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

|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 TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 4, 4, 4, 4, 2, 1, 1 $ $10$ $4$ $( 3, 8,20,15)( 4, 7,19,16)( 5,14,17,10)( 6,13,18, 9)(11,12)$
$ 4, 4, 4, 4, 2, 1, 1 $ $10$ $4$ $( 3,15,20, 8)( 4,16,19, 7)( 5,10,17,14)( 6, 9,18,13)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 3,20)( 4,19)( 5,17)( 6,18)( 7,16)( 8,15)( 9,13)(10,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3,19)( 4,20)( 5,18)( 6,17)( 7,15)( 8,16)( 9,14)(10,13)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 3)( 2, 4)( 5,20)( 6,19)( 7,17)( 8,18)( 9,15)(10,16)(11,14)(12,13)$
$ 10, 10 $ $4$ $10$ $( 1, 3, 5, 7,10,11,14,16,17,20)( 2, 4, 6, 8, 9,12,13,15,18,19)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 3, 9, 8)( 2, 4,10, 7)( 5,16, 6,15)(11,13,19,17)(12,14,20,18)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 3,18,15)( 2, 4,17,16)( 5,11,13, 8)( 6,12,14, 7)( 9,19,10,20)$
$ 10, 10 $ $4$ $10$ $( 1, 4, 5, 8,10,12,14,15,17,19)( 2, 3, 6, 7, 9,11,13,16,18,20)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1, 5,10,14,17)( 2, 6, 9,13,18)( 3, 7,11,16,20)( 4, 8,12,15,19)$
$ 10, 10 $ $4$ $10$ $( 1, 6,10,13,17, 2, 5, 9,14,18)( 3, 8,11,15,20, 4, 7,12,16,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1,11)( 2,12)( 3,14)( 4,13)( 5,16)( 6,15)( 7,17)( 8,18)( 9,19)(10,20)$

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 2ζ52+1+2ζ52 2ζ5212ζ52
80.34.4c2 R 4 4 0 0 0 0 0 0 0 0 1 1 2ζ5212ζ52 2ζ52+1+2ζ52

magma: CharacterTable(G);