Properties

Label 10.1
Order \( 2 \cdot 5 \)
Exponent \( 2 \cdot 5 \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2 \)
$\card{Z(G)}$ \( 1 \)
$\card{\Aut(G)}$ \( 2^{2} \cdot 5 \)
$\card{\mathrm{Out}(G)}$ \( 2 \)
Perm deg. $5$
Trans deg. $5$
Rank $2$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define group as a dihedral group
 
Copy content magma:G := DihedralGroup(5);
 
Copy content gap:G := DihedralGroup(10);
 
Copy content sage:G = DihedralGroup(5)
 
Copy content sage_gap:G = libgap.eval('DihedralGroup(10)')
 
Copy content oscar:G = dihedral_group(10)
 

Group information

Description:$D_5$
Order: \(10\)\(\medspace = 2 \cdot 5 \)
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()
 
Copy content oscar:order(G)
 
Exponent: \(10\)\(\medspace = 2 \cdot 5 \)
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()
 
Copy content oscar:exponent(G)
 
Automorphism group:$F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \)
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:$C_2$, $C_5$
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()
 
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, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), 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 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 5
Elements 1 5 4 10
Conjugacy classes   1 1 2 4
Divisions 1 1 1 3
Autjugacy classes 1 1 1 3

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
Irr. complex chars.   2 2 0 4
Irr. rational chars. 2 0 1 3

Minimal presentations

Permutation degree:$5$
Transitive degree:$5$
Rank: $2$
Inequivalent generating pairs: $3$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 2 2 4
Arbitrary 2 2 4

Constructions

Show commands: Gap / Magma / Oscar / SageMath


Groups of Lie type:$\GSOMinus(2,4)$
Copy content magma:G := CSOMinus(2,4);
 
Copy content gap:F = GF(4); al = F.0; MS = MatrixSpace(F, 2, 2) G = MatrixGroup([MS([[1, 0], [1, 1]]), MS([[al^1, al^2], [1, 1]])])
 
Copy content sage:F = GF(4); al = F.0; MS = MatrixSpace(F, 2, 2) G = MatrixGroup([MS([[1, 0], [1, 1]]), MS([[al^1, al^2], [1, 1]])])
 
Copy content oscar:F = GF(4); al = F.0; MS = MatrixSpace(F, 2, 2) G = MatrixGroup([MS([[1, 0], [1, 1]]), MS([[al^1, al^2], [1, 1]])])
 
Presentation: $\langle a, b \mid a^{2}=b^{5}=1, b^{a}=b^{4} \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([2, -2, -5, 33]); a,b := Explode([G.1, G.2]); AssignNames(~G, ["a", "b"]);
 
Copy content gap:G := PcGroupCode(227,10); 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(227,10)'); 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(227,10)'); a = G.1; b = G.2;
 
Permutation group: $\langle(2,5)(3,4), (1,5,4,3,2)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 5 | (2,5)(3,4), (1,5,4,3,2) >;
 
Copy content gap:G := Group( (2,5)(3,4), (1,5,4,3,2) );
 
Copy content sage:G = PermutationGroup(['(2,5)(3,4)', '(1,5,4,3,2)'])
 
Copy content sage_gap:G = gap.new('Group( (2,5)(3,4), (1,5,4,3,2) )')
 
Copy content oscar:G = @permutation_group(5, (2,5)(3,4), (1,5,4,3,2))
 
Matrix group:$\left\langle \left(\begin{array}{rrrr} 0 & 1 & 1 & 1 \\ 0 & -1 & -1 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 1 & 1 \\ -1 & -1 & -1 & -1 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{4}(\Z)$
Copy content comment:Define the group as a matrix group with coefficients in Z
 
Copy content magma:G := MatrixGroup< 4, Integers() | [[0, 1, 1, 1, 0, -1, -1, 0, 0, 0, 1, 0, 1, 1, 0, 0], [0, 1, 1, 1, -1, -1, -1, -1, 1, 1, 0, 0, 0, 0, 1, 0]] >;
 
Copy content gap:G := Group([[[0, 1, 1, 1], [0, -1, -1, 0], [0, 0, 1, 0], [1, 1, 0, 0]], [[0, 1, 1, 1], [-1, -1, -1, -1], [1, 1, 0, 0], [0, 0, 1, 0]]]);
 
Copy content sage:MS = MatrixSpace(Integers(), 4, 4) G = MatrixGroup([MS([[0, 1, 1, 1], [0, -1, -1, 0], [0, 0, 1, 0], [1, 1, 0, 0]]), MS([[0, 1, 1, 1], [-1, -1, -1, -1], [1, 1, 0, 0], [0, 0, 1, 0]])])
 
Copy content sage_gap:G = gap.new('Group([[[0, 1, 1, 1], [0, -1, -1, 0], [0, 0, 1, 0], [1, 1, 0, 0]], [[0, 1, 1, 1], [-1, -1, -1, -1], [1, 1, 0, 0], [0, 0, 1, 0]]])')
 
