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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

|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

Cycle 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:    
      2  4  3  3  4  4  4  3   2  3  3   2  2   2  3
      5  1  .  .  .  1  .  .   1  .  .   1  1   1  1

        1a 4a 4b 2a 2b 2c 2d 10a 4c 4d 10b 5a 10c 2e
     2P 1a 2a 2a 1a 1a 1a 1a  5a 2c 2c  5a 5a  5a 1a
     3P 1a 4b 4a 2a 2b 2c 2d 10b 4d 4c 10a 5a 10c 2e
     5P 1a 4a 4b 2a 2b 2c 2d  2e 4c 4d  2e 1a  2b 2e
     7P 1a 4b 4a 2a 2b 2c 2d 10b 4d 4c 10a 5a 10c 2e

X.1      1  1  1  1  1  1  1   1  1  1   1  1   1  1
X.2      1 -1 -1  1  1  1 -1  -1  1  1  -1  1   1 -1
X.3      1 -1 -1  1  1  1  1   1 -1 -1   1  1   1  1
X.4      1  1  1  1  1  1 -1  -1 -1 -1  -1  1   1 -1
X.5      1  A -A -1  1 -1 -1   1  A -A   1  1   1  1
X.6      1 -A  A -1  1 -1 -1   1 -A  A   1  1   1  1
X.7      1  A -A -1  1 -1  1  -1 -A  A  -1  1   1 -1
X.8      1 -A  A -1  1 -1  1  -1  A -A  -1  1   1 -1
X.9      2  .  . -2 -2  2  .   .  .  .   .  2  -2  .
X.10     2  .  .  2 -2 -2  .   .  .  .   .  2  -2  .
X.11     4  .  .  .  4  .  .  -1  .  .  -1 -1  -1  4
X.12     4  .  .  .  4  .  .   1  .  .   1 -1  -1 -4
X.13     4  .  .  . -4  .  .   B  .  .  -B -1   1  .
X.14     4  .  .  . -4  .  .  -B  .  .   B -1   1  .

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(5)+E(5)^2+E(5)^3-E(5)^4
  = -Sqrt(5) = -r5

magma: CharacterTable(G);