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Properties

Label 20T22
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, 22);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $22$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{10}:C_4$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
Nilpotency class:  $-1$ (not nilpotent)
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,14,5,12)(2,13,6,11)(3,17,4,18)(7,15,9,19)(8,16,10,20), (1,5,10,3,7,2,6,9,4,8)(11,15,20,14,18)(12,16,19,13,17)
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: $F_5$

Low degree siblings

20T19 x 2, 20T22, 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $10$ $2$ $( 3, 9)( 4,10)( 5, 8)( 6, 7)(11,19)(12,20)(13,18)(14,17)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 3, 9)( 4,10)( 5, 8)( 6, 7)(11,20)(12,19)(13,17)(14,18)$
$ 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 $ $5$ $2$ $( 1, 2)( 3,10)( 4, 9)( 5, 7)( 6, 8)(11,19)(12,20)(13,18)(14,17)(15,16)$
$ 10, 10 $ $4$ $10$ $( 1, 3, 6, 8,10, 2, 4, 5, 7, 9)(11,13,15,17,20,12,14,16,18,19)$
$ 10, 5, 5 $ $4$ $10$ $( 1, 3, 6, 8,10, 2, 4, 5, 7, 9)(11,14,15,18,20)(12,13,16,17,19)$
$ 10, 5, 5 $ $4$ $10$ $( 1, 4, 6, 7,10)( 2, 3, 5, 8, 9)(11,13,15,17,20,12,14,16,18,19)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1, 4, 6, 7,10)( 2, 3, 5, 8, 9)(11,14,15,18,20)(12,13,16,17,19)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1,11, 9,17)( 2,12,10,18)( 3,16, 7,14)( 4,15, 8,13)( 5,19, 6,20)$
$ 4, 4, 4, 4, 2, 2 $ $10$ $4$ $( 1,11,10,18)( 2,12, 9,17)( 3,16, 8,13)( 4,15, 7,14)( 5,19)( 6,20)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1,11, 5,13)( 2,12, 6,14)( 3,17, 4,18)( 7,20, 9,16)( 8,19,10,15)$
$ 4, 4, 4, 4, 2, 2 $ $10$ $4$ $( 1,11, 6,14)( 2,12, 5,13)( 3,17)( 4,18)( 7,20,10,15)( 8,19, 9,16)$

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);
 
Label:  80.34
magma: IdentifyGroup(G);
 
Character table:    
      2  4  3  3  4  4  4   2   2   2  2  3  3  3  3
      5  1  1  .  .  1  .   1   1   1  1  .  .  .  .

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

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

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

magma: CharacterTable(G);