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Properties

Label 128.2319
Order \( 2^{7} \)
Exponent \( 2^{2} \)
Abelian yes
$\card{\Aut(G)}$ \( 2^{21} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \)
Perm deg. $14$
Trans deg. $128$
Rank $6$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SmallGroup(128, 2319);
 
Copy content gap:G := SmallGroup(128, 2319);
 
Copy content sage_gap:G = libgap.SmallGroup(128, 2319)
 
Copy content comment:Define the group as a permutation group
 
Copy content sage:G = PermutationGroup(['(11,14,12,13)', '(1,2)', '(3,4)', '(5,6)', '(7,8)', '(9,10)', '(11,12)(13,14)'])
 

Group information

Description:$C_2^5\times C_4$
Order: \(128\)\(\medspace = 2^{7} \)
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()
 
Exponent: \(4\)\(\medspace = 2^{2} \)
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()
 
Automorphism group:$C_2^6.C_2^5.\GL(5,2)$, of order \(20478689280\)\(\medspace = 2^{21} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$ x 7
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()
 
Nilpotency class:$1$
Copy content comment:Nilpotency class of the group
 
Copy content magma:NilpotencyClass(G);
 
Copy content gap:NilpotencyClassOfGroup(G);
 
Copy content sage_gap:G.NilpotencyClassOfGroup()
 
Derived length:$1$
Copy content comment:Derived length of the group
 
Copy content magma:DerivedLength(G);
 
Copy content gap:DerivedLength(G);
 
Copy content sage_gap:G.DerivedLength()
 

This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).

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 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 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 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 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 comment:Determine if the group G is simple
 
Copy content magma:IsSimple(G);
 
Copy content gap:IsSimpleGroup(G);
 
Copy content sage_gap:G.IsSimpleGroup()
 

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))
 

Order 1 2 4
Elements 1 63 64 128
Conjugacy classes   1 63 64 128
Divisions 1 63 32 96
Autjugacy classes 1 2 1 4

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:G.CharacterDegrees()
 

Dimension 1 2
Irr. complex chars.   128 0 128
Irr. rational chars. 64 32 96

Minimal presentations

Permutation degree:$14$
Transitive degree:$128$
Rank: $6$
Inequivalent generating 6-tuples: $63$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible none none none
Arbitrary 6 not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Presentation:Abelian group $\langle a, b, c, d, e, f \mid a^{2}=b^{2}=c^{2}=d^{2}=e^{2}=f^{4}=1 \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([7, -2, 2, 2, 2, 2, 2, -2, 124]); a,b,c,d,e,f := Explode([G.1, G.2, G.3, G.4, G.5, G.6]); AssignNames(~G, ["a", "b", "c", "d", "e", "f", "f2"]);
 
Copy content gap:G := PcGroupCode(134217760,128); a := G.1; b := G.2; c := G.3; d := G.4; e := G.5; f := G.6;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(134217760,128)'); a = G.1; b = G.2; c = G.3; d = G.4; e = G.5; f = G.6;
 
Copy content sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(134217760,128)'); a = G.1; b = G.2; c = G.3; d = G.4; e = G.5; f = G.6;
 
Permutation group:Degree $14$ $\langle(11,14,12,13), (1,2), (3,4), (5,6), (7,8), (9,10), (11,12)(13,14)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 14 | (11,14,12,13), (1,2), (3,4), (5,6), (7,8), (9,10), (11,12)(13,14) >;
 
Copy content gap:G := Group( (11,14,12,13), (1,2), (3,4), (5,6), (7,8), (9,10), (11,12)(13,14) );
 
Copy content sage:G = PermutationGroup(['(11,14,12,13)', '(1,2)', '(3,4)', '(5,6)', '(7,8)', '(9,10)', '(11,12)(13,14)'])
 
Matrix group:$\left\langle \left(\begin{array}{rr} 7 & 8 \\ 8 & 9 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 8 & 9 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 15 & 0 \\ 8 & 15 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/16\Z)$
Copy content comment:Define the group as a matrix group with coefficients in GLZq
 
Copy content magma:G := MatrixGroup< 2, Integers(16) | [[7, 8, 8, 9], [9, 0, 0, 1], [1, 8, 8, 9], [9, 0, 0, 9], [15, 0, 8, 15], [3, 0, 0, 1], [1, 8, 0, 1]] >;
 
Copy content gap:G := Group([[[ZmodnZObj(7,16), ZmodnZObj(8,16)], [ZmodnZObj(8,16), ZmodnZObj(9,16)]],[[ZmodnZObj(9,16), ZmodnZObj(0,16)], [ZmodnZObj(0,16), ZmodnZObj(1,16)]],[[ZmodnZObj(1,16), ZmodnZObj(8,16)], [ZmodnZObj(8,16), ZmodnZObj(9,16)]],[[ZmodnZObj(9,16), ZmodnZObj(0,16)], [ZmodnZObj(0,16), ZmodnZObj(9,16)]],[[ZmodnZObj(15,16), ZmodnZObj(0,16)], [ZmodnZObj(8,16), ZmodnZObj(15,16)]],[[ZmodnZObj(3,16), ZmodnZObj(0,16)], [ZmodnZObj(0,16), ZmodnZObj(1,16)]],[[ZmodnZObj(1,16), ZmodnZObj(8,16)], [ZmodnZObj(0,16), ZmodnZObj(1,16)]]]);
 
