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Properties

Label 15T5
Degree $15$
Order $60$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $A_5$

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Show commands: Magma

magma: G := TransitiveGroup(15, 5);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $5$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_5$
CHM label:  $A_{5}(15)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
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)
magma: Generators(G);
 

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $60=2^{2} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  60.5
magma: IdentifyGroup(G);
 
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

magma: CharacterTable(G);