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Properties

Label 20T8
Degree $20$
Order $40$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times D_{10}$

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Show commands: Magma

magma: G := TransitiveGroup(20, 8);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $C_2^3$
$10$:  $D_{5}$
$20$:  $D_{10}$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $D_{5}$

Degree 10: $D_{10}$ x 3

Low degree siblings

20T8 x 3, 40T10

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $40=2^{3} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  40.13
magma: IdentifyGroup(G);
 
Character table:    
      2  3  3  3  3  3   2  3   2   2   2   2   2   2   2  3  3
      5  1  .  1  .  .   1  .   1   1   1   1   1   1   1  1  1

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

X.1      1  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  1 -1
X.3      1 -1 -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  1  1
X.5      1 -1  1 -1  1  -1  1  -1   1   1  -1  -1   1   1 -1 -1
X.6      1  1 -1 -1 -1  -1  1   1  -1   1  -1   1  -1   1 -1  1
X.7      1  1 -1 -1  1   1 -1  -1  -1   1   1  -1  -1   1  1 -1
X.8      1  1  1  1 -1  -1 -1  -1   1   1  -1  -1   1   1 -1 -1
X.9      2  . -2  .  .   A  .  -A  *A -*A  *A -*A   A  -A -2  2
X.10     2  . -2  .  .  *A  . -*A   A  -A   A  -A  *A -*A -2  2
X.11     2  . -2  .  . -*A  .  *A   A  -A  -A   A  *A -*A  2 -2
X.12     2  . -2  .  .  -A  .   A  *A -*A -*A  *A   A  -A  2 -2
X.13     2  .  2  .  .   A  .   A -*A -*A  *A  *A  -A  -A -2 -2
X.14     2  .  2  .  .  *A  .  *A  -A  -A   A   A -*A -*A -2 -2
X.15     2  .  2  .  . -*A  . -*A  -A  -A  -A  -A -*A -*A  2  2
X.16     2  .  2  .  .  -A  .  -A -*A -*A -*A -*A  -A  -A  2  2

A = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5

magma: CharacterTable(G);