gap:G.CommutatorLength()
 

Subgroups

Copy content comment:List of subgroups of the group
 
Copy content magma:Subgroups(G);
 
Copy content gap:AllSubgroups(G);
 
Copy content sage:G.subgroups()
 
Copy content sage_gap:G.AllSubgroups()
 
Copy content oscar:subgroups(G)
 

There are 386 subgroups in 12 conjugacy classes, 2 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

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

Subgroup diagram and profile

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
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Subgroup information

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Series

Derived series $\SL(2,8)$
Copy content comment:Derived series of the group G
 
Copy content magma:DerivedSeries(G);
 
Copy content gap:DerivedSeriesOfGroup(G);
 
Copy content sage:G.derived_series()
 
Copy content sage_gap:G.DerivedSeriesOfGroup()
 
Copy content oscar:derived_series(G)
 
Chief series $\SL(2,8)$ $\rhd$ $C_1$
Copy content comment:Chief series of the group G
 
Copy content magma:ChiefSeries(G);
 
Copy content gap:ChiefSeries(G);
 
Copy content sage:libgap(G).ChiefSeries()
 
Copy content sage_gap:G.ChiefSeries()
 
Copy content oscar:chief_series(G)
 
Lower central series $\SL(2,8)$
Copy content comment:The lower central series of the group G
 
Copy content magma:LowerCentralSeries(G);
 
Copy content gap:LowerCentralSeriesOfGroup(G);
 
Copy content sage:G.lower_central_series()
 
Copy content sage_gap:G.LowerCentralSeriesOfGroup()
 
Copy content oscar:lower_central_series(G)
 
Upper central series $C_1$
Copy content comment:The upper central series of the group G
 
Copy content magma:UpperCentralSeries(G);
 
Copy content gap:UpperCentralSeriesOfGroup(G);
 
Copy content sage:G.upper_central_series()
 
Copy content sage_gap:G.UpperCentralSeriesOfGroup()
 
Copy content oscar:upper_central_series(G)
 

Supergroups

This group is a maximal subgroup of 10 larger groups in the database.

This group is a maximal quotient of 11 larger groups in the database.

Character theory

Copy content comment:Character table
 
Copy content magma:CharacterTable(G); // Output not guaranteed to exactly match the LMFDB table
 
Copy content gap:CharacterTable(G); # Output not guaranteed to exactly match the LMFDB table
 
Copy content sage:G.character_table() # Output not guaranteed to exactly match the LMFDB table
 
Copy content sage_gap:G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table
 
Copy content oscar:character_table(G) # Output not guaranteed to exactly match the LMFDB table
 

Complex character table

1A 2A 3A 7A1 7A2 7A3 9A1 9A2 9A4
Size 1 63 56 72 72 72 56 56 56
2 P 1A 1A 3A 7A2 7A3 7A1 9A2 9A4 9A1
3 P 1A 2A 1A 7A3 7A1 7A2 3A 3A 3A
7 P 1A 2A 3A 1A 1A 1A 9A2 9A4 9A1
Type
504.156.1a R 1 1 1 1 1 1 1 1 1
504.156.7a R 7 −1 −2 0 0 0 1 1 1
504.156.7b1 R 7 −1 1 0 0 0 −ζ9−1−ζ9 −ζ9−2−ζ92 −ζ9−4−ζ94
504.156.7b2 R 7 −1 1 0 0 0 −ζ9−2−ζ92 −ζ9−4−ζ94 −ζ9−1−ζ9
504.156.7b3 R 7 −1 1 0 0 0 −ζ9−4−ζ94 −ζ9−1−ζ9 −ζ9−2−ζ92
504.156.8a R 8 0 −1 1 1 1 −1 −1 −1
504.156.9a1 R 9 1 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 0 0 0
504.156.9a2 R 9 1 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 0 0 0
504.156.9a3 R 9 1 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 0 0 0

Rational character table

1A 2A 3A 7A 9A
Size 1 63 56 216 168
2 P 1A 1A 3A 7A 9A
3 P 1A 2A 1A 7A 3A
7 P 1A 2A 3A 1A 9A
504.156.1a 1 1 1 1 1
504.156.7a 7 −1 −2 0 1
504.156.7b 21 −3 3 0 0
504.156.8a 8 0 −1 1 −1
504.156.9a 27 3 0 −1 0