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Properties

Label 20T20
20T20 1 2 1->2 7 1->7 12 1->12 8 2->8 11 2->11 3 5 3->5 3->8 13 3->13 4 6 4->6 4->7 14 4->14 5->13 19 5->19 20 5->20 6->14 6->19 6->20 7->6 7->14 7->19 8->5 8->13 8->20 9 9->5 9->7 9->12 10 10->6 10->8 10->11 11->1 11->12 18 11->18 12->2 17 12->17 13->4 16 13->16 13->17 14->3 15 14->15 14->18 15->3 15->9 15->10 16->4 16->9 16->10 17->4 17->9 17->16 18->3 18->10 18->15 19->1 19->16 19->18 20->2 20->15 20->17
Degree $20$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_4\times F_5$

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

Group invariants

Abstract group:  $C_4\times F_5$
Copy content magma:IdentifyGroup(G);
 
Order:  $80=2^{4} \cdot 5$
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 $t$:  $20$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,12,2,11)(3,5,19,18)(4,6,20,17)(7,14,15,9)(8,13,16,10)$, $(1,7,6,19)(2,8,5,20)(3,13,4,14)(9,12,17,16)(10,11,18,15)$, $(1,2)(3,8,20,15)(4,7,19,16)(5,13,17,9)(6,14,18,10)(11,12)$
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 6, $C_2^2$
$8$:  $C_4\times C_2$ x 3
$16$:  $C_4^2$
$20$:  $F_5$
$40$:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $F_5$

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

Low degree siblings

20T20 x 3, 40T53 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 $2^{10}$ $5$ $2$ $10$ $( 1, 2)( 3,19)( 4,20)( 5,18)( 6,17)( 7,15)( 8,16)( 9,14)(10,13)(11,12)$
2C $2^{8},1^{4}$ $5$ $2$ $8$ $( 1,17)( 2,18)( 3,16)( 4,15)( 5,14)( 6,13)( 7,11)( 8,12)$
4A1 $4^{5}$ $1$ $4$ $15$ $( 1,11, 2,12)( 3,13, 4,14)( 5,16, 6,15)( 7,18, 8,17)( 9,19,10,20)$
4A-1 $4^{5}$ $1$ $4$ $15$ $( 1,12, 2,11)( 3,14, 4,13)( 5,15, 6,16)( 7,17, 8,18)( 9,20,10,19)$
4B1 $4^{5}$ $5$ $4$ $15$ $( 1,11, 2,12)( 3,18,19, 5)( 4,17,20, 6)( 7, 9,15,14)( 8,10,16,13)$
4B-1 $4^{5}$ $5$ $4$ $15$ $( 1,12, 2,11)( 3, 5,19,18)( 4, 6,20,17)( 7,14,15, 9)( 8,13,16,10)$
4C1 $4^{5}$ $5$ $4$ $15$ $( 1,15, 2,16)( 3,14, 4,13)( 5,12, 6,11)( 7,10, 8, 9)(17,19,18,20)$
4C-1 $4^{5}$ $5$ $4$ $15$ $( 1, 3, 2, 4)( 5,20, 6,19)( 7,18, 8,17)( 9,15,10,16)(11,13,12,14)$
4D1 $4^{5}$ $5$ $4$ $15$ $( 1,19,13,16)( 2,20,14,15)( 3, 5,12, 9)( 4, 6,11,10)( 7,17, 8,18)$
4D-1 $4^{5}$ $5$ $4$ $15$ $( 1,20, 6, 8)( 2,19, 5, 7)( 3,13, 4,14)( 9,15,17,11)(10,16,18,12)$
4E1 $4^{4},2^{2}$ $5$ $4$ $14$ $( 1, 6,17,13)( 2, 5,18,14)( 3,12,16, 8)( 4,11,15, 7)( 9,10)(19,20)$
4E-1 $4^{4},2^{2}$ $5$ $4$ $14$ $( 1,13,17, 6)( 2,14,18, 5)( 3, 8,16,12)( 4, 7,15,11)( 9,10)(19,20)$
4F1 $4^{4},1^{4}$ $5$ $4$ $12$ $( 1,14,10,17)( 2,13, 9,18)( 3,20, 7,11)( 4,19, 8,12)$
4F-1 $4^{4},1^{4}$ $5$ $4$ $12$ $( 1, 5,14,10)( 2, 6,13, 9)( 3,20,11,16)( 4,19,12,15)$
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 $20$ $4$ $20$ $19$ $( 1,16, 9, 4,17,11, 6,19,14, 7, 2,15,10, 3,18,12, 5,20,13, 8)$
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 magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 4C1 4C-1 4D1 4D-1 4E1 4E-1 4F1 4F-1 5A 10A 20A1 20A-1
Size 1 1 5 5 1 1 5 5 5 5 5 5 5 5 5 5 4 4 4 4
2 P 1A 1A 1A 1A 2A 2A 2B 2B 2A 2A 2B 2B 2C 2C 2C 2C 5A 5A 10A 10A
5 P 1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 4C1 4C-1 4D1 4D-1 4E1 4E-1 4F1 4F-1 1A 2A 4A1 4A-1
Type
80.30.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
80.30.1b R 1 1 1 1 −1 −1 −1 −1 1 1 −1 1 −1 1 −1 −1 1 1 −1 −1
80.30.1c R 1 1 1 1 −1 −1 1 1 −1 −1 1 −1 −1 −1 −1 1 1 1 −1 −1
80.30.1d R 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 −1 1 −1 1 1 1 1
80.30.1e1 C 1 −1 −1 1 −i i −i i −1 −1 i 1 −i 1 i −i 1 −1 −i i
80.30.1e2 C 1 −1 −1 1 i −i i −i −1 −1 −i 1 i 1 −i i 1 −1 i −i
80.30.1f1 C 1 −1 −1 1 −i i i −i 1 1 −i −1 −i −1 i i 1 −1 −i i
80.30.1f2 C 1 −1 −1 1 i −i −i i 1 1 i −1 i −1 −i −i 1 −1 i −i
80.30.1g1 C 1 −1 1 −1 −i i −1 −1 i −i 1 −i i i −i 1 1 −1 −i i
80.30.1g2 C 1 −1 1 −1 i −i −1 −1 −i i 1 i −i −i i 1 1 −1 i −i
80.30.1h1 C 1 −1 1 −1 −i i 1 1 −i i −1 i i −i −i −1 1 −1 −i i
80.30.1h2 C 1 −1 1 −1 i −i 1 1 i −i −1 −i −i i i −1 1 −1 i −i
80.30.1i1 C 1 1 −1 −1 −1 −1 −i i i −i −i i 1 −i 1 i 1 1 −1 −1
80.30.1i2 C 1 1 −1 −1 −1 −1 i −i −i i i −i 1 i 1 −i 1 1 −1 −1
80.30.1j1 C 1 1 −1 −1 1 1 −i i −i i −i −i −1 i −1 i 1 1 1 1
80.30.1j2 C 1 1 −1 −1 1 1 i −i i −i i i −1 −i −1 −i 1 1 1 1
80.30.4a R 4 4 0 0 4 4 0 0 0 0 0 0 0 0 0 0 −1 −1 −1 −1
80.30.4b R 4 4 0 0 −4 −4 0 0 0 0 0 0 0 0 0 0 −1 −1 1 1
80.30.4c1 C 4 −4 0 0 −4i 4i 0 0 0 0 0 0 0 0 0 0 −1 1 i −i
80.30.4c2 C 4 −4 0 0 4i −4i 0 0 0 0 0 0 0 0 0 0 −1 1 −i i

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed