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Properties

Label 128.68
Order \( 2^{7} \)
Exponent \( 2^{4} \)
Nilpotent yes
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{4} \)
$\card{Z(G)}$ \( 2^{2} \)
$\card{\Aut(G)}$ \( 2^{8} \)
$\card{\mathrm{Out}(G)}$ \( 2^{3} \)
Perm deg. $16$
Trans deg. $16$
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SmallGroup(128, 68);
 
Copy content gap:G := SmallGroup(128, 68);
 
Copy content sage_gap:G = libgap.SmallGroup(128, 68)
 
Copy content comment:Define the group as a permutation group
 
Copy content sage:G = PermutationGroup(['(1,11,8,14,4,9,6,15,2,12,7,13,3,10,5,16)', '(1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)', '(1,7,4,5,2,8,3,6)(9,14,11,16,10,13,12,15)', '(1,8,4,6,2,7,3,5)(9,15,12,13,10,16,11,14)', '(1,4,2,3)(5,8,6,7)(9,11,10,12)(13,15,14,16)', '(1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15)', '(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)'])
 

Group information

Description:$D_8:C_8$
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: \(16\)\(\medspace = 2^{4} \)
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:$C_4^2.C_2^4$, of order \(256\)\(\medspace = 2^{8} \)
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:$4$
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 16
Elements 1 11 20 64 32 128
Conjugacy classes   1 3 8 16 4 32
Divisions 1 3 5 7 1 17
Autjugacy classes 1 3 4 6 1 15

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.   16 12 4 0 32
Irr. rational chars. 4 4 7 2 17

Minimal presentations

Permutation degree:$16$
Transitive degree:$16$
Rank: $2$
Inequivalent generating pairs: $24$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 4 8 8
Arbitrary 4 8 8

Constructions

Show commands: Gap / Magma / SageMath


Presentation: $\langle a, b, c \mid a^{2}=b^{8}=c^{8}=[a,c]=1, b^{a}=b^{3}c, c^{b}=c^{5} \rangle$ 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, 309, 36, 2270, 58, 2019, 1411, 102, 124]); a,b,c := Explode([G.1, G.2, G.5]); AssignNames(~G, ["a", "b", "b2", "b4", "c", "c2", "c4"]);
 
Copy content gap:G := PcGroupCode(76199608944428719292918,128); a := G.1; b := G.2; 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(76199608944428719292918,128)'); a = G.1; b = G.2; 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(76199608944428719292918,128)'); a = G.1; b = G.2; c = G.5;
 
Permutation group:Degree $16$ $\langle(1,11,8,14,4,9,6,15,2,12,7,13,3,10,5,16), (1,9)(2,10)(3,11)(4,12)(5,13) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 16 | (1,11,8,14,4,9,6,15,2,12,7,13,3,10,5,16), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,7,4,5,2,8,3,6)(9,14,11,16,10,13,12,15), (1,8,4,6,2,7,3,5)(9,15,12,13,10,16,11,14), (1,4,2,3)(5,8,6,7)(9,11,10,12)(13,15,14,16), (1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16) >;
 
Copy content gap:G := Group( (1,11,8,14,4,9,6,15,2,12,7,13,3,10,5,16), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,7,4,5,2,8,3,6)(9,14,11,16,10,13,12,15), (1,8,4,6,2,7,3,5)(9,15,12,13,10,16,11,14), (1,4,2,3)(5,8,6,7)(9,11,10,12)(13,15,14,16), (1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16) );
 
Copy content sage:G = PermutationGroup(['(1,11,8,14,4,9,6,15,2,12,7,13,3,10,5,16)', '(1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)', '(1,7,4,5,2,8,3,6)(9,14,11,16,10,13,12,15)', '(1,8,4,6,2,7,3,5)(9,15,12,13,10,16,11,14)', '(1,4,2,3)(5,8,6,7)(9,11,10,12)(13,15,14,16)', '(1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15)', '(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)'])
 
Matrix group:$\left\langle \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 3 & 4 & 4 & 4 \\ 3 & 2 & 1 & 3 \\ 0 & 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 4 & 1 & 2 & 3 \\ 1 & 0 & 2 & 4 \\ 0 & 0 & 3 & 0 \\ 4 & 2 & 3 & 3 \end{array}\right), \left(\begin{array}{rrrr} 2 & 2 & 3 & 3 \\ 4 & 2 & 1 & 4 \\ 1 & 3 & 2 & 3 \\ 4 & 1 & 2 & 4 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 3 & 3 & 4 & 4 \\ 3 & 2 & 0 & 3 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 4 & 1 & 0 & 4 \\ 0 & 2 & 0 & 0 \\ 2 & 3 & 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{5})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 4, GF(5) | [[4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4], [2, 0, 0, 0, 3, 4, 4, 4, 3, 2, 1, 3, 0, 0, 0, 3], [4, 1, 2, 3, 1, 0, 2, 4, 0, 0, 3, 0, 4, 2, 3, 3], [2, 2, 3, 3, 4, 2, 1, 4, 1, 3, 2, 3, 4, 1, 2, 4], [1, 0, 0, 0, 3, 3, 4, 4, 3, 2, 0, 3, 0, 0, 0, 2], [2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2], [4, 0, 0, 0, 4, 1, 0, 4, 0, 2, 0, 0, 2, 3, 1, 0]] >;
 
