# Properties

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

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

## 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