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Properties

Label 20T18
20T18 1 8 1->8 15 1->15 2 7 2->7 16 2->16 3 3->2 17 3->17 4 4->1 18 4->18 5 5->15 19 5->19 6 6->16 20 6->20 7->1 9 7->9 8->2 10 8->10 9->3 9->3 10->4 10->4 11 11->5 11->18 12 12->6 12->17 13 13->7 13->11 14 14->8 14->12 15->9 16->10 17->12 17->19 18->11 18->20 19->13 19->14 20->13 20->14
Degree $20$
Order $80$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_{20}:C_4$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(20, 18);
 
Copy content sage:G = TransitiveGroup(20, 18)
 
Copy content oscar:G = transitive_group(20, 18)
 
Copy content gap:G := TransitiveGroup(20, 18);
 

Group invariants

Abstract group:  $C_{20}:C_4$
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 $t$:  $18$
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:  $(1,15,9,3,17,12,6,20,14,8,2,16,10,4,18,11,5,19,13,7)$, $(1,8,10,4)(2,7,9,3)(5,15)(6,16)(11,18,20,13)(12,17,19,14)$
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 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$, $C_4\times C_2$, $Q_8$
$16$:  $C_4:C_4$
$20$:  $F_5$
$40$:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $F_5$

Degree 10: $F_{5}\times C_2$

Low degree siblings

20T18, 40T52, 40T54

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 $2^{10}$ $5$ $2$ $10$ $( 1, 9)( 2,10)( 3, 8)( 4, 7)( 5, 6)(11,19)(12,20)(13,17)(14,18)(15,16)$
2C $2^{8},1^{4}$ $5$ $2$ $8$ $( 1,14)( 2,13)( 3,11)( 4,12)( 5,10)( 6, 9)(15,19)(16,20)$
4A $4^{5}$ $2$ $4$ $15$ $( 1,12, 2,11)( 3,14, 4,13)( 5,15, 6,16)( 7,17, 8,18)( 9,20,10,19)$
4B $4^{5}$ $10$ $4$ $15$ $( 1, 3, 2, 4)( 5,20, 6,19)( 7,18, 8,17)( 9,15,10,16)(11,13,12,14)$
4C1 $4^{4},2,1^{2}$ $10$ $4$ $13$ $( 1, 9,14, 6)( 2,10,13, 5)( 3,16,11,20)( 4,15,12,19)(17,18)$
4C-1 $4^{4},2,1^{2}$ $10$ $4$ $13$ $( 1, 6,14, 9)( 2, 5,13,10)( 3,20,11,16)( 4,19,12,15)(17,18)$
4D1 $4^{4},2^{2}$ $10$ $4$ $14$ $( 1,20, 5, 7)( 2,19, 6, 8)( 3,14)( 4,13)( 9,15,18,12)(10,16,17,11)$
4D-1 $4^{4},2^{2}$ $10$ $4$ $14$ $( 1, 7, 5,20)( 2, 8, 6,19)( 3,14)( 4,13)( 9,12,18,15)(10,11,17,16)$
5A $5^{4}$ $4$ $5$ $16$ $( 1,17,14,10, 5)( 2,18,13, 9, 6)( 3,20,16,11, 7)( 4,19,15,12, 8)$
10A $10^{2}$ $4$ $10$ $18$ $( 1, 9,17, 6,14, 2,10,18, 5,13)( 3,12,20, 8,16, 4,11,19, 7,15)$
20A1 $20$ $4$ $20$ $19$ $( 1,15, 9, 3,17,12, 6,20,14, 8, 2,16,10, 4,18,11, 5,19,13, 7)$
20A-1 $20$ $4$ $20$ $19$ $( 1, 4, 6, 7,10,12,13,16,17,19, 2, 3, 5, 8, 9,11,14,15,18,20)$

Malle's constant $a(G)$:     $1/8$

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 4A 4B 4C1 4C-1 4D1 4D-1 5A 10A 20A1 20A-1
Size 1 1 5 5 2 10 10 10 10 10 4 4 4 4
2 P 1A 1A 1A 1A 2A 2A 2C 2C 2C 2C 5A 5A 10A 10A
5 P 1A 2A 2B 2C 4A 4B 4C1 4C-1 4D1 4D-1 1A 2A 4A 4A
Type
80.31.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
80.31.1b R 1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1
80.31.1c R 1 1 1 1 −1 1 1 −1 −1 −1 1 1 −1 −1
80.31.1d R 1 1 1 1 1 −1 −1 1 −1 −1 1 1 1 1
80.31.1e1 C 1 1 −1 −1 −1 −i i 1 −i i 1 1 −1 −1
80.31.1e2 C 1 1 −1 −1 −1 i −i 1 i −i 1 1 −1 −1
80.31.1f1 C 1 1 −1 −1 1 −i i −1 i −i 1 1 1 1
80.31.1f2 C 1 1 −1 −1 1 i −i −1 −i i 1 1 1 1
80.31.2a R 2 −2 −2 2 0 0 0 0 0 0 2 −2 0 0
80.31.2b S 2 −2 2 −2 0 0 0 0 0 0 2 −2 0 0
80.31.4a R 4 4 0 0 4 0 0 0 0 0 −1 −1 −1 −1
80.31.4b R 4 4 0 0 −4 0 0 0 0 0 −1 −1 1 1
80.31.4c1 C 4 −4 0 0 0 0 0 0 0 0 −1 1 −2ζ203+ζ205−2ζ207 2ζ203−ζ205+2ζ207
80.31.4c2 C 4 −4 0 0 0 0 0 0 0 0 −1 1 2ζ203−ζ205+2ζ207 −2ζ203+ζ205−2ζ207

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