Copy content sage:MS = MatrixSpace(Integers(16), 2, 2) G = MatrixGroup([MS([[7, 8], [8, 9]]), MS([[9, 0], [0, 1]]), MS([[1, 8], [8, 9]]), MS([[9, 0], [0, 9]]), MS([[15, 0], [8, 15]]), MS([[3, 0], [0, 1]]), MS([[1, 8], [0, 1]])])
 
$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 19 & 10 \\ 10 & 9 \end{array}\right), \left(\begin{array}{rr} 7 & 10 \\ 10 & 7 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/20\Z)$
Copy content comment:Define the group as a matrix group with coefficients in GLZN
 
Copy content magma:G := MatrixGroup< 2, Integers(20) | [[9, 0, 0, 1], [17, 0, 0, 17], [1, 10, 0, 1], [9, 0, 0, 9], [19, 10, 10, 9], [7, 10, 10, 7], [3, 0, 0, 3]] >;
 
Copy content gap:G := Group([[[ZmodnZObj(9,20), ZmodnZObj(0,20)], [ZmodnZObj(0,20), ZmodnZObj(1,20)]],[[ZmodnZObj(17,20), ZmodnZObj(0,20)], [ZmodnZObj(0,20), ZmodnZObj(17,20)]],[[ZmodnZObj(1,20), ZmodnZObj(10,20)], [ZmodnZObj(0,20), ZmodnZObj(1,20)]],[[ZmodnZObj(9,20), ZmodnZObj(0,20)], [ZmodnZObj(0,20), ZmodnZObj(9,20)]],[[ZmodnZObj(19,20), ZmodnZObj(10,20)], [ZmodnZObj(10,20), ZmodnZObj(9,20)]],[[ZmodnZObj(7,20), ZmodnZObj(10,20)], [ZmodnZObj(10,20), ZmodnZObj(7,20)]],[[ZmodnZObj(3,20), ZmodnZObj(0,20)], [ZmodnZObj(0,20), ZmodnZObj(3,20)]]]);
 
Copy content sage:MS = MatrixSpace(Integers(20), 2, 2) G = MatrixGroup([MS([[9, 0], [0, 1]]), MS([[17, 0], [0, 17]]), MS([[1, 10], [0, 1]]), MS([[9, 0], [0, 9]]), MS([[19, 10], [10, 9]]), MS([[7, 10], [10, 7]]), MS([[3, 0], [0, 3]])])
 
Direct product: $C_2$ ${}^5$ $\, \times\, $ $C_4$
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: $C_2^6$ . $C_2$ $C_2$ . $C_2^6$ $C_2^5$ . $C_2^2$ $C_2^4$ . $C_2^3$ all 6

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

Homology

Primary decomposition: $C_{2}^{5} \times C_{4}$
Copy content comment:The primary decomposition of the group
 
Copy content magma:PrimaryInvariants(G);
 
Copy content gap:AbelianInvariants(G);
 
Copy content sage_gap:G.AbelianInvariants()
 
Schur multiplier: $C_{2}^{15}$
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: $0$
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()
 

There are 5276 subgroups, all normal (4 characteristic).

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

Special subgroups

Center: $Z \simeq$ $C_2^5\times C_4$ $G/Z \simeq$ $C_1$
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()
 
Commutator: $G' \simeq$ $C_1$ $G/G' \simeq$ $C_2^5\times C_4$
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()
 
Frattini: $\Phi \simeq$ $C_2$ $G/\Phi \simeq$ $C_2^6$
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()
 
Fitting: $\operatorname{Fit} \simeq$ $C_2^5\times C_4$ $G/\operatorname{Fit} \simeq$ $C_1$
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()
 
Radical: $R \simeq$ $C_2^5\times C_4$ $G/R \simeq$ $C_1$
Copy content comment:Radical of the group G
 
Copy content magma:Radical(G);
 
Copy content gap:SolvableRadical(G);
 
Copy content sage_gap:G.SolvableRadical()
 
Socle: $\operatorname{soc} \simeq$ $C_2^6$ $G/\operatorname{soc} \simeq$ $C_2$
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()
 
2-Sylow subgroup: $P_{ 2 } \simeq$ $C_2^5\times C_4$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $C_2^5\times C_4$ $\rhd$ $C_1$
Copy content comment:Derived series of the group GF
 
Copy content magma:DerivedSeries(G);
 
Copy content gap:DerivedSeriesOfGroup(G);
 
Copy content sage:G.derived_series()
 
Copy content sage_gap:G.DerivedSeriesOfGroup()
 
Chief series $C_2^5\times C_4$ $\rhd$ $C_2^4\times C_4$ $\rhd$ $C_2^3\times C_4$ $\rhd$ $C_2^2\times C_4$ $\rhd$ $C_2\times C_4$ $\rhd$ $C_4$ $\rhd$ $C_2$ $\rhd$ $C_1$
Copy content comment:Chief series of the group G
 
Copy content magma:ChiefSeries(G);
 
Copy content gap:ChiefSeries(G);
 
Copy content sage_gap:G.ChiefSeries()
 
Lower central series $C_2^5\times C_4$ $\rhd$ $C_1$
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()
 
Upper central series $C_1$ $\lhd$ $C_2^5\times C_4$
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()
 

Supergroups

This group is a maximal subgroup of 45 larger groups in the database.

This group is a maximal quotient of 29 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
 

Complex character table

See the $128 \times 128$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $96 \times 96$ rational character table.