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Properties

Label 12T33
Degree $12$
Order $60$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $A_5$

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Group action invariants

Degree $n$:  $12$
Transitive number $t$:  $33$
Group:  $A_5$
CHM label:  $A_{5}(12)$
Parity:  $1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$\card{\Aut(F/K)}$:  $2$
Generators:  (1,3,5,7,9)(2,4,6,8,12), (1,11,5)(2,7,9)(3,6,8)(4,12,10)

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: $\PSL(2,5)$

Low degree siblings

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

Group invariants

Order:  $60=2^{2} \cdot 3 \cdot 5$
Cyclic:  no
Abelian:  no
Solvable:  no
Label:  60.5
Character table:    
     2  2  .  .  2  .
     3  1  .  .  .  1
     5  1  1  1  .  .

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

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

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