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Properties

Label 20T21
20T21 1 11 1->11 13 1->13 14 1->14 2 12 2->12 2->13 2->14 3 3->12 3->13 15 3->15 4 4->11 4->14 16 4->16 5 5->16 18 5->18 20 5->20 6 6->15 17 6->17 19 6->19 7 7->17 7->18 7->20 8 8->17 8->18 8->19 9 9->11 9->16 9->20 10 10->12 10->15 10->19 11->3 11->4 11->6 12->3 12->4 12->5 13->1 13->5 13->7 14->2 14->6 14->8 15->8 15->9 15->9 16->7 16->10 16->10 17->1 17->7 17->10 18->2 18->8 18->9 19->1 19->4 19->5 20->2 20->3 20->6
Degree $20$
Order $80$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $D_4\times D_5$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(20, 21);
 
Copy content sage:G = TransitiveGroup(20, 21)
 
Copy content oscar:G = transitive_group(20, 21)
 
Copy content gap:G := TransitiveGroup(20, 21);
 

Group invariants

Abstract group:  $D_4\times D_5$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Copy content oscar:small_group_identification(G)
 
Copy content gap:IdGroup(G);
 
Order:  $80=2^{4} \cdot 5$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Copy content gap:Order(G);
 
Cyclic:  no
Copy content comment:Determine if group is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content oscar:is_cyclic(G)
 
Copy content gap:IsCyclic(G);
 
Abelian:  no
Copy content comment:Determine if group is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content oscar:is_abelian(G)
 
Copy content gap:IsAbelian(G);
 
Solvable:  yes
Copy content comment:Determine if group is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content sage:G.is_solvable()
 
Copy content oscar:is_solvable(G)
 
Copy content gap:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content comment:Nilpotency class
 
Copy content magma:NilpotencyClass(G);
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 
Copy content gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;
 

Group action invariants

Degree $n$:  $20$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Copy content gap:NrMovedPoints(G);
 
Transitive number $t$:  $21$
Copy content comment:Transitive number
 
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Copy content sage:G.transitive_number()
 
Copy content oscar:transitive_group_identification(G)[2]
 
Copy content gap:TransitiveIdentification(G);
 
Parity:  $-1$
Copy content comment:Parity
 
Copy content magma:IsEven(G);
 
Copy content sage:all(g.SignPerm() == 1 for g in libgap(G).GeneratorsOfGroup())
 
Copy content oscar:is_even(G)
 
Copy content gap:ForAll(GeneratorsOfGroup(G), g -> SignPerm(g) = 1);
 
Transitivity:  1
Primitive:  no
Copy content comment:Determine if group is primitive
 
Copy content magma:IsPrimitive(G);
 
Copy content sage:G.is_primitive()
 
Copy content oscar:is_primitive(G)
 
Copy content gap:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(20).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(20), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(20), G));
 
Generators:  $(1,13,7,20,3,15,9,11,6,17)(2,14,8,19,4,16,10,12,5,18)$, $(1,14,2,13)(3,12,4,11)(5,20,6,19)(7,18,8,17)(9,16,10,15)$, $(1,11,4,14,6,15,8,18,9,20,2,12,3,13,5,16,7,17,10,19)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 
Copy content gap:GeneratorsOfGroup(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_2^3$
$10$:  $D_{5}$
$16$:  $D_4\times C_2$
$20$:  $D_{10}$ x 3
$40$:  20T8

