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Properties

Label 128.144
Order \( 2^{7} \)
Exponent \( 2^{3} \)
Nilpotent yes
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{3} \)
$\card{Z(G)}$ \( 2 \)
$\card{\Aut(G)}$ \( 2^{9} \)
$\card{\mathrm{Out}(G)}$ \( 2^{3} \)
Perm deg. $16$
Trans deg. $16$
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SmallGroup(128, 144);
 
Copy content gap:G := SmallGroup(128, 144);
 
Copy content sage_gap:G = libgap.SmallGroup(128, 144)
 
Copy content comment:Define the group as a permutation group
 
Copy content sage:G = PermutationGroup(['(1,9)(2,10)(3,12,4,11)(5,15,7,13)(6,16,8,14)', '(1,6,2,5)(3,8,4,7)(9,13)(10,14)(11,15)(12,16)', '(5,6)(7,8)(9,12,10,11)(13,16,14,15)', '(3,4)(5,7)(6,8)(11,12)(13,15)(14,16)', '(1,4,2,3)(5,7,6,8)(9,11,10,12)(13,16,14,15)', '(1,2)(3,4)(5,6)(7,8)', '(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)'])
 

Group information

Description:$C_4^2.(C_2\times C_4)$
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: \(8\)\(\medspace = 2^{3} \)
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:$(D_4\times C_2^3).D_4$, of order \(512\)\(\medspace = 2^{9} \)
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:$5$
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:$2$
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 nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.

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 8
Elements 1 19 92 16 128
Conjugacy classes   1 4 7 2 14
Divisions 1 4 5 1 11
Autjugacy classes 1 3 5 1 10

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 4 8
Irr. complex chars.   8 2 3 1 14
Irr. rational chars. 4 4 1 2 11

Minimal presentations

Permutation degree:$16$
Transitive degree:$16$
Rank: $2$
Inequivalent generating pairs: $12$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 8 8 8
Arbitrary 8 8 8

Constructions

Show commands: Gap / Magma / SageMath


Presentation: $\langle a, b, c \mid a^{4}=b^{4}=c^{8}=1, b^{a}=bc^{6}, c^{a}=bc, c^{b}=c^{5} \rangle$ 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, 14, 2102, 135, 58, 2019, 704, 2111, 718, 102, 4373, 2028, 124]); a,b,c := Explode([G.1, G.3, G.5]); AssignNames(~G, ["a", "a2", "b", "b2", "c", "c2", "c4"]);
 
Copy content gap:G := PcGroupCode(20454561330425315986176559339445,128); a := G.1; b := G.3; c := G.5;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(20454561330425315986176559339445,128)'); a = G.1; b = G.3; c = G.5;
 
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(20454561330425315986176559339445,128)'); a = G.1; b = G.3; c = G.5;
 
Permutation group:Degree $16$ $\langle(1,9)(2,10)(3,12,4,11)(5,15,7,13)(6,16,8,14), (1,6,2,5)(3,8,4,7)(9,13)(10,14) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 16 | (1,9)(2,10)(3,12,4,11)(5,15,7,13)(6,16,8,14), (1,6,2,5)(3,8,4,7)(9,13)(10,14)(11,15)(12,16), (5,6)(7,8)(9,12,10,11)(13,16,14,15), (3,4)(5,7)(6,8)(11,12)(13,15)(14,16), (1,4,2,3)(5,7,6,8)(9,11,10,12)(13,16,14,15), (1,2)(3,4)(5,6)(7,8), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16) >;
 
Copy content gap:G := Group( (1,9)(2,10)(3,12,4,11)(5,15,7,13)(6,16,8,14), (1,6,2,5)(3,8,4,7)(9,13)(10,14)(11,15)(12,16), (5,6)(7,8)(9,12,10,11)(13,16,14,15), (3,4)(5,7)(6,8)(11,12)(13,15)(14,16), (1,4,2,3)(5,7,6,8)(9,11,10,12)(13,16,14,15), (1,2)(3,4)(5,6)(7,8), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16) );
 
Copy content sage:G = PermutationGroup(['(1,9)(2,10)(3,12,4,11)(5,15,7,13)(6,16,8,14)', '(1,6,2,5)(3,8,4,7)(9,13)(10,14)(11,15)(12,16)', '(5,6)(7,8)(9,12,10,11)(13,16,14,15)', '(3,4)(5,7)(6,8)(11,12)(13,15)(14,16)', '(1,4,2,3)(5,7,6,8)(9,11,10,12)(13,16,14,15)', '(1,2)(3,4)(5,6)(7,8)', '(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)'])
 
Transitive group: 16T375 32T850 32T851 32T1824 all 5
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: $(C_4.Q_8)$ $\,\rtimes\,$ $C_4$ $(C_8:C_4)$ $\,\rtimes\,$ $C_4$ more information
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $(C_2\times D_4)$ . $D_4$ (2) $(C_4:D_4)$ . $C_2^2$ $(C_4^2:C_4)$ . $C_2$ (2) $C_4^2$ . $(C_2\times C_4)$ all 8

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

Homology

Abelianization: $C_{2} \times C_{4} $
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())
 
Schur multiplier: $C_{4}$
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()
 

There are 184 subgroups in 47 conjugacy classes, 14 normal (10 characteristic).

