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Properties

Label 20T31
Degree $20$
Order $120$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_2\times A_5$

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Show commands: Magma

magma: G := TransitiveGroup(20, 31);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $31$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times A_5$
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:  (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16), (1,12,3,15,5,2,11,4,16,6)(7,10,17,20,13,8,9,18,19,14)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$60$:  $A_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: None

Degree 10: $A_{5}$

Low degree siblings

10T11, 12T75, 12T76, 20T36, 24T203, 30T29, 30T30, 40T61

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Label:  120.35
magma: IdentifyGroup(G);
 
Character table:    
      2  3  1  3  3  1  3  1  1   1   1
      3  1  1  .  1  1  .  .  .   .   .
      5  1  .  .  1  .  .  1  1   1   1

        1a 3a 2a 2b 6a 2c 5a 5b 10a 10b
     2P 1a 3a 1a 1a 3a 1a 5b 5a  5b  5a
     3P 1a 1a 2a 2b 2b 2c 5b 5a 10b 10a
     5P 1a 3a 2a 2b 6a 2c 1a 1a  2b  2b
     7P 1a 3a 2a 2b 6a 2c 5b 5a 10b 10a

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

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

magma: CharacterTable(G);