Properties

Label 20T15
Order \(60\)
n \(20\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $A_5$

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $15$
Group :  $A_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,17)(2,18)(3,4)(5,7)(6,8)(9,20)(10,19)(11,14)(12,13)(15,16), (1,6,10,13,17)(2,5,9,14,18)(3,8,12,15,20)(4,7,11,16,19)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $A_5$

Degree 10: $A_{5}$

Low degree siblings

5T4, 6T12, 10T7, 12T33, 15T5, 30T9

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

Group invariants

Order:  $60=2^{2} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [60, 5]
Character table:   
     2  2  .  2  .  .
     3  1  1  .  .  .
     5  1  .  .  1  1

       1a 3a 2a 5a 5b
    2P 1a 3a 1a 5b 5a
    3P 1a 1a 2a 5b 5a
    5P 1a 3a 2a 1a 1a

X.1     1  1  1  1  1
X.2     3  . -1  A *A
X.3     3  . -1 *A  A
X.4     4  1  . -1 -1
X.5     5 -1  1  .  .

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