Group action invariants
Degree $n$: | $6$ | |
Transitive number $t$: | $12$ | |
Group: | $\PSL(2,5)$ | |
CHM label: | $L(6) = PSL(2,5) = A_{5}(6)$ | |
Parity: | $1$ | |
Primitive: | yes | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $1$ | |
Generators: | (1,4)(5,6), (1,2,3,4,6) |
Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: None
Low degree siblings
5T4, 10T7, 12T33, 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 1, 1 $ | $15$ | $2$ | $(3,6)(4,5)$ |
$ 5, 1 $ | $12$ | $5$ | $(2,3,5,4,6)$ |
$ 5, 1 $ | $12$ | $5$ | $(2,4,3,6,5)$ |
$ 3, 3 $ | $20$ | $3$ | $(1,2,3)(4,5,6)$ |
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 5b 3a 2P 1a 1a 5b 5a 3a 3P 1a 2a 5b 5a 1a 5P 1a 2a 1a 1a 3a 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 |