Properties

Label 30T9
Order \(60\)
n \(30\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $A_5$

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Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $9$
Group :  $A_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,6,7,3)(2,11,5,8,4)(9,21,30,27,17)(10,22,29,28,18)(13,25,15,20,23)(14,26,16,19,24), (1,13,5)(2,14,6)(3,19,9)(4,20,10)(7,18,15)(8,17,16)(11,28,21)(12,27,22)(23,29,25)(24,30,26)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 5: $A_5$

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

Degree 10: $A_{5}$

Degree 15: $A_5$

Low degree siblings

5T4, 6T12, 10T7, 12T33, 15T5, 20T15

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, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $15$ $2$ $( 3,12)( 4,11)( 5, 8)( 6, 7)( 9,29)(10,30)(13,19)(14,20)(15,26)(16,25)(17,28) (18,27)(21,22)(23,24)$
$ 5, 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 3, 7, 6,12)( 2, 4, 8, 5,11)( 9,17,27,30,21)(10,18,28,29,22) (13,23,20,15,25)(14,24,19,16,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $20$ $3$ $( 1, 4,27)( 2, 3,28)( 5,24,20)( 6,23,19)( 7,16,18)( 8,15,17)( 9,25,21) (10,26,22)(11,29,14)(12,30,13)$
$ 5, 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 4,29,25,18)( 2, 3,30,26,17)( 5,15,22, 9,14)( 6,16,21,10,13) ( 7,23,20,11,27)( 8,24,19,12,28)$

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