↑

Properties

Label 6T12
Degree $6$
Order $60$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\PSL(2,5)$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(6, 12);
 

Group action invariants

Degree $n$:  $6$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $12$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PSL(2,5)$
CHM label:  $L(6) = PSL(2,5) = A_{5}(6)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,4)(5,6), (1,2,3,4,6)
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Low degree siblings

5T4, 10T7, 12T33, 15T5, 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 $ $1$ $1$ $()$
$ 2, 2, 1, 1 $ $15$ $2$ $(3,6)(4,5)$
$ 5, 1 $ $12$ $5$ $(2,3,5,4,6)$
$ 5, 1 $ $12$ $5$ $(2,4,3,6,5)$
$ 3, 3 $ $20$ $3$ $(1,2,3)(4,5,6)$

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);