# Properties

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

# Related objects

## 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 Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $5, 5, 1, 1$ $12$ $5$ $( 2, 5,11, 7, 8)( 3, 4,10, 6, 9)$ $5, 5, 1, 1$ $12$ $5$ $( 2, 7, 5, 8,11)( 3, 6, 4, 9,10)$ $2, 2, 2, 2, 2, 2$ $15$ $2$ $( 1, 2)( 3,12)( 4, 9)( 5, 8)( 6, 7)(10,11)$ $3, 3, 3, 3$ $20$ $3$ $( 1, 2, 5)( 3, 4,12)( 6,11, 8)( 7,10, 9)$

## 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