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Properties

Label 20T9
Degree $20$
Order $40$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times F_5$

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Show commands: Magma

magma: G := TransitiveGroup(20, 9);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $9$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times F_5$
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:  (1,17,13,9,6,2,18,14,10,5)(3,20,16,11,8,4,19,15,12,7), (1,15,17,3)(2,16,18,4)(5,8,13,11)(6,7,14,12)(9,19,10,20)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$20$:  $F_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $F_5$

Degree 10: $F_5$

Low degree siblings

10T5 x 2, 20T13, 40T14

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

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:  40.12
magma: IdentifyGroup(G);
 
Character table:    
      2  3  3  3  3  3  3  3  3   1  1
      5  1  .  1  .  .  .  .  .   1  1

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

X.1      1  1  1  1  1  1  1  1   1  1
X.2      1 -1 -1  1 -1  1  1 -1  -1  1
X.3      1 -1 -1  1  1 -1 -1  1  -1  1
X.4      1  1  1  1 -1 -1 -1 -1   1  1
X.5      1 -1  1 -1  A -A  A -A   1  1
X.6      1 -1  1 -1 -A  A -A  A   1  1
X.7      1  1 -1 -1  A  A -A -A  -1  1
X.8      1  1 -1 -1 -A -A  A  A  -1  1
X.9      4  . -4  .  .  .  .  .   1 -1
X.10     4  .  4  .  .  .  .  .  -1 -1

A = -E(4)
  = -Sqrt(-1) = -i

magma: CharacterTable(G);