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Properties

Label 960.3829
Order \( 2^{6} \cdot 3 \cdot 5 \)
Exponent \( 2^{3} \cdot 3 \cdot 5 \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{3} \cdot 3 \)
$\card{Z(G)}$ \( 2^{2} \cdot 3 \)
$\card{\Aut(G)}$ \( 2^{11} \cdot 5 \)
$\card{\mathrm{Out}(G)}$ \( 2^{7} \)
Perm deg. $24$
Trans deg. $480$
Rank $3$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SmallGroup(960, 3829);
 
Copy content gap:G := SmallGroup(960, 3829);
 
Copy content sage_gap:G = libgap.SmallGroup(960, 3829)
 
Copy content comment:Define the group as a permutation group
 
Copy content sage:G = PermutationGroup(['(2,5)(3,7)(6,8)(9,10)(11,15)(12,13)(14,16)(18,19)(20,21)', '(2,6)(5,8)(10,13)(11,16)', '(1,2,3,5,4,6,7,8)(9,11)(10,14)(12,16)(13,15)', '(22,23,24)', '(1,3,4,7)(2,5,6,8)(9,12)(10,13)(11,16)(14,15)', '(9,12)(10,13)(11,16)(14,15)', '(1,4)(2,6)(3,7)(5,8)', '(17,18,20,21,19)'])
 

Group information

Description:$C_3\times C_2^3.D_{20}$
Order: \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \)
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: \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
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:$C_5:(C_2^5\times C_4\times C_2\times D_4)$, of order \(10240\)\(\medspace = 2^{11} \cdot 5 \)
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 6, $C_3$, $C_5$
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, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, 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 3 4 5 6 8 10 12 15 20 24 30 40 60 120
Elements 1 47 2 128 4 94 16 28 256 8 32 32 56 64 64 128 960
Conjugacy classes   1 5 2 9 2 10 4 10 18 4 12 8 20 16 24 32 177
Divisions 1 5 1 6 1 5 2 4 6 1 3 2 4 2 3 2 48
Autjugacy classes 1 5 1 6 1 5 2 4 6 1 3 2 4 2 3 2 48

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 16 32 64
Irr. complex chars.   24 126 27 0 0 0 0 177
Irr. rational chars. 8 10 9 12 6 2 1 48

Minimal presentations

Permutation degree:$24$
Transitive degree:$480$
Rank: $3$
Inequivalent generating triples: $26208$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible none none none
Arbitrary 4 8 14

Constructions

Show commands: Gap / Magma / SageMath


Presentation: $\langle a, b, c \mid a^{2}=b^{4}=c^{120}=1, b^{a}=b^{3}, c^{a}=b^{2}c^{61}, c^{b}=c^{79} \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([8, -2, -2, -2, 2, -2, -2, -3, -5, 97, 41, 15747, 10123, 91, 6092, 116, 14605, 141, 7182, 222, 24591]); a,b,c := Explode([G.1, G.2, G.4]); AssignNames(~G, ["a", "b", "b2", "c", "c2", "c4", "c8", "c24"]);
 
Copy content gap:G := PcGroupCode(4313063399413719859890583247009584685657039175687,960); a := G.1; b := G.2; c := G.4;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(4313063399413719859890583247009584685657039175687,960)'); a = G.1; b = G.2; c = G.4;
 
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(4313063399413719859890583247009584685657039175687,960)'); a = G.1; b = G.2; c = G.4;
 
Permutation group:Degree $24$ $\langle(2,5)(3,7)(6,8)(9,10)(11,15)(12,13)(14,16)(18,19)(20,21), (2,6)(5,8)(10,13) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 24 | (2,5)(3,7)(6,8)(9,10)(11,15)(12,13)(14,16)(18,19)(20,21), (2,6)(5,8)(10,13)(11,16), (1,2,3,5,4,6,7,8)(9,11)(10,14)(12,16)(13,15), (22,23,24), (1,3,4,7)(2,5,6,8)(9,12)(10,13)(11,16)(14,15), (9,12)(10,13)(11,16)(14,15), (1,4)(2,6)(3,7)(5,8), (17,18,20,21,19) >;
 
Copy content gap:G := Group( (2,5)(3,7)(6,8)(9,10)(11,15)(12,13)(14,16)(18,19)(20,21), (2,6)(5,8)(10,13)(11,16), (1,2,3,5,4,6,7,8)(9,11)(10,14)(12,16)(13,15), (22,23,24), (1,3,4,7)(2,5,6,8)(9,12)(10,13)(11,16)(14,15), (9,12)(10,13)(11,16)(14,15), (1,4)(2,6)(3,7)(5,8), (17,18,20,21,19) );
 
Copy content sage:G = PermutationGroup(['(2,5)(3,7)(6,8)(9,10)(11,15)(12,13)(14,16)(18,19)(20,21)', '(2,6)(5,8)(10,13)(11,16)', '(1,2,3,5,4,6,7,8)(9,11)(10,14)(12,16)(13,15)', '(22,23,24)', '(1,3,4,7)(2,5,6,8)(9,12)(10,13)(11,16)(14,15)', '(9,12)(10,13)(11,16)(14,15)', '(1,4)(2,6)(3,7)(5,8)', '(17,18,20,21,19)'])
 
Direct product: $C_3$ $\, \times\, $ $(C_2^3.D_{20})$
Semidirect product: $(D_{20}:C_4)$ $\,\rtimes\,$ $C_6$ (2) $(C_{40}:C_4)$ $\,\rtimes\,$ $C_6$ $(C_{40}:C_4)$ $\,\rtimes\,$ $C_6$ $(D_{20}:C_{12})$ $\,\rtimes\,$ $C_2$ (2) all 14
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $C_{12}$ . $(D_4:D_5)$ (2) $(C_{60}:D_4)$ . $C_2$ $(C_{20}:D_4)$ . $C_6$ $(C_2\times C_{60})$ . $D_4$ all 46

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

Homology

Abelianization: $C_{2}^{2} \times C_{6} \simeq C_{2}^{3} \times C_{3}$
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 988 subgroups in 212 conjugacy classes, 78 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_2\times C_6$ $G/Z \simeq$ $C_2\times D_{20}$
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: $G' \simeq$ $C_2\times C_{20}$ $G/G' \simeq$ $C_2^2\times C_6$
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: $\Phi \simeq$ $C_2\times C_4$ $G/\Phi \simeq$ $C_6\times D_{10}$
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: $\operatorname{Fit} \simeq$ $C_2^2:C_{120}$ $G/\operatorname{Fit} \simeq$ $C_2$
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: $R \simeq$ $C_3\times C_2^3.D_{20}$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_2\times C_{30}$ $G/\operatorname{soc} \simeq$ $C_2\times D_4$
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: $P_{ 2 } \simeq$ $C_2^3.D_4$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup: $P_{ 5 } \simeq$ $C_5$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $C_3\times C_2^3.D_{20}$ $\rhd$ $C_2\times C_{20}$ $\rhd$ $C_1$
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 $C_3\times C_2^3.D_{20}$ $\rhd$ $C_{60}:D_4$ $\rhd$ $C_2^2\times C_{60}$ $\rhd$ $C_2\times C_{60}$ $\rhd$ $C_2\times C_{30}$ $\rhd$ $C_{30}$ $\rhd$ $C_{15}$ $\rhd$ $C_5$ $\rhd$ $C_1$
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 $C_3\times C_2^3.D_{20}$ $\rhd$ $C_2\times C_{20}$ $\rhd$ $C_{10}$ $\rhd$ $C_5$
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 $C_1$ $\lhd$ $C_2\times C_6$ $\lhd$ $C_2^2\times C_{12}$ $\lhd$ $C_2^2:C_{24}$
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()
 

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 $177 \times 177$ 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 $48 \times 48$ rational character table.