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Properties

Label 20T42
20T42 1 2 1->2 18 1->18 20 1->20 17 2->17 19 2->19 3 7 3->7 14 3->14 16 3->16 4 8 4->8 13 4->13 15 4->15 5 5->3 5->13 5->18 6 6->4 6->14 6->17 9 7->9 11 7->11 7->11 10 8->10 12 8->12 8->12 9->5 9->16 9->19 10->6 10->15 10->20 11->8 11->8 11->13 12->7 12->7 12->14 13->2 13->6 13->20 14->1 14->5 14->19 15->3 15->6 16->4 16->5 17->1 17->10 17->12 18->2 18->9 18->11 19->4 19->10 19->17 20->3 20->9 20->18
Degree $20$
Order $160$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $D_4\times F_5$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(20, 42);
 
Copy content sage:G = TransitiveGroup(20, 42)
 
Copy content oscar:G = transitive_group(20, 42)
 
Copy content gap:G := TransitiveGroup(20, 42);
 

Group invariants

Abstract group:  $D_4\times F_5$
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:  $160=2^{5} \cdot 5$
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 $t$:  $42$
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:  $(1,18,2,17)(3,16,4,15)(5,13,6,14)(7,11,8,12)(9,19,10,20)$, $(1,2)(3,7,9,5)(4,8,10,6)(11,13,20,18)(12,14,19,17)$, $(1,20,3,14)(2,19,4,13)(5,18,9,16)(6,17,10,15)(7,11,8,12)$
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
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$, $Q_8:C_2$, $C_4\times C_2^2$
$20$:  $F_5$
$32$:  $C_4 \times D_4$
$40$:  $F_{5}\times C_2$ x 3
$80$:  20T16

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

20T42 x 3, 40T109 x 2, 40T110 x 2, 40T112, 40T118 x 2, 40T119 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 $2^{5},1^{10}$ $2$ $2$ $5$ $(11,12)(13,14)(15,16)(17,18)(19,20)$
2C $2^{10}$ $2$ $2$ $10$ $( 1,15)( 2,16)( 3,17)( 4,18)( 5,20)( 6,19)( 7,11)( 8,12)( 9,14)(10,13)$
2D $2^{8},1^{4}$ $5$ $2$ $8$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)(11,20)(12,19)(13,18)(14,17)$
2E $2^{10}$ $5$ $2$ $10$ $( 1, 9)( 2,10)( 3, 7)( 4, 8)( 5, 6)(11,17)(12,18)(13,16)(14,15)(19,20)$
2F $2^{9},1^{2}$ $10$ $2$ $9$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)(11,19)(12,20)(13,17)(14,18)(15,16)$
2G $2^{10}$ $10$ $2$ $10$ $( 1,18)( 2,17)( 3,16)( 4,15)( 5,13)( 6,14)( 7,11)( 8,12)( 9,19)(10,20)$
4A $4^{5}$ $2$ $4$ $15$ $( 1,15, 2,16)( 3,17, 4,18)( 5,20, 6,19)( 7,11, 8,12)( 9,14,10,13)$
4B1 $4^{4},1^{4}$ $5$ $4$ $12$ $( 3, 8, 9, 6)( 4, 7,10, 5)(11,13,20,18)(12,14,19,17)$
4B-1 $4^{4},1^{4}$ $5$ $4$ $12$ $( 3, 6, 9, 8)( 4, 5,10, 7)(11,18,20,13)(12,17,19,14)$
4C1 $4^{4},2^{2}$ $5$ $4$ $14$ $( 1, 6, 7, 3)( 2, 5, 8, 4)( 9,10)(11,17,15,19)(12,18,16,20)(13,14)$
4C-1 $4^{4},2^{2}$ $5$ $4$ $14$ $( 1, 6, 4, 9)( 2, 5, 3,10)( 7, 8)(11,12)(13,16,20,17)(14,15,19,18)$
4D $4^{5}$ $10$ $4$ $15$ $( 1,18, 2,17)( 3,16, 4,15)( 5,13, 6,14)( 7,11, 8,12)( 9,19,10,20)$
4E1 $4^{5}$ $10$ $4$ $15$ $( 1,18, 9,12)( 2,17,10,11)( 3,14, 7,15)( 4,13, 8,16)( 5,20, 6,19)$
4E-1 $4^{5}$ $10$ $4$ $15$ $( 1,18, 8,19)( 2,17, 7,20)( 3,12, 5,15)( 4,11, 6,16)( 9,14,10,13)$
4F1 $4^{4},2,1^{2}$ $10$ $4$ $13$ $( 3, 8, 9, 6)( 4, 7,10, 5)(11,14,20,17)(12,13,19,18)(15,16)$
4F-1 $4^{4},2,1^{2}$ $10$ $4$ $13$ $( 3, 6, 9, 8)( 4, 5,10, 7)(11,17,20,14)(12,18,19,13)(15,16)$
4G1 $4^{4},2^{2}$ $10$ $4$ $14$ $( 1,18,10,11)( 2,17, 9,12)( 3,14, 8,16)( 4,13, 7,15)( 5,20)( 6,19)$
4G-1 $4^{4},2^{2}$ $10$ $4$ $14$ $( 1,18, 7,20)( 2,17, 8,19)( 3,12, 6,16)( 4,11, 5,15)( 9,14)(10,13)$
5A $5^{4}$ $4$ $5$ $16$ $( 1, 7, 4,10, 5)( 2, 8, 3, 9, 6)(11,18,13,20,15)(12,17,14,19,16)$
10A $10^{2}$ $4$ $10$ $18$ $( 1, 6,10, 3, 7, 2, 5, 9, 4, 8)(11,16,20,14,18,12,15,19,13,17)$
10B $10,5^{2}$ $8$ $10$ $17$ $( 1,10, 7, 5, 4)( 2, 9, 8, 6, 3)(11,19,18,16,13,12,20,17,15,14)$
10C $10^{2}$ $8$ $10$ $18$ $( 1,18, 5,11,10,15, 4,20, 7,13)( 2,17, 6,12, 9,16, 3,19, 8,14)$
20A $20$ $8$ $20$ $19$ $( 1,18, 6,12,10,15, 3,19, 7,13, 2,17, 5,11, 9,16, 4,20, 8,14)$

