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Properties

Label 31416.c
Order \( 2^{3} \cdot 3 \cdot 7 \cdot 11 \cdot 17 \)
Exponent \( 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 17 \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{2} \cdot 3 \cdot 17 \)
$\card{Z(G)}$ \( 2 \cdot 3 \cdot 17 \)
$\card{\Aut(G)}$ \( 2^{10} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
$\card{\mathrm{Out}(G)}$ \( 2^{8} \cdot 3 \cdot 5 \)
Perm deg. $46$
Trans deg. $31416$
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := DicyclicGroup(7854);
 
Copy content gap:G := DicyclicGroup(31416);
 
Copy content sage:G = DiCyclicGroup(7854)
 
Copy content comment:Define the group as a permutation group
 
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(5500981821065330660484879038941621950305714206961613144248554911,31416)'); a = G.1; b = G.2;
 

Group information

Description:$C_{3927}:Q_8$
Order: \(31416\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 11 \cdot 17 \)
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: \(15708\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 17 \)
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:$C_{154}.C_{240}.C_2^5$, of order \(1182720\)\(\medspace = 2^{10} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
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 3, $C_3$, $C_7$, $C_{11}$, $C_{17}$
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()
 
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, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.

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 3 4 6 7 11 12 14 17 21 22 28 33 34 42 44 51 66 68 77 84 102 119 132 154 187 204 231 238 308 357 374 462 476 561 714 748 924 1122 1309 1428 2244 2618 3927 5236 7854 15708
Elements 1 1 2 310 2 6 10 620 6 16 12 10 12 20 16 12 20 32 20 4960 60 24 32 96 40 60 160 9920 120 96 120 192 160 120 192 320 192 320 240 320 960 384 640 960 1920 1920 1920 3840 31416
Conjugacy classes   1 1 2 3 2 3 5 6 3 16 6 5 6 10 16 6 10 32 10 48 30 12 32 48 20 30 80 96 60 48 60 96 80 60 96 160 96 160 120 160 480 192 320 480 960 960 960 1920 8007
Divisions 1 1 1 3 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 56
Autjugacy classes 1 1 1 2 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 52

Minimal presentations

Permutation degree:$46$
Transitive degree:$31416$
Rank: $2$
Inequivalent generating pairs: $216$

Minimal degrees of linear representations for this group have not been computed

Constructions

Show commands: Gap / Magma / SageMath


Presentation: $\langle a, b \mid b^{15708}=1, a^{2}=b^{14938}, b^{a}=b^{307} \rangle$ 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, -3, -7, -11, -17, 209132, 8597, 36, 25790, 58, 68771, 108, 257884, 277, 846725, 502]); a,b := Explode([G.1, G.2]); AssignNames(~G, ["a", "b", "b2", "b4", "b12", "b84", "b924"]);
 
Copy content gap:G := PcGroupCode(5500981821065330660484879038941621950305714206961613144248554911,31416); a := G.1; b := G.2;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(5500981821065330660484879038941621950305714206961613144248554911,31416)'); a = G.1; b = G.2;
 
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(5500981821065330660484879038941621950305714206961613144248554911,31416)'); a = G.1; b = G.2;
 
Permutation group:Degree $46$ $\langle(1,2,4,7)(3,6,8,5)(9,10,11)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 46 | (1,2,4,7)(3,6,8,5)(9,10,11)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39)(41,42)(43,44)(45,46), (1,3,4,8)(2,5,7,6)(9,10,11)(12,13,15,17,19,21,22,20,18,16,14)(23,25,27,29,31,33,35,37,39,24,26,28,30,32,34,36,38)(40,41,43,45,46,44,42) >;
 
Copy content gap:G := Group( (1,2,4,7)(3,6,8,5)(9,10,11)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39)(41,42)(43,44)(45,46), (1,3,4,8)(2,5,7,6)(9,10,11)(12,13,15,17,19,21,22,20,18,16,14)(23,25,27,29,31,33,35,37,39,24,26,28,30,32,34,36,38)(40,41,43,45,46,44,42) );
 