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

Special subgroups

Center: $Z \simeq$ $C_2$ $G/Z \simeq$ $C_4^2:C_4$
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: $G' \simeq$ $C_4^2$ $G/G' \simeq$ $C_2\times C_4$
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: $\Phi \simeq$ $C_4:D_4$ $G/\Phi \simeq$ $C_2^2$
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: $\operatorname{Fit} \simeq$ $C_4^2.(C_2\times C_4)$ $G/\operatorname{Fit} \simeq$ $C_1$
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: $R \simeq$ $C_4^2.(C_2\times C_4)$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_2$ $G/\operatorname{soc} \simeq$ $C_4^2:C_4$
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: $P_{ 2 } \simeq$ $C_4^2.(C_2\times C_4)$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $C_4^2.(C_2\times C_4)$ $\rhd$ $C_4^2$ $\rhd$ $C_1$
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 $C_4^2.(C_2\times C_4)$ $\rhd$ $C_4^2.C_2^2$ $\rhd$ $C_4:D_4$ $\rhd$ $C_4^2$ $\rhd$ $C_2\times C_4$ $\rhd$ $C_2^2$ $\rhd$ $C_2$ $\rhd$ $C_1$
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 $C_4^2.(C_2\times C_4)$ $\rhd$ $C_4^2$ $\rhd$ $C_2\times C_4$ $\rhd$ $C_2^2$ $\rhd$ $C_2$ $\rhd$ $C_1$
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 $C_1$ $\lhd$ $C_2$ $\lhd$ $C_2^2$ $\lhd$ $C_2\times C_4$ $\lhd$ $C_4:D_4$ $\lhd$ $C_4^2.(C_2\times C_4)$
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 13 larger groups in the database.

This group is a maximal quotient of 12 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

1A 2A 2B 2C 2D 4A 4B 4C 4D1 4D-1 4E1 4E-1 8A1 8A-1
Size 1 1 2 8 8 4 8 16 16 16 16 16 8 8
2 P 1A 1A 1A 1A 1A 2A 2B 2B 2C 2C 2D 2D 4A 4A
Type
128.144.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
128.144.1b R 1 1 1 1 1 1 1 −1 −1 1 −1 1 −1 −1
128.144.1c R 1 1 1 1 1 1 1 −1 1 −1 −1 −1 1 1
128.144.1d R 1 1 1 1 1 1 1 1 −1 −1 1 −1 −1 −1
128.144.1e1 C 1 1 1 −1 −1 1 1 −i −1 i i −i 1 1
128.144.1e2 C 1 1 1 −1 −1 1 1 i −1 −i −i i 1 1
128.144.1f1 C 1 1 1 −1 −1 1 1 −i 1 −i i i −1 −1
128.144.1f2 C 1 1 1 −1 −1 1 1 i 1 i −i −i −1 −1
128.144.2a R 2 2 2 −2 2 2 −2 0 0 0 0 0 0 0
128.144.2b R 2 2 2 2 −2 2 −2 0 0 0 0 0 0 0
128.144.4a R 4 4 4 0 0 −4 0 0 0 0 0 0 0 0
128.144.4b1 C 4 4 −4 0 0 0 0 0 0 0 0 0 −2i 2i
128.144.4b2 C 4 4 −4 0 0 0 0 0 0 0 0 0 2i −2i
128.144.8a R 8 −8 0 0 0 0 0 0 0 0 0 0 0 0

Rational character table

1A 2A 2B 2C 2D 4A 4B 4C 4D 4E 8A
Size 1 1 2 8 8 4 8 16 32 32 16
2 P 1A 1A 1A 1A 1A 2A 2B 2B 2C 2D 4A
128.144.1a 1 1 1 1 1 1 1 1 1 1 1
128.144.1b 1 1 1 1 1 1 1 −1 −1 −1 −1
128.144.1c 1 1 1 1 1 1 1 −1 1 −1 1
128.144.1d 1 1 1 1 1 1 1 1 −1 1 −1
128.144.1e 2 2 2 −2 −2 2 2 0 −2 0 2
128.144.1f 2 2 2 −2 −2 2 2 0 2 0 −2
128.144.2a 2 2 2 −2 2 2 −2 0 0 0 0
128.144.2b 2 2 2 2 −2 2 −2 0 0 0 0
128.144.4a 4 4 4 0 0 −4 0 0 0 0 0
128.144.4b 8 8 −8 0 0 0 0 0 0 0 0
128.144.8a 8 −8 0 0 0 0 0 0 0 0 0