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Properties

Label 20T134
20T134 1 13 1->13 19 1->19 2 14 2->14 20 2->20 3 7 3->7 11 3->11 4 8 4->8 12 4->12 5 16 5->16 5->16 6 15 6->15 6->15 7->2 7->4 8->1 8->3 9 9->13 9->19 10 10->14 10->20 11->6 11->9 12->5 12->10 13->5 14->6 15->2 15->3 16->1 16->4 17 17->7 17->12 18 18->8 18->11 19->17 19->18 20->17 20->18
Degree $20$
Order $640$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\wr F_5$

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Copy content magma:G := TransitiveGroup(20, 134);
 

Group invariants

Abstract group:  $C_2\wr F_5$
Copy content magma:IdentifyGroup(G);
 
Order:  $640=2^{7} \cdot 5$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $20$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $134$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,19,18,8)(2,20,17,7)(3,11,6,15)(4,12,5,16)(9,13)(10,14)$, $(1,13,5,16)(2,14,6,15)(3,7,4,8)(9,19,17,12,10,20,18,11)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$20$:  $F_5$
$40$:  $F_{5}\times C_2$
$320$:  $(C_2^4 : C_5):C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: $F_5$

Degree 10: $F_5$

Low degree siblings

10T29 x 2, 20T129, 20T131 x 2, 20T132, 20T133, 20T135, 20T137 x 2, 20T140, 32T34608 x 2, 40T460, 40T462, 40T473, 40T474, 40T475, 40T476, 40T487, 40T488, 40T489, 40T490, 40T557, 40T558 x 2, 40T561, 40T562, 40T563, 40T564, 40T565, 40T566, 40T567 x 2, 40T576, 40T577, 40T578, 40T579, 40T586