Copy content sage:G = PermutationGroup(['(1,2,4,7)(3,6,8,5)(9,10,11)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39)(41,42)(43,44)(45,46)', '(1,3,4,8)(2,5,7,6)(9,10,11)(12,13,15,17,19,21,22,20,18,16,14)(23,25,27,29,31,33,35,37,39,24,26,28,30,32,34,36,38)(40,41,43,45,46,44,42)'])
 
Matrix group:$\left\langle \left(\begin{array}{rr} 69 & 285 \\ 57 & 69 \end{array}\right), \left(\begin{array}{rr} 162 & 181 \\ 148 & 145 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{307})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 2, GF(307) | [[69, 285, 57, 69], [162, 181, 148, 145]] >;
 
Copy content gap:G := Group([[[ Z(307)^260, Z(307)^250 ], [ Z(307)^249, Z(307)^260 ]], [[ Z(307)^111, Z(307)^218 ], [ Z(307)^64, Z(307)^264 ]]]);
 
Copy content sage:MS = MatrixSpace(GF(307), 2, 2) G = MatrixGroup([MS([[69, 285], [57, 69]]), MS([[162, 181], [148, 145]])])
 
Direct product: $C_3$ $\, \times\, $ $C_{17}$ $\, \times\, $ $(C_{77}:Q_8)$
Semidirect product: $C_{3927}$ $\,\rtimes\,$ $Q_8$ $C_{561}$ $\,\rtimes\,$ $(C_7:Q_8)$ $C_{77}$ $\,\rtimes\,$ $(Q_8\times C_{51})$ $C_7$ $\,\rtimes\,$ $(C_{561}:Q_8)$ all 12
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $C_{2244}$ . $D_7$ $C_{714}$ . $D_{22}$ $C_{204}$ . $D_{77}$ $C_{5236}$ . $C_6$ all 36

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

Homology

Abelianization: $C_{2} \times C_{102} \simeq C_{2}^{2} \times C_{3} \times C_{17}$
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: $C_1$
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: not computed
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 1200 subgroups in 96 conjugacy classes, 60 normal (52 characteristic).

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

Special subgroups

Center: $Z \simeq$ $C_{102}$ $G/Z \simeq$ $D_{154}$
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: $G' \simeq$ $C_{154}$ $G/G' \simeq$ $C_2\times C_{102}$
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: $\Phi \simeq$ $C_2$ $G/\Phi \simeq$ $C_{102}\times D_{77}$
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: $\operatorname{Fit} \simeq$ $C_{15708}$ $G/\operatorname{Fit} \simeq$ $C_2$
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: $R \simeq$ $C_{3927}:Q_8$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_{7854}$ $G/\operatorname{soc} \simeq$ $C_2^2$
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: $P_{ 2 } \simeq$ $Q_8$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup: $P_{ 7 } \simeq$ $C_7$
11-Sylow subgroup: $P_{ 11 } \simeq$ $C_{11}$
17-Sylow subgroup: $P_{ 17 } \simeq$ $C_{17}$

Subgroup diagram and profile

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Subgroup information

Click on a subgroup in the diagram to see information about it.

Series

Derived series $C_{3927}:Q_8$ $\rhd$ $C_{154}$ $\rhd$ $C_1$
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 $C_{3927}:Q_8$ $\rhd$ $C_{77}:C_{204}$ $\rhd$ $C_{7854}$ $\rhd$ $C_{3927}$ $\rhd$ $C_{1309}$ $\rhd$ $C_{187}$ $\rhd$ $C_{17}$ $\rhd$ $C_1$
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 $C_{3927}:Q_8$ $\rhd$ $C_{154}$ $\rhd$ $C_{77}$
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 $C_1$ $\lhd$ $C_{102}$ $\lhd$ $C_{204}$
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 2 larger groups in the database.

This group is a maximal quotient of 1 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

The $8007 \times 8007$ character table is not available for this group.

Rational character table

The $56 \times 56$ rational character table is not available for this group.