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $D_{5}$

Degree 10: $D_{10}$

Low degree siblings

20T21 x 3, 40T22 x 2, 40T39 x 2, 40T40 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{20}$ $1$ $1$ $0$ $()$
2A $2^{10}$ $1$ $2$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
2B $2^{10}$ $2$ $2$ $10$ $( 1,15)( 2,16)( 3,17)( 4,18)( 5,19)( 6,20)( 7,11)( 8,12)( 9,13)(10,14)$
2C $2^{5},1^{10}$ $2$ $2$ $5$ $(11,12)(13,14)(15,16)(17,18)(19,20)$
2D $2^{8},1^{4}$ $5$ $2$ $8$ $( 3, 9)( 4,10)( 5, 8)( 6, 7)(11,20)(12,19)(13,17)(14,18)$
2E $2^{10}$ $5$ $2$ $10$ $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,18)(12,17)(13,16)(14,15)(19,20)$
2F $2^{10}$ $10$ $2$ $10$ $( 1,11)( 2,12)( 3,20)( 4,19)( 5,18)( 6,17)( 7,15)( 8,16)( 9,13)(10,14)$
2G $2^{9},1^{2}$ $10$ $2$ $9$ $( 1, 6)( 2, 5)( 7, 9)( 8,10)(11,14)(12,13)(15,19)(16,20)(17,18)$
4A $4^{5}$ $2$ $4$ $15$ $( 1,16, 2,15)( 3,18, 4,17)( 5,20, 6,19)( 7,12, 8,11)( 9,14,10,13)$
4B $4^{5}$ $10$ $4$ $15$ $( 1,11, 2,12)( 3,20, 4,19)( 5,18, 6,17)( 7,15, 8,16)( 9,13,10,14)$
5A1 $5^{4}$ $2$ $5$ $16$ $( 1, 3, 6, 7, 9)( 2, 4, 5, 8,10)(11,13,15,17,20)(12,14,16,18,19)$
5A2 $5^{4}$ $2$ $5$ $16$ $( 1, 6, 9, 3, 7)( 2, 5,10, 4, 8)(11,15,20,13,17)(12,16,19,14,18)$
10A1 $10^{2}$ $2$ $10$ $18$ $( 1,10, 7, 5, 3, 2, 9, 8, 6, 4)(11,19,17,16,13,12,20,18,15,14)$
10A3 $10^{2}$ $2$ $10$ $18$ $( 1, 5, 9, 4, 7, 2, 6,10, 3, 8)(11,16,20,14,17,12,15,19,13,18)$
10B1 $10^{2}$ $4$ $10$ $18$ $( 1,11, 3,13, 6,15, 7,17, 9,20)( 2,12, 4,14, 5,16, 8,18,10,19)$
10B3 $10^{2}$ $4$ $10$ $18$ $( 1,14, 7,19, 3,16, 9,12, 6,18)( 2,13, 8,20, 4,15,10,11, 5,17)$
10C1 $10,5^{2}$ $4$ $10$ $17$ $( 1, 6, 9, 3, 7)( 2, 5,10, 4, 8)(11,16,20,14,17,12,15,19,13,18)$
10C3 $10,5^{2}$ $4$ $10$ $17$ $( 1, 4, 6, 8, 9, 2, 3, 5, 7,10)(11,13,15,17,20)(12,14,16,18,19)$
20A1 $20$ $4$ $20$ $19$ $( 1,19,10,17, 7,16, 5,13, 3,12, 2,20, 9,18, 8,15, 6,14, 4,11)$
20A3 $20$ $4$ $20$ $19$ $( 1,17, 5,12, 9,15, 4,19, 7,13, 2,18, 6,11,10,16, 3,20, 8,14)$

Malle's constant $a(G)$:     $1/5$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 
Copy content gap:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A1 5A2 10A1 10A3 10B1 10B3 10C1 10C3 20A1 20A3
Size 1 1 2 2 5 5 10 10 2 10 2 2 2 2 4 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 5A2 5A1 5A2 5A1 5A1 5A2 5A1 5A2 10A1 10A3
5 P 1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 1A 1A 2A 2A 2B 2B 2C 2C 4A 4A
Type
80.39.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
80.39.1b R 1 1 −1 −1 −1 −1 1 1 1 −1 1 1 1 1 −1 −1 −1 −1 1 1
80.39.1c R 1 1 −1 −1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 1 1
80.39.1d R 1 1 −1 1 −1 −1 1 −1 −1 1 1 1 1 1 −1 −1 1 1 −1 −1
80.39.1e R 1 1 −1 1 1 1 −1 1 −1 −1 1 1 1 1 −1 −1 1 1 −1 −1
80.39.1f R 1 1 1 −1 −1 −1 −1 1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1
80.39.1g R 1 1 1 −1 1 1 1 −1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1
80.39.1h R 1 1 1 1 −1 −1 −1 −1 1 −1 1 1 1 1 1 1 1 1 1 1
80.39.2a R 2 −2 0 0 −2 2 0 0 0 0 2 2 −2 −2 0 0 0 0 0 0
80.39.2b R 2 −2 0 0 2 −2 0 0 0 0 2 2 −2 −2 0 0 0 0 0 0
80.39.2c1 R 2 2 2 2 0 0 0 0 2 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
80.39.2c2 R 2 2 2 2 0 0 0 0 2 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
80.39.2d1 R 2 2 −2 −2 0 0 0 0 2 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−1+ζ5 ζ5−2+ζ52
80.39.2d2 R 2 2 −2 −2 0 0 0 0 2 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2+ζ52 ζ5−1+ζ5
80.39.2e1 R 2 2 −2 2 0 0 0 0 −2 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
80.39.2e2 R 2 2 −2 2 0 0 0 0 −2 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
80.39.2f1 R 2 2 2 −2 0 0 0 0 −2 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
80.39.2f2 R 2 2 2 −2 0 0 0 0 −2 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
80.39.4a1 R 4 −4 0 0 0 0 0 0 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 0 0 0 0 0 0
80.39.4a2 R 4 −4 0 0 0 0 0 0 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 0 0 0 0 0 0

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 
Copy content gap:CharacterTable(G);
 

Regular extensions

Data not computed