Properties

Label 384.13243
Order \( 2^{7} \cdot 3 \)
Exponent \( 2^{3} \cdot 3 \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{4} \)
$\card{Z(G)}$ \( 2^{2} \)
$\card{\Aut(G)}$ \( 2^{11} \cdot 3 \)
$\card{\mathrm{Out}(G)}$ \( 2^{6} \)
Perm deg. $27$
Trans deg. $96$
Rank $4$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Construction of abstract group
 
Copy content magma:G := SmallGroup(384, 13243);
 
Copy content gap:G := SmallGroup(384, 13243);
 
Copy content sage:G = PermutationGroup(['(2,5)(6,7)(10,11)(12,13)(14,23)(15,26)(16,21)(17,19)(18,24)(20,27)(22,25)', '(1,2)(3,7)(4,5)(6,8)(12,14)(13,18)(15,17)(16,24)(19,20)(21,23)(22,26)(25,27)', '(1,3)(2,6)(4,8)(5,7)(10,11)(13,19)(14,24)(17,25)(18,26)(20,23)(21,22)', '(1,2)(3,6)(4,5)(7,8)(10,11)(12,15)(13,20)(14,25)(16,27)(17,24)(18,22)(19,23)(21,26)', '(1,4)(2,5)(3,8)(6,7)(12,16)(13,21)(14,24)(15,27)(17,25)(18,23)(19,22)(20,26)', '(12,17,16,25)(13,22,21,19)(14,15,24,27)(18,26,23,20)', '(12,16)(13,21)(14,24)(15,27)(17,25)(18,23)(19,22)(20,26)', '(9,10,11)'])
 
Copy content sage_gap:G = libgap.SmallGroup(384, 13243)
 
Copy content oscar:G = small_group(384, 13243)
 

Group information

Description:$(C_2^2\times C_8):D_6$
Order: \(384\)\(\medspace = 2^{7} \cdot 3 \)
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()
 
Copy content oscar:order(G)
 
Exponent: \(24\)\(\medspace = 2^{3} \cdot 3 \)
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()
 
Copy content oscar:exponent(G)
 
Automorphism group:$C_3:(C_2^7\times C_2\times D_4)$, of order \(6144\)\(\medspace = 2^{11} \cdot 3 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage:libgap(G).AutomorphismGroup()
 
Copy content sage_gap:G.AutomorphismGroup()
 
Copy content oscar:automorphism_group(G)
 
Composition factors:$C_2$ x 7, $C_3$
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()
 
Copy content oscar:composition_series(G)
 
Derived length:$2$
Copy content comment:Derived length of the group
 
Copy content magma:DerivedLength(G);
 
Copy content gap:DerivedLength(G);
 
Copy content sage:libgap(G).DerivedLength()
 
Copy content sage_gap:G.DerivedLength()
 
Copy content oscar:derived_length(G)
 

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 oscar:is_abelian(G)
 
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 oscar:is_cyclic(G)
 
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 oscar:is_nilpotent(G)
 
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 oscar:is_solvable(G)
 
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 oscar:is_supersolvable(G)
 
Copy content comment:Determine if the group G is simple
 
Copy content magma:IsSimple(G);
 
Copy content gap:IsSimpleGroup(G);
 
Copy content sage:G.is_simple()
 
Copy content sage_gap:G.IsSimpleGroup()
 
Copy content oscar:is_simple(G)
 

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))
 
Copy content sage_gap:# Sage code (using the GAP interface) to output the first two rows of the group statistics table element_orders = [g.Order() for g in G.Elements()] orders = sorted(list(set(element_orders))) print("Orders:", orders) print("Elements:", [element_orders.count(n) for n in orders], G.Order()) cc_orders = [cc.Representative().Order() for cc in G.ConjugacyClasses()] print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))
 
