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
noneResolvents 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, 30T9Siblings 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 |