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Properties

Label 112.39
Order \( 2^{4} \cdot 7 \)
Exponent \( 2^{2} \cdot 7 \)
Nilpotent yes
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{3} \cdot 7 \)
$\card{Z(G)}$ \( 2^{2} \cdot 7 \)
$\card{\Aut(G)}$ \( 2^{7} \cdot 3^{2} \)
$\card{\mathrm{Out}(G)}$ \( 2^{5} \cdot 3^{2} \)
Perm deg. $17$
Trans deg. $112$
Rank $3$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SmallGroup(112, 39);
 
Copy content gap:G := SmallGroup(112, 39);
 
Copy content sage_gap:G = libgap.SmallGroup(112, 39)
 
Copy content comment:Define the group as a permutation group
 
Copy content sage:G = PermutationGroup(['(1,2,4,6)(3,7,8,5)', '(1,3,4,8)(2,5,6,7)', '(9,10)', '(11,17,16,15,14,13,12)', '(1,4)(2,6)(3,8)(5,7)'])
 

Group information

Description:$Q_8\times C_{14}$
Order: \(112\)\(\medspace = 2^{4} \cdot 7 \)
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()
 
Exponent: \(28\)\(\medspace = 2^{2} \cdot 7 \)
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()
 
Automorphism group:$C_6\times C_2^3:S_4$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)
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 4, $C_7$
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:$2$
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, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, 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 7 14 28
Elements 1 3 12 6 18 72 112
Conjugacy classes   1 3 6 6 18 36 70
Divisions 1 3 6 1 3 6 20
Autjugacy classes 1 2 1 1 2 1 8

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 6 12
Irr. complex chars.   56 14 0 0 70
Irr. rational chars. 8 2 8 2 20

Minimal presentations

Permutation degree:$17$
Transitive degree:$112$
Rank: $3$
Inequivalent generating triples: $399$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible none none none
Arbitrary 3 5 10

Constructions

Show commands: Gap / Magma / SageMath


Presentation: $\langle a, b, c \mid a^{2}=c^{28}=[a,b]=[a,c]=1, b^{2}=c^{14}, c^{b}=c^{15} \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([5, -2, -2, -2, -2, -7, 286, 457, 42, 58]); a,b,c := Explode([G.1, G.2, G.3]); AssignNames(~G, ["a", "b", "c", "c2", "c4"]);
 
Copy content gap:G := PcGroupCode(1791605435533637,112); a := G.1; b := G.2; c := G.3;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(1791605435533637,112)'); a = G.1; b = G.2; c = G.3;
 
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(1791605435533637,112)'); a = G.1; b = G.2; c = G.3;
 
Permutation group:Degree $17$ $\langle(1,2,4,6)(3,7,8,5), (1,3,4,8)(2,5,6,7), (9,10), (11,17,16,15,14,13,12), (1,4)(2,6)(3,8)(5,7)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 17 | (1,2,4,6)(3,7,8,5), (1,3,4,8)(2,5,6,7), (9,10), (11,17,16,15,14,13,12), (1,4)(2,6)(3,8)(5,7) >;
 
Copy content gap:G := Group( (1,2,4,6)(3,7,8,5), (1,3,4,8)(2,5,6,7), (9,10), (11,17,16,15,14,13,12), (1,4)(2,6)(3,8)(5,7) );
 
Copy content sage:G = PermutationGroup(['(1,2,4,6)(3,7,8,5)', '(1,3,4,8)(2,5,6,7)', '(9,10)', '(11,17,16,15,14,13,12)', '(1,4)(2,6)(3,8)(5,7)'])
 
Matrix group:$\left\langle \left(\begin{array}{rr} 1 & 3 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 14 & 8 \end{array}\right), \left(\begin{array}{rr} 8 & 0 \\ 0 & 8 \end{array}\right), \left(\begin{array}{rr} 1 & 7 \\ 7 & 8 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/21\Z)$
Copy content comment:Define the group as a matrix group with coefficients in GLZN
 
Copy content magma:G := MatrixGroup< 2, Integers(21) | [[1, 3, 0, 1], [13, 0, 0, 13], [1, 14, 14, 8], [8, 0, 0, 8], [1, 7, 7, 8]] >;
 
Copy content gap:G := Group([[[ZmodnZObj(1,21), ZmodnZObj(3,21)], [ZmodnZObj(0,21), ZmodnZObj(1,21)]],[[ZmodnZObj(13,21), ZmodnZObj(0,21)], [ZmodnZObj(0,21), ZmodnZObj(13,21)]],[[ZmodnZObj(1,21), ZmodnZObj(14,21)], [ZmodnZObj(14,21), ZmodnZObj(8,21)]],[[ZmodnZObj(8,21), ZmodnZObj(0,21)], [ZmodnZObj(0,21), ZmodnZObj(8,21)]],[[ZmodnZObj(1,21), ZmodnZObj(7,21)], [ZmodnZObj(7,21), ZmodnZObj(8,21)]]]);
 
Copy content sage:MS = MatrixSpace(Integers(21), 2, 2) G = MatrixGroup([MS([[1, 3], [0, 1]]), MS([[13, 0], [0, 13]]), MS([[1, 14], [14, 8]]), MS([[8, 0], [0, 8]]), MS([[1, 7], [7, 8]])])
 
Direct product: $C_2$ $\, \times\, $ $C_7$ $\, \times\, $ $Q_8$
Semidirect product: not computed
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $C_{28}$ . $C_2^2$ (6) $C_{14}$ . $C_2^3$ $(C_2\times C_{28})$ . $C_2$ (3) $(C_2\times C_4)$ . $C_{14}$ (3) all 8

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

Homology

Abelianization: $C_{2}^{2} \times C_{14} \simeq C_{2}^{3} \times C_{7}$
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: $C_{2}^{2}$
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 38 subgroups, all normal (8 characteristic).

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

Special subgroups

Center: $Z \simeq$ $C_2\times C_{14}$ $G/Z \simeq$ $C_2^2$
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: $G' \simeq$ $C_2$ $G/G' \simeq$ $C_2^2\times C_{14}$
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: $\Phi \simeq$ $C_2$ $G/\Phi \simeq$ $C_2^2\times C_{14}$
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: $\operatorname{Fit} \simeq$ $Q_8\times C_{14}$ $G/\operatorname{Fit} \simeq$ $C_1$
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: $R \simeq$ $Q_8\times C_{14}$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_2\times C_{14}$ $G/\operatorname{soc} \simeq$ $C_2^2$
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: $P_{ 2 } \simeq$ $C_2\times Q_8$
7-Sylow subgroup: $P_{ 7 } \simeq$ $C_7$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $Q_8\times C_{14}$ $\rhd$ $C_2$ $\rhd$ $C_1$
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 $Q_8\times C_{14}$ $\rhd$ $C_2\times C_{28}$ $\rhd$ $C_2\times C_{14}$ $\rhd$ $C_{14}$ $\rhd$ $C_7$ $\rhd$ $C_1$
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 $Q_8\times C_{14}$ $\rhd$ $C_2$ $\rhd$ $C_1$
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 $C_1$ $\lhd$ $C_2\times C_{14}$ $\lhd$ $Q_8\times C_{14}$
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 48 larger groups in the database.

This group is a maximal quotient of 26 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 $70 \times 70$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.