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 $2^{6},1^{8}$ $5$ $2$ $6$ $( 1, 2)( 3, 4)( 9,10)(11,12)(15,16)(19,20)$
2C $2^{4},1^{12}$ $5$ $2$ $4$ $( 3, 4)( 5, 6)(11,12)(15,16)$
2D $2^{4},1^{12}$ $10$ $2$ $4$ $( 3, 4)( 7, 8)( 9,10)(17,18)$
2E $2^{6},1^{8}$ $10$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(17,18)(19,20)$
2F $2^{10}$ $20$ $2$ $10$ $( 1,17)( 2,18)( 3, 5)( 4, 6)( 7,19)( 8,20)( 9,10)(11,16)(12,15)(13,14)$
2G $2^{8},1^{4}$ $20$ $2$ $8$ $( 1,17)( 2,18)( 3, 6)( 4, 5)( 7,19)( 8,20)(11,16)(12,15)$
4A $4^{2},2^{4},1^{4}$ $20$ $4$ $10$ $( 1,17)( 2,18)( 3, 5, 4, 6)( 7,20)( 8,19)(11,15,12,16)$
4B $4^{2},2^{6}$ $20$ $4$ $12$ $( 1, 3, 2, 4)( 5, 9)( 6,10)( 7,11)( 8,12)(13,19,14,20)(15,16)(17,18)$
4C $4^{3},2^{3},1^{2}$ $40$ $4$ $12$ $( 1,17, 2,18)( 3, 6, 4, 5)( 7,20, 8,19)( 9,10)(11,15)(12,16)$
4D $4^{3},2^{3},1^{2}$ $40$ $4$ $12$ $( 1,17, 2,18)( 3, 5, 4, 6)( 7,20, 8,19)(11,15)(12,16)(13,14)$
4E1 $4^{4},2^{2}$ $40$ $4$ $14$ $( 1, 8,17,20)( 2, 7,18,19)( 3,15, 6,12)( 4,16, 5,11)( 9,14)(10,13)$
4E-1 $4^{4},2^{2}$ $40$ $4$ $14$ $( 1,20,17, 8)( 2,19,18, 7)( 3,12, 6,15)( 4,11, 5,16)( 9,14)(10,13)$
4F1 $4^{4},2^{2}$ $40$ $4$ $14$ $( 1, 8,18,19)( 2, 7,17,20)( 3,15, 5,12)( 4,16, 6,11)( 9,14)(10,13)$
4F-1 $4^{4},2^{2}$ $40$ $4$ $14$ $( 1,19,18, 8)( 2,20,17, 7)( 3,12, 5,15)( 4,11, 6,16)( 9,14)(10,13)$
5A $5^{4}$ $64$ $5$ $16$ $( 1, 9,18, 6, 4)( 2,10,17, 5, 3)( 7,11,13,16,20)( 8,12,14,15,19)$
8A1 $8,4^{3}$ $40$ $8$ $16$ $( 1,14, 3,20, 2,13, 4,19)( 5,12, 9, 8)( 6,11,10, 7)(15,18,16,17)$
8A-1 $8,4^{3}$ $40$ $8$ $16$ $( 1,19, 4,13, 2,20, 3,14)( 5, 8, 9,12)( 6, 7,10,11)(15,17,16,18)$
8B1 $8,4^{3}$ $40$ $8$ $16$ $( 1,19, 4,13, 2,20, 3,14)( 5, 7, 9,11)( 6, 8,10,12)(15,18,16,17)$
8B-1 $8,4^{3}$ $40$ $8$ $16$ $( 1,14, 3,20, 2,13, 4,19)( 5,11, 9, 7)( 6,12,10, 8)(15,17,16,18)$
10A $10^{2}$ $64$ $10$ $18$ $( 1, 5, 9, 3,18, 2, 6,10, 4,17)( 7,15,11,19,13, 8,16,12,20,14)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E1 4E-1 4F1 4F-1 5A 8A1 8A-1 8B1 8B-1 10A
Size 1 1 5 5 10 10 20 20 20 20 40 40 40 40 40 40 64 40 40 40 40 64
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2C 2C 2E 2E 2G 2G 2G 2G 5A 4B 4B 4B 4B 5A
5 P 1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E1 4E-1 4F1 4F-1 1A 8A1 8A-1 8B1 8B-1 2A
Type
640.21536.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
640.21536.1b R 1 −1 −1 1 −1 1 −1 1 −1 1 1 −1 1 −1 −1 1 1 −1 −1 1 1 −1
640.21536.1c R 1 −1 −1 1 −1 1 −1 1 −1 1 1 1 −1 −1 1 −1 1 1 1 −1 −1 −1
640.21536.1d R 1 1 1 1 1 1 1 1 1 1 1 −1 −1 1 −1 −1 1 −1 −1 −1 −1 1
640.21536.1e1 C 1 −1 −1 1 −1 1 1 −1 1 −1 −1 −i i 1 i −i 1 −i i −i i −1
640.21536.1e2 C 1 −1 −1 1 −1 1 1 −1 1 −1 −1 i −i 1 −i i 1 i −i i −i −1
640.21536.1f1 C 1 1 1 1 1 1 −1 −1 −1 −1 −1 −i −i −1 i i 1 −i i i −i 1
640.21536.1f2 C 1 1 1 1 1 1 −1 −1 −1 −1 −1 i i −1 −i −i 1 i −i −i i 1
640.21536.4a R 4 4 4 4 4 4 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 −1
640.21536.4b R 4 −4 −4 4 −4 4 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 1
640.21536.5a R 5 5 −3 −3 1 1 1 1 1 1 −1 1 1 −1 1 1 0 −1 −1 −1 −1 0
640.21536.5b R 5 5 −3 −3 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 0 1 1 1 1 0
640.21536.5c R 5 −5 3 −3 −1 1 −1 1 −1 1 −1 −1 1 1 −1 1 0 1 1 −1 −1 0
640.21536.5d R 5 −5 3 −3 −1 1 −1 1 −1 1 −1 1 −1 1 1 −1 0 −1 −1 1 1 0
640.21536.5e1 C 5 5 −3 −3 1 1 −1 −1 −1 −1 1 −i −i 1 i i 0 i −i −i i 0
640.21536.5e2 C 5 5 −3 −3 1 1 −1 −1 −1 −1 1 i i 1 −i −i 0 −i i i −i 0
640.21536.5f1 C 5 −5 3 −3 −1 1 1 −1 1 −1 1 −i i −1 i −i 0 i −i i −i 0
640.21536.5f2 C 5 −5 3 −3 −1 1 1 −1 1 −1 1 i −i −1 −i i 0 −i i −i i 0
640.21536.10a R 10 10 2 2 −2 −2 2 2 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0
640.21536.10b R 10 10 2 2 −2 −2 −2 −2 2 2 0 0 0 0 0 0 0 0 0 0 0 0
640.21536.10c R 10 −10 −2 2 2 −2 −2 2 2 −2 0 0 0 0 0 0 0 0 0 0 0 0
640.21536.10d R 10 −10 −2 2 2 −2 2 −2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed