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Properties

Label 720.61
Order \( 2^{4} \cdot 3^{2} \cdot 5 \)
Exponent \( 2^{3} \cdot 3^{2} \cdot 5 \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{2} \cdot 5 \)
$\card{Z(G)}$ \( 2 \cdot 5 \)
$\card{\Aut(G)}$ \( 2^{7} \cdot 3^{3} \)
$\card{\mathrm{Out}(G)}$ \( 2^{4} \cdot 3 \)
Perm deg. $22$
Trans deg. $360$
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SmallGroup(720, 61);
 
Copy content gap:G := SmallGroup(720, 61);
 
Copy content sage_gap:G = libgap.SmallGroup(720, 61)
 
Copy content comment:Define the group as a permutation group
 
Copy content sage:G = PermutationGroup(['(1,2)(3,5)(4,7)(6,8)(15,17)(16,20)(18,21)(19,22)', '(2,5)(3,8)(6,7)', '(9,10,11,12,13)', '(1,3,4,8)(2,6,7,5)', '(1,4)(2,7)(3,8)(5,6)', '(14,15,18,16,19,22,20,21,17)', '(14,16,20)(15,19,21)(17,18,22)'])
 

Group information

Description:$C_{45}:D_8$
Order: \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \)
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: \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \)
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:$C_{18}.(C_2^4\times C_{12})$, of order \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \)
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, $C_3$ x 2, $C_5$
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 9 10 12 15 18 20 30 36 40 45 60 90 180
Elements 1 41 2 2 4 10 36 6 164 4 8 30 8 40 12 144 24 16 120 48 720
Conjugacy classes   1 3 1 1 4 3 2 3 12 1 4 9 4 12 3 8 12 4 36 12 135
Divisions 1 3 1 1 1 2 1 1 3 1 1 2 1 2 1 1 1 1 2 1 28
Autjugacy classes 1 3 1 1 1 2 1 1 3 1 1 2 1 2 1 1 1 1 2 1 28

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 6 8 12 16 24 48
Irr. complex chars.   20 95 20 0 0 0 0 0 0 135
Irr. rational chars. 4 3 7 2 3 2 3 2 2 28

Minimal presentations

Permutation degree:$22$
Transitive degree:$360$
Rank: $2$
Inequivalent generating pairs: $36$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 4 8 48
Arbitrary 4 6 14

Constructions

Show commands: Gap / Magma / SageMath


Presentation: $\langle a, b, c \mid a^{2}=b^{8}=c^{45}=[a,c]=1, b^{a}=b^{7}, c^{b}=c^{26} \rangle$ 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, -3, -3, -5, 197, 36, 254, 58, 7291, 137, 11100, 166]); a,b,c := Explode([G.1, G.2, G.5]); AssignNames(~G, ["a", "b", "b2", "b4", "c", "c3", "c9"]);
 
Copy content gap:G := PcGroupCode(1772894909794341293632362121101335,720); a := G.1; b := G.2; c := 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(1772894909794341293632362121101335,720)'); a = G.1; b = G.2; c = 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(1772894909794341293632362121101335,720)'); a = G.1; b = G.2; c = G.5;
 
Permutation group:Degree $22$ $\langle(1,2)(3,5)(4,7)(6,8)(15,17)(16,20)(18,21)(19,22), (2,5)(3,8)(6,7), (9,10,11,12,13) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 22 | (1,2)(3,5)(4,7)(6,8)(15,17)(16,20)(18,21)(19,22), (2,5)(3,8)(6,7), (9,10,11,12,13), (1,3,4,8)(2,6,7,5), (1,4)(2,7)(3,8)(5,6), (14,15,18,16,19,22,20,21,17), (14,16,20)(15,19,21)(17,18,22) >;
 
Copy content gap:G := Group( (1,2)(3,5)(4,7)(6,8)(15,17)(16,20)(18,21)(19,22), (2,5)(3,8)(6,7), (9,10,11,12,13), (1,3,4,8)(2,6,7,5), (1,4)(2,7)(3,8)(5,6), (14,15,18,16,19,22,20,21,17), (14,16,20)(15,19,21)(17,18,22) );
 
Copy content sage:G = PermutationGroup(['(1,2)(3,5)(4,7)(6,8)(15,17)(16,20)(18,21)(19,22)', '(2,5)(3,8)(6,7)', '(9,10,11,12,13)', '(1,3,4,8)(2,6,7,5)', '(1,4)(2,7)(3,8)(5,6)', '(14,15,18,16,19,22,20,21,17)', '(14,16,20)(15,19,21)(17,18,22)'])
 
Direct product: $C_5$ $\, \times\, $ $(C_9:D_8)$
Semidirect product: $C_{45}$ $\,\rtimes\,$ $D_8$ $D_{36}$ $\,\rtimes\,$ $C_{10}$ $(C_5\times D_4)$ $\,\rtimes\,$ $D_9$ $D_4$ $\,\rtimes\,$ $(C_5\times D_9)$ all 10
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $C_{90}$ . $D_4$ $C_{60}$ . $D_6$ $C_{20}$ . $D_{18}$ $C_{180}$ . $C_2^2$ all 16

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

Homology

Abelianization: $C_{2} \times C_{10} \simeq C_{2}^{2} \times C_{5}$
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: $C_{2}$
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 294 subgroups in 66 conjugacy classes, 30 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_{10}$ $G/Z \simeq$ $C_9:D_4$
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: $G' \simeq$ $C_{36}$ $G/G' \simeq$ $C_2\times C_{10}$
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: $\Phi \simeq$ $C_{12}$ $G/\Phi \simeq$ $S_3\times C_{10}$
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: $\operatorname{Fit} \simeq$ $D_4\times C_{45}$ $G/\operatorname{Fit} \simeq$ $C_2$
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: $R \simeq$ $C_{45}:D_8$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_{30}$ $G/\operatorname{soc} \simeq$ $C_3:D_4$
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: $P_{ 2 } \simeq$ $D_8$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_9$
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 $C_{45}:D_8$ $\rhd$ $C_{36}$ $\rhd$ $C_1$
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 $C_{45}:D_8$ $\rhd$ $D_4\times C_{45}$ $\rhd$ $C_{180}$ $\rhd$ $C_{90}$ $\rhd$ $C_{45}$ $\rhd$ $C_{15}$ $\rhd$ $C_5$ $\rhd$ $C_1$
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 $C_{45}:D_8$ $\rhd$ $C_{36}$ $\rhd$ $C_{18}$ $\rhd$ $C_9$
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 $C_1$ $\lhd$ $C_{10}$ $\lhd$ $C_{20}$ $\lhd$ $C_5\times D_4$
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()
 

Supergroups

This group is a maximal subgroup of 16 larger groups in the database.

This group is a maximal quotient of 8 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 $135 \times 135$ 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 $28 \times 28$ rational character table.