Copy content oscar:# Oscar code to output the first two rows of the group statistics table element_orders = [order(g) for g in elements(G)] orders = sort(unique(element_orders)) println("Orders: ", orders) element_counts = [count(==(n), element_orders) for n in orders] println("Elements: ", element_counts, " ", order(G)) ccs = conjugacy_classes(G) cc_orders = [order(representative(cc)) for cc in ccs] cc_counts = [count(==(n), cc_orders) for n in orders] println("Conjugacy classes: ", cc_counts, " ", length(ccs))
 

Order 1 2 3 4 6 8 12 24
Elements 1 87 2 104 30 64 64 32 384
Conjugacy classes   1 9 1 10 6 6 7 8 48
Divisions 1 9 1 10 5 4 6 2 38
Autjugacy classes 1 9 1 10 5 4 6 2 38

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:# 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 G.CharacterDegrees()
 
Copy content oscar:# Outputs an MSet containing the absolutely irreducible degrees of G and their multiplicities. character_degrees(G)
 

Dimension 1 2 4 8 16
Irr. complex chars.   16 12 20 0 0 48
Irr. rational chars. 16 12 4 4 2 38

Minimal presentations

Permutation degree:$27$
Transitive degree:$96$
Rank: $4$
Inequivalent generating quadruples: $1048320$

Minimal degrees of faithful linear representations

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

Constructions

Show commands: Gap / Magma / Oscar / SageMath


Presentation: $\langle a, b, c, d \mid a^{2}=b^{2}=c^{4}=d^{24}=[a,b]=1, c^{a}=c^{3}d^{12}, d^{a}=d^{23}, c^{b}=cd^{12}, d^{b}=c^{2}d^{17}, d^{c}=d^{23} \rangle$ 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, -2, -3, 4898, 2362, 66, 14724, 5612, 3700, 116, 16901, 3853, 4245, 141, 17926, 8974, 4502, 166, 16391, 8207, 4119]); a,b,c,d := Explode([G.1, G.2, G.3, G.5]); AssignNames(~G, ["a", "b", "c", "c2", "d", "d2", "d4", "d8"]);
 
Copy content gap:G := PcGroupCode(758912038008271358037188403123151889047173546966523933697,384); a := G.1; b := G.2; c := G.3; d := 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(758912038008271358037188403123151889047173546966523933697,384)'); a = G.1; b = G.2; c = G.3; d = 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(758912038008271358037188403123151889047173546966523933697,384)'); a = G.1; b = G.2; c = G.3; d = G.5;
 
Permutation group:Degree $27$ $\langle(2,5)(6,7)(10,11)(12,13)(14,23)(15,26)(16,21)(17,19)(18,24)(20,27)(22,25) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 27 | (2,5)(6,7)(10,11)(12,13)(14,23)(15,26)(16,21)(17,19)(18,24)(20,27)(22,25), (1,2)(3,7)(4,5)(6,8)(12,14)(13,18)(15,17)(16,24)(19,20)(21,23)(22,26)(25,27), (1,3)(2,6)(4,8)(5,7)(10,11)(13,19)(14,24)(17,25)(18,26)(20,23)(21,22), (1,2)(3,6)(4,5)(7,8)(10,11)(12,15)(13,20)(14,25)(16,27)(17,24)(18,22)(19,23)(21,26), (1,4)(2,5)(3,8)(6,7)(12,16)(13,21)(14,24)(15,27)(17,25)(18,23)(19,22)(20,26), (12,17,16,25)(13,22,21,19)(14,15,24,27)(18,26,23,20), (12,16)(13,21)(14,24)(15,27)(17,25)(18,23)(19,22)(20,26), (9,10,11) >;
 
Copy content gap:G := Group( (2,5)(6,7)(10,11)(12,13)(14,23)(15,26)(16,21)(17,19)(18,24)(20,27)(22,25), (1,2)(3,7)(4,5)(6,8)(12,14)(13,18)(15,17)(16,24)(19,20)(21,23)(22,26)(25,27), (1,3)(2,6)(4,8)(5,7)(10,11)(13,19)(14,24)(17,25)(18,26)(20,23)(21,22), (1,2)(3,6)(4,5)(7,8)(10,11)(12,15)(13,20)(14,25)(16,27)(17,24)(18,22)(19,23)(21,26), (1,4)(2,5)(3,8)(6,7)(12,16)(13,21)(14,24)(15,27)(17,25)(18,23)(19,22)(20,26), (12,17,16,25)(13,22,21,19)(14,15,24,27)(18,26,23,20), (12,16)(13,21)(14,24)(15,27)(17,25)(18,23)(19,22)(20,26), (9,10,11) );
 
Copy content sage:G = PermutationGroup(['(2,5)(6,7)(10,11)(12,13)(14,23)(15,26)(16,21)(17,19)(18,24)(20,27)(22,25)', '(1,2)(3,7)(4,5)(6,8)(12,14)(13,18)(15,17)(16,24)(19,20)(21,23)(22,26)(25,27)', '(1,3)(2,6)(4,8)(5,7)(10,11)(13,19)(14,24)(17,25)(18,26)(20,23)(21,22)', '(1,2)(3,6)(4,5)(7,8)(10,11)(12,15)(13,20)(14,25)(16,27)(17,24)(18,22)(19,23)(21,26)', '(1,4)(2,5)(3,8)(6,7)(12,16)(13,21)(14,24)(15,27)(17,25)(18,23)(19,22)(20,26)', '(12,17,16,25)(13,22,21,19)(14,15,24,27)(18,26,23,20)', '(12,16)(13,21)(14,24)(15,27)(17,25)(18,23)(19,22)(20,26)', '(9,10,11)'])
 
Copy content sage_gap:G = gap.new('Group( (2,5)(6,7)(10,11)(12,13)(14,23)(15,26)(16,21)(17,19)(18,24)(20,27)(22,25), (1,2)(3,7)(4,5)(6,8)(12,14)(13,18)(15,17)(16,24)(19,20)(21,23)(22,26)(25,27), (1,3)(2,6)(4,8)(5,7)(10,11)(13,19)(14,24)(17,25)(18,26)(20,23)(21,22), (1,2)(3,6)(4,5)(7,8)(10,11)(12,15)(13,20)(14,25)(16,27)(17,24)(18,22)(19,23)(21,26), (1,4)(2,5)(3,8)(6,7)(12,16)(13,21)(14,24)(15,27)(17,25)(18,23)(19,22)(20,26), (12,17,16,25)(13,22,21,19)(14,15,24,27)(18,26,23,20), (12,16)(13,21)(14,24)(15,27)(17,25)(18,23)(19,22)(20,26), (9,10,11) )')
 
Copy content oscar:G = @permutation_group(27, (2,5)(6,7)(10,11)(12,13)(14,23)(15,26)(16,21)(17,19)(18,24)(20,27)(22,25), (1,2)(3,7)(4,5)(6,8)(12,14)(13,18)(15,17)(16,24)(19,20)(21,23)(22,26)(25,27), (1,3)(2,6)(4,8)(5,7)(10,11)(13,19)(14,24)(17,25)(18,26)(20,23)(21,22), (1,2)(3,6)(4,5)(7,8)(10,11)(12,15)(13,20)(14,25)(16,27)(17,24)(18,22)(19,23)(21,26), (1,4)(2,5)(3,8)(6,7)(12,16)(13,21)(14,24)(15,27)(17,25)(18,23)(19,22)(20,26), (12,17,16,25)(13,22,21,19)(14,15,24,27)(18,26,23,20), (12,16)(13,21)(14,24)(15,27)(17,25)(18,23)(19,22)(20,26), (9,10,11))
 