Copy content oscar:G = matrix_group([matrix(ZZ, [[0, 1, 1, 1], [0, -1, -1, 0], [0, 0, 1, 0], [1, 1, 0, 0]]), matrix(ZZ, [[0, 1, 1, 1], [-1, -1, -1, -1], [1, 1, 0, 0], [0, 0, 1, 0]])])
 
$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 4 & 0 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{5})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 2, GF(5) | [[1, 1, 0, 1], [4, 0, 0, 1]] >;
 
Copy content gap:G := Group([[[ Z(5)^0, Z(5)^0 ], [ 0*Z(5), Z(5)^0 ]], [[ Z(5)^2, 0*Z(5) ], [ 0*Z(5), Z(5)^0 ]]]);
 
Copy content sage:MS = MatrixSpace(GF(5), 2, 2) G = MatrixGroup([MS([[1, 1], [0, 1]]), MS([[4, 0], [0, 1]])])
 
Copy content sage_gap:G = gap.new('Group([[[ Z(5)^0, Z(5)^0 ], [ 0*Z(5), Z(5)^0 ]], [[ Z(5)^2, 0*Z(5) ], [ 0*Z(5), Z(5)^0 ]]])')
 
Copy content oscar:G = matrix_group([matrix(GF(5), [[1, 1], [0, 1]]), matrix(GF(5), [[4, 0], [0, 1]])])
 
Transitive group: 5T2 10T2 more information
Copy content magma:G := TransitiveGroup(5, 2);
 
Copy content gap:G := TransitiveGroup(5, 2);
 
Copy content sage:G = TransitiveGroup(5, 2)
 
Copy content sage_gap:G = libgap.TransitiveGroup(5, 2)
 
Copy content oscar:G = transitive_group(5, 2)
 
Copy content magma:G := TransitiveGroup(10, 2);
 
Copy content gap:G := TransitiveGroup(10, 2);
 
Copy content sage:G = TransitiveGroup(10, 2)
 
Copy content sage_gap:G = libgap.TransitiveGroup(10, 2)
 
Copy content oscar:G = transitive_group(10, 2)
 
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: $C_5$ $\,\rtimes\,$ $C_2$ more information
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

Abelianization: $C_{2} $
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: $C_1$
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: $1$
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 8 subgroups in 4 conjugacy classes, 3 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_1$ $G/Z \simeq$ $D_5$
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: $G' \simeq$ $C_5$ $G/G' \simeq$ $C_2$
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: $\Phi \simeq$ $C_1$ $G/\Phi \simeq$ $D_5$
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: $\operatorname{Fit} \simeq$ $C_5$ $G/\operatorname{Fit} \simeq$ $C_2$
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: $R \simeq$ $D_5$ $G/R \simeq$ $C_1$
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: $\operatorname{soc} \simeq$ $C_5$ $G/\operatorname{soc} \simeq$ $C_2$
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()
 
Copy content oscar:socle(G)
 
2-Sylow subgroup: $P_{ 2 } \simeq$ $C_2$
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 $D_5$ $\rhd$ $C_5$ $\rhd$ $C_1$
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 $D_5$ $\rhd$ $C_5$ $\rhd$ $C_1$
Copy content comment:Chief series of the group G
 
Copy content magma:ChiefSeries(G);
 
Copy content gap:ChiefSeries(G);
 
Copy content sage:libgap(G).ChiefSeries()
 
Copy content sage_gap:G.ChiefSeries()
 
Copy content oscar:chief_series(G)
 
Lower central series $D_5$ $\rhd$ $C_5$
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()
 
Copy content oscar:lower_central_series(G)
 
Upper central series $C_1$
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)
 

Supergroups

This group is a maximal subgroup of 99 larger groups in the database.

This group is a maximal quotient of 127 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
 
Copy content oscar:character_table(G) # Output not guaranteed to exactly match the LMFDB table
 

Complex character table

1A 2A 5A1 5A2
Size 1 5 2 2
2 P 1A 1A 5A2 5A1
5 P 1A 2A 1A 1A
Type
10.1.1a R 1 1 1 1
10.1.1b R 1 −1 1 1
10.1.2a1 R 2 0 ζ5−2+ζ52 ζ5−1+ζ5
10.1.2a2 R 2 0 ζ5−1+ζ5 ζ5−2+ζ52

Rational character table

1A 2A 5A
Size 1 5 4
2 P 1A 1A 5A
5 P 1A 2A 1A
10.1.1a 1 1 1
10.1.1b 1 −1 1
10.1.2a 4 0 −1

Additional information

This is the smallest group so that the natural map into its automorphism group ($F_5$) is injective and not surjective. The kernel of this map, which sends $g$ to the inner automorphism $x \mapsto g^{-1} x g$, is always the center, which is trivial for $D_5$. And while $S_3$ is smaller and has trivial center, $S_3$ is its own automorphism group.