Properties

Label 15T5
Order \(60\)
n \(15\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $A_5$

Related objects

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Group action invariants

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

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $A_5$

Low degree siblings

5T4, 6T12, 10T7, 12T33, 20T15, 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $15$ $2$ $( 2, 5)( 3,13)( 4,10)( 7,14)( 9,15)(11,12)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 2, 7,14, 5)( 3,13,15, 8, 9)( 4,11,12,10, 6)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 2,10,13,11)( 3, 6, 4,15,14)( 5,12, 7, 8, 9)$
$ 3, 3, 3, 3, 3 $ $20$ $3$ $( 1, 2,12)( 3,11, 7)( 4,13, 6)( 5,10,15)( 8, 9,14)$

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

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