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Properties

Label 20T19
Degree $20$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{10}:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(20, 19);
 

Group action invariants

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

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

20T19, 20T22 x 2, 40T26, 40T45, 40T55 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 $2^{10}$ $2$ $2$ $10$ $( 1,12)( 2,11)( 3,13)( 4,14)( 5,15)( 6,16)( 7,18)( 8,17)( 9,20)(10,19)$
2C $2^{10}$ $5$ $2$ $10$ $( 1, 9)( 2,10)( 3, 8)( 4, 7)( 5, 6)(11,19)(12,20)(13,17)(14,18)(15,16)$
2D $2^{8},1^{4}$ $5$ $2$ $8$ $( 1,10)( 2, 9)( 3, 7)( 4, 8)(11,20)(12,19)(13,18)(14,17)$
2E $2^{10}$ $10$ $2$ $10$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,20)(10,19)(11,18)(12,17)(13,16)(14,15)$
4A1 $4^{4},2,1^{2}$ $10$ $4$ $13$ $( 1,13,10,18)( 2,14, 9,17)( 3,20, 7,11)( 4,19, 8,12)( 5, 6)$
4A-1 $4^{4},2,1^{2}$ $10$ $4$ $13$ $( 1,18,10,13)( 2,17, 9,14)( 3,11, 7,20)( 4,12, 8,19)( 5, 6)$
4B1 $4^{5}$ $10$ $4$ $15$ $( 1,12, 2,11)( 3, 5,19,18)( 4, 6,20,17)( 7,14,15, 9)( 8,13,16,10)$
4B-1 $4^{5}$ $10$ $4$ $15$ $( 1,19,13,16)( 2,20,14,15)( 3, 5,12, 9)( 4, 6,11,10)( 7,17, 8,18)$
5A $5^{4}$ $4$ $5$ $16$ $( 1, 5,10,14,17)( 2, 6, 9,13,18)( 3, 7,11,16,20)( 4, 8,12,15,19)$
10A $10^{2}$ $4$ $10$ $18$ $( 1, 6,10,13,17, 2, 5, 9,14,18)( 3, 8,11,15,20, 4, 7,12,16,19)$
10B1 $10^{2}$ $4$ $10$ $18$ $( 1, 4, 5, 8,10,12,14,15,17,19)( 2, 3, 6, 7, 9,11,13,16,18,20)$
10B3 $10^{2}$ $4$ $10$ $18$ $( 1, 8,14,19, 5,12,17, 4,10,15)( 2, 7,13,20, 6,11,18, 3, 9,16)$

Malle's constant $a(G)$:     $1/8$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $80=2^{4} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  80.34
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 4A1 4A-1 4B1 4B-1 5A 10A 10B1 10B3
Size 1 1 2 5 5 10 10 10 10 10 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 2D 2D 2C 2C 5A 5A 5A 5A
5 P 1A 2A 2B 2C 2D 2E 4B-1 4B1 4A-1 4A1 1A 2A 2B 2B
Type
80.34.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
80.34.1b R 1 1 −1 1 1 −1 −1 −1 1 1 1 1 −1 −1
80.34.1c R 1 1 −1 1 1 −1 1 1 −1 −1 1 1 −1 −1
80.34.1d R 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1
80.34.1e1 C 1 1 −1 −1 −1 1 −i i −i i 1 1 −1 −1
80.34.1e2 C 1 1 −1 −1 −1 1 i −i i −i 1 1 −1 −1
80.34.1f1 C 1 1 1 −1 −1 −1 −i i i −i 1 1 1 1
80.34.1f2 C 1 1 1 −1 −1 −1 i −i −i i 1 1 1 1
80.34.2a R 2 −2 0 −2 2 0 0 0 0 0 2 −2 0 0
80.34.2b R 2 −2 0 2 −2 0 0 0 0 0 2 −2 0 0
80.34.4a R 4 4 4 0 0 0 0 0 0 0 −1 −1 −1 −1
80.34.4b R 4 4 −4 0 0 0 0 0 0 0 −1 −1 1 1
80.34.4c1 R 4 −4 0 0 0 0 0 0 0 0 −1 1 2ζ5−2+1+2ζ52 −2ζ5−2−1−2ζ52
80.34.4c2 R 4 −4 0 0 0 0 0 0 0 0 −1 1 −2ζ5−2−1−2ζ52 2ζ5−2+1+2ζ52

magma: CharacterTable(G);