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Properties

Label 192.1543
Order \( 2^{6} \cdot 3 \)
Exponent \( 2 \cdot 3 \)
Abelian yes
$\card{\Aut(G)}$ \( 2^{16} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 31 \)
Perm deg. $15$
Trans deg. $192$
Rank $6$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SmallGroup(192, 1543);
 
Copy content gap:G := SmallGroup(192, 1543);
 
Copy content sage_gap:G = libgap.SmallGroup(192, 1543)
 
Copy content comment:Define the group as a permutation group
 
Copy content sage:G = PermutationGroup(['(1,2)', '(3,4)', '(5,6)', '(7,8)', '(9,10)', '(11,12)', '(13,15,14)'])
 

Group information

Description:$C_2^5\times C_6$
Order: \(192\)\(\medspace = 2^{6} \cdot 3 \)
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()
 
Exponent: \(6\)\(\medspace = 2 \cdot 3 \)
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()
 
Automorphism group:$C_2\times \GL(6,2)$, of order \(40317419520\)\(\medspace = 2^{16} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 31 \)
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, $C_3$
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:$1$
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:$1$
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 abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).

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 6
Elements 1 63 2 126 192
Conjugacy classes   1 63 2 126 192
Divisions 1 63 1 63 128
Autjugacy classes 1 1 1 1 4

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
Irr. complex chars.   192 0 192
Irr. rational chars. 64 64 128

Minimal presentations

Permutation degree:$15$
Transitive degree:$192$
Rank: $6$
Inequivalent generating 6-tuples: $364$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible none none none
Arbitrary 6 not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Presentation:Abelian group $\langle a, b, c, d, e, f \mid a^{2}=b^{2}=c^{2}=d^{2}=e^{2}=f^{6}=1 \rangle$ 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, -3, 124]); a,b,c,d,e,f := Explode([G.1, G.2, G.3, G.4, G.5, G.6]); AssignNames(~G, ["a", "b", "c", "d", "e", "f", "f2"]);
 
Copy content gap:G := PcGroupCode(17179873281,192); a := G.1; b := G.2; c := G.3; d := G.4; e := G.5; f := G.6;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(17179873281,192)'); a = G.1; b = G.2; c = G.3; d = G.4; e = G.5; f = G.6;
 
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(17179873281,192)'); a = G.1; b = G.2; c = G.3; d = G.4; e = G.5; f = G.6;
 
Permutation group:Degree $15$ $\langle(1,2), (3,4), (5,6), (7,8), (9,10), (11,12), (13,15,14)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 15 | (1,2), (3,4), (5,6), (7,8), (9,10), (11,12), (13,15,14) >;
 
Copy content gap:G := Group( (1,2), (3,4), (5,6), (7,8), (9,10), (11,12), (13,15,14) );
 
Copy content sage:G = PermutationGroup(['(1,2)', '(3,4)', '(5,6)', '(7,8)', '(9,10)', '(11,12)', '(13,15,14)'])
 
Matrix group:$\left\langle \left(\begin{array}{ll}\alpha^{18} & \alpha^{39} \\ \alpha^{6} & \alpha^{18} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{21} & 0 \\ 0 & \alpha^{21} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{9} & \alpha^{32} \\ \alpha^{62} & \alpha^{9} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{51} & \alpha^{25} \\ \alpha^{55} & \alpha^{51} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{38} & \alpha^{58} \\ \alpha^{25} & \alpha^{38} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{45} & \alpha^{21} \\ \alpha^{51} & \alpha^{45} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{60} & \alpha^{23} \\ \alpha^{53} & \alpha^{60} \\ \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{64}) = \GL_{2}(\F_{2}[\alpha]/(\alpha^{6} + \alpha^{4} + \alpha^{3} + \alpha + 1))$
Copy content comment:Define the group as a matrix group with coefficients in GLFq
 
Copy content magma:F:=GF(64); al:=F.1; G := MatrixGroup< 2, F | [[al^18, al^39], [al^6, al^18]],[[al^21, 0], [0, al^21]],[[al^9, al^32], [al^62, al^9]],[[al^51, al^25], [al^55, al^51]],[[al^38, al^58], [al^25, al^38]],[[al^45, al^21], [al^51, al^45]],[[al^60, al^23], [al^53, al^60]] >;
 
Copy content gap:G := Group([[[Z(64)^18, Z(64)^39], [Z(64)^6, Z(64)^18]],[[Z(64)^21, 0*Z(64)], [0*Z(64), Z(64)^21]],[[Z(64)^9, Z(64)^32], [Z(64)^62, Z(64)^9]],[[Z(64)^51, Z(64)^25], [Z(64)^55, Z(64)^51]],[[Z(64)^38, Z(64)^58], [Z(64)^25, Z(64)^38]],[[Z(64)^45, Z(64)^21], [Z(64)^51, Z(64)^45]],[[Z(64)^60, Z(64)^23], [Z(64)^53, Z(64)^60]]]);
 
Copy content sage:F = GF(64); al = F.0; MS = MatrixSpace(F, 2, 2) G = MatrixGroup([MS([[al^18, al^39], [al^6, al^18]]), MS([[al^21, 0], [0, al^21]]), MS([[al^9, al^32], [al^62, al^9]]), MS([[al^51, al^25], [al^55, al^51]]), MS([[al^38, al^58], [al^25, al^38]]), MS([[al^45, al^21], [al^51, al^45]]), MS([[al^60, al^23], [al^53, al^60]])])
 
Direct product: $C_2$ ${}^6$ $\, \times\, $ $C_3$
Semidirect product: not isomorphic to a non-trivial semidirect product
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product

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

Homology

Primary decomposition: $C_{2}^{6} \times C_{3}$
Copy content comment:The primary decomposition of the group
 
Copy content magma:PrimaryInvariants(G);
 
Copy content gap:AbelianInvariants(G);
 
Copy content sage_gap:G.AbelianInvariants()
 
Schur multiplier: $C_{2}^{15}$
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: $0$
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 5650 subgroups, all normal (4 characteristic).

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

Special subgroups

Center: $Z \simeq$ $C_2^5\times C_6$ $G/Z \simeq$ $C_1$
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: $G' \simeq$ $C_1$ $G/G' \simeq$ $C_2^5\times C_6$
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: $\Phi \simeq$ $C_1$ $G/\Phi \simeq$ $C_2^5\times C_6$
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()
 
Fitting: $\operatorname{Fit} \simeq$ $C_2^5\times C_6$ $G/\operatorname{Fit} \simeq$ $C_1$
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: $R \simeq$ $C_2^5\times C_6$ $G/R \simeq$ $C_1$
Copy content comment:Radical of the group G
 
Copy content magma:Radical(G);
 
Copy content gap:SolvableRadical(G);
 
Copy content sage_gap:G.SolvableRadical()
 
Socle: $\operatorname{soc} \simeq$ $C_2^5\times C_6$ $G/\operatorname{soc} \simeq$ $C_1$
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: $P_{ 2 } \simeq$ $C_2^6$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $C_2^5\times C_6$ $\rhd$ $C_1$
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 $C_2^5\times C_6$ $\rhd$ $C_2^4\times C_6$ $\rhd$ $C_2^3\times C_6$ $\rhd$ $C_2^2\times C_6$ $\rhd$ $C_2\times C_6$ $\rhd$ $C_6$ $\rhd$ $C_3$ $\rhd$ $C_1$
Copy content comment:Chief series of the group G
 
Copy content magma:ChiefSeries(G);
 
Copy content gap:ChiefSeries(G);
 
Copy content sage_gap:G.ChiefSeries()
 
Lower central series $C_2^5\times C_6$ $\rhd$ $C_1$
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 $C_1$ $\lhd$ $C_2^5\times C_6$
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 34 larger groups in the database.

This group is a maximal quotient of 20 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 $192 \times 192$ 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 $128 \times 128$ rational character table (warning: may be slow to load).