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: $(C_8:D_4)$ $\,\rtimes\,$ $S_3$ $(Q_8:C_4)$ $\,\rtimes\,$ $D_6$ $(D_6.D_8)$ $\,\rtimes\,$ $C_2$ $(C_8:C_4)$ $\,\rtimes\,$ $D_6$ all 36
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $(C_6:Q_8)$ . $D_4$ $(C_6:D_4)$ . $D_4$ (2) $(C_4:D_4)$ . $D_6$ $(D_4:C_4)$ . $D_6$ all 56

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

Homology

Abelianization: $C_{2}^{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())
 
Copy content sage_gap:G.FactorGroup(G.DerivedSubgroup())
 
Copy content oscar:quo(G, derived_subgroup(G)[1])
 
Schur multiplier: $C_{2}^{4}$
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()
 
Copy content oscar:subgroups(G)
 

There are 1830 subgroups in 422 conjugacy classes, 109 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_2^2$ $G/Z \simeq$ $D_4\times D_6$
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()
 
Copy content oscar:center(G)
 
Commutator: $G' \simeq$ $C_2\times C_{12}$ $G/G' \simeq$ $C_2^4$
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()
 
Copy content oscar:derived_subgroup(G)
 
Frattini: $\Phi \simeq$ $C_2\times C_4$ $G/\Phi \simeq$ $C_2^2\times D_6$
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()
 
Copy content oscar:frattini_subgroup(G)
 
Fitting: $\operatorname{Fit} \simeq$ $C_{24}:D_4$ $G/\operatorname{Fit} \simeq$ $C_2$
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()
 
Copy content oscar:fitting_subgroup(G)
 
Radical: $R \simeq$ $(C_2^2\times C_8):D_6$ $G/R \simeq$ $C_1$
Copy content comment:Radical of the group G
 
Copy content magma:Radical(G);
 
Copy content gap:SolvableRadical(G);
 
Copy content sage_gap:G.SolvableRadical()
 
Copy content oscar:solvable_radical(G)
 
Socle: $\operatorname{soc} \simeq$ $C_2\times C_6$ $G/\operatorname{soc} \simeq$ $C_2^2\times D_4$
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()
 
Copy content oscar:socle(G)
 
2-Sylow subgroup: $P_{ 2 } \simeq$ $(C_2^2\times C_8):C_2^2$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3$

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

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

Derived series $(C_2^2\times C_8):D_6$ $\rhd$ $C_2\times C_{12}$ $\rhd$ $C_1$
Copy content comment:Derived series of the group G
 
Copy content magma:DerivedSeries(G);
 
Copy content gap:DerivedSeriesOfGroup(G);
 
Copy content sage:G.derived_series()
 
Copy content sage_gap:G.DerivedSeriesOfGroup()
 
Copy content oscar:derived_series(G)
 
Chief series $(C_2^2\times C_8):D_6$ $\rhd$ $(C_4\times S_3):D_4$ $\rhd$ $D_4\times D_6$ $\rhd$ $C_2\times D_{12}$ $\rhd$ $C_2\times C_{12}$ $\rhd$ $C_2\times C_6$ $\rhd$ $C_6$ $\rhd$ $C_3$ $\rhd$ $C_1$
Copy content comment:Chief series of the group G
 
Copy content magma:ChiefSeries(G);
 
Copy content gap:ChiefSeries(G);
 
Copy content sage:libgap(G).ChiefSeries()
 
Copy content sage_gap:G.ChiefSeries()
 
Copy content oscar:chief_series(G)
 
Lower central series $(C_2^2\times C_8):D_6$ $\rhd$ $C_2\times C_{12}$ $\rhd$ $C_6$ $\rhd$ $C_3$
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()
 
Copy content oscar:lower_central_series(G)
 
Upper central series $C_1$ $\lhd$ $C_2^2$ $\lhd$ $C_2^2\times C_4$ $\lhd$ $C_8:D_4$
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()
 
Copy content oscar:upper_central_series(G)
 

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
 
Copy content oscar:character_table(G) # Output not guaranteed to exactly match the LMFDB table
 

Complex character table

See the $48 \times 48$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $38 \times 38$ rational character table.