Copy content gap:G := Group([[[ Z(5)^2, 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^2, 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5)^2, 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^2 ]], [[ Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ], [ Z(5)^3, Z(5)^2, Z(5)^2, Z(5)^2 ], [ Z(5)^3, Z(5), Z(5)^0, Z(5)^3 ], [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^3 ]], [[ Z(5)^2, Z(5)^0, Z(5), Z(5)^3 ], [ Z(5)^0, 0*Z(5), Z(5), Z(5)^2 ], [ 0*Z(5), 0*Z(5), Z(5)^3, 0*Z(5) ], [ Z(5)^2, Z(5), Z(5)^3, Z(5)^3 ]], [[ Z(5), Z(5), Z(5)^3, Z(5)^3 ], [ Z(5)^2, Z(5), Z(5)^0, Z(5)^2 ], [ Z(5)^0, Z(5)^3, Z(5), Z(5)^3 ], [ Z(5)^2, Z(5)^0, Z(5), Z(5)^2 ]], [[ Z(5)^0, 0*Z(5), 0*Z(5), 0*Z(5) ], [ Z(5)^3, Z(5)^3, Z(5)^2, Z(5)^2 ], [ Z(5)^3, Z(5), 0*Z(5), Z(5)^3 ], [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5) ]], [[ Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5) ]], [[ Z(5)^2, 0*Z(5), 0*Z(5), 0*Z(5) ], [ Z(5)^2, Z(5)^0, 0*Z(5), Z(5)^2 ], [ 0*Z(5), Z(5), 0*Z(5), 0*Z(5) ], [ Z(5), Z(5)^3, Z(5)^0, 0*Z(5) ]]]);
 
Copy content sage:MS = MatrixSpace(GF(5), 4, 4) G = MatrixGroup([MS([[4, 0, 0, 0], [0, 4, 0, 0], [0, 0, 4, 0], [0, 0, 0, 4]]), MS([[2, 0, 0, 0], [3, 4, 4, 4], [3, 2, 1, 3], [0, 0, 0, 3]]), MS([[4, 1, 2, 3], [1, 0, 2, 4], [0, 0, 3, 0], [4, 2, 3, 3]]), MS([[2, 2, 3, 3], [4, 2, 1, 4], [1, 3, 2, 3], [4, 1, 2, 4]]), MS([[1, 0, 0, 0], [3, 3, 4, 4], [3, 2, 0, 3], [0, 0, 0, 2]]), MS([[2, 0, 0, 0], [0, 2, 0, 0], [0, 0, 2, 0], [0, 0, 0, 2]]), MS([[4, 0, 0, 0], [4, 1, 0, 4], [0, 2, 0, 0], [2, 3, 1, 0]])])
 
Transitive group: 16T260 32T600 more information
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: $D_8$ $\,\rtimes\,$ $C_8$ $Q_{16}$ $\,\rtimes\,$ $C_8$ $(C_8:C_8)$ $\,\rtimes\,$ $C_2$ $(C_8.C_8)$ $\,\rtimes\,$ $C_2$ more information
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $C_8$ . $D_8$ $C_4^2$ . $D_4$ $C_8$ . $\SD_{16}$ $C_8$ . $\OD_{16}$ all 16

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

Homology

Abelianization: $C_{2} \times C_{8} $
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: $C_{2}$
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 100 subgroups in 48 conjugacy classes, 22 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_4$ $G/Z \simeq$ $D_4:C_4$
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: $G' \simeq$ $C_8$ $G/G' \simeq$ $C_2\times C_8$
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: $\Phi \simeq$ $C_4\times C_8$ $G/\Phi \simeq$ $C_2^2$
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: $\operatorname{Fit} \simeq$ $D_8:C_8$ $G/\operatorname{Fit} \simeq$ $C_1$
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: $R \simeq$ $D_8:C_8$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_2$ $G/\operatorname{soc} \simeq$ $D_4:C_8$
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: $P_{ 2 } \simeq$ $D_8:C_8$

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 $D_8:C_8$ $\rhd$ $C_8$ $\rhd$ $C_1$
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 $D_8:C_8$ $\rhd$ $C_8:C_8$ $\rhd$ $C_4\times C_8$ $\rhd$ $C_2\times C_8$ $\rhd$ $C_8$ $\rhd$ $C_4$ $\rhd$ $C_2$ $\rhd$ $C_1$
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 $D_8:C_8$ $\rhd$ $C_8$ $\rhd$ $C_4$ $\rhd$ $C_2$ $\rhd$ $C_1$
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 $C_1$ $\lhd$ $C_4$ $\lhd$ $C_2\times C_8$ $\lhd$ $C_4\times C_8$ $\lhd$ $D_8:C_8$
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 16 larger groups in the database.

This group is a maximal quotient of 16 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 $32 \times 32$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $17 \times 17$ rational character table.