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 4B1 4B-1 4C1 4C-1 4D 4E1 4E-1 4F1 4F-1 4G1 4G-1 5A 10A 10B 10C 20A
Size 1 1 2 2 5 5 10 10 2 5 5 5 5 10 10 10 10 10 10 10 4 4 8 8 8
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2A 2D 2D 2D 2D 2A 2E 2E 2D 2D 2D 2D 5A 5A 5A 5A 10A
5 P 1A 2A 2B 2C 2D 2E 2F 2G 4A 4B1 4B-1 4C1 4C-1 4D 4E1 4E-1 4F1 4F-1 4G1 4G-1 1A 2A 2B 2C 4A
Type
160.207.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
160.207.1b R 1 1 −1 −1 1 1 −1 −1 1 −1 −1 −1 −1 1 −1 −1 1 1 1 1 1 1 −1 −1 1
160.207.1c R 1 1 −1 −1 1 1 −1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 1
160.207.1d R 1 1 −1 1 1 1 −1 1 −1 −1 −1 −1 −1 −1 1 1 1 1 −1 −1 1 1 −1 1 −1
160.207.1e R 1 1 −1 1 1 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 −1 1 1 1 1 −1 1 −1
160.207.1f R 1 1 1 −1 1 1 1 −1 −1 −1 −1 −1 −1 −1 1 1 −1 −1 1 1 1 1 1 −1 −1
160.207.1g R 1 1 1 −1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 1 1 −1 −1 1 1 1 −1 −1
160.207.1h R 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 −1 −1 −1 −1 −1 −1 1 1 1 1 1
160.207.1i1 C 1 1 −1 −1 −1 −1 1 1 1 −i i −i i −1 −i i i −i i −i 1 1 −1 −1 1
160.207.1i2 C 1 1 −1 −1 −1 −1 1 1 1 i −i i −i −1 i −i −i i −i i 1 1 −1 −1 1
160.207.1j1 C 1 1 −1 1 −1 −1 1 −1 −1 −i i −i i 1 i −i i −i −i i 1 1 −1 1 −1
160.207.1j2 C 1 1 −1 1 −1 −1 1 −1 −1 i −i i −i 1 −i i −i i i −i 1 1 −1 1 −1
160.207.1k1 C 1 1 1 −1 −1 −1 −1 1 −1 −i i −i i 1 i −i −i i i −i 1 1 1 −1 −1
160.207.1k2 C 1 1 1 −1 −1 −1 −1 1 −1 i −i i −i 1 −i i i −i −i i 1 1 1 −1 −1
160.207.1l1 C 1 1 1 1 −1 −1 −1 −1 1 −i i −i i −1 −i i −i i −i i 1 1 1 1 1
160.207.1l2 C 1 1 1 1 −1 −1 −1 −1 1 i −i i −i −1 i −i i −i i −i 1 1 1 1 1
160.207.2a R 2 −2 0 0 2 −2 0 0 0 −2 −2 2 2 0 0 0 0 0 0 0 2 −2 0 0 0
160.207.2b R 2 −2 0 0 2 −2 0 0 0 2 2 −2 −2 0 0 0 0 0 0 0 2 −2 0 0 0
160.207.2c1 C 2 −2 0 0 −2 2 0 0 0 −2i 2i 2i −2i 0 0 0 0 0 0 0 2 −2 0 0 0
160.207.2c2 C 2 −2 0 0 −2 2 0 0 0 2i −2i −2i 2i 0 0 0 0 0 0 0 2 −2 0 0 0
160.207.4a R 4 4 4 4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 −1 −1 −1 −1 −1
160.207.4b R 4 4 −4 −4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 −1 −1 1 1 −1
160.207.4c R 4 4 −4 4 0 0 0 0 −4 0 0 0 0 0 0 0 0 0 0 0 −1 −1 1 −1 1
160.207.4d R 4 4 4 −4 0 0 0 0 −4 0 0 0 0 0 0 0 0 0 0 0 −1 −1 −1 1 1
160.207.8a R 8 −8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −2 2 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