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) | |
| Generators: | (1,4)(5,6), (1,2,3,4,6) | |
| $|\Aut(F/K)|$: | $1$ |
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
|