Properties

 Label 5T4 Order $$60$$ n $$5$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $A_5$

Related objects

Group action invariants

 Degree $n$ : $5$ Transitive number $t$ : $4$ Group : $A_5$ CHM label : $A5$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,2,3), (3,4,5) $|\Aut(F/K)|$: $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

6T12, 10T7, 12T33, 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 $1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 1, 1$ $20$ $3$ $(3,4,5)$ $2, 2, 1$ $15$ $2$ $(2,3)(4,5)$ $5$ $12$ $5$ $(1,2,3,4,5)$ $5$ $12$ $5$ $(1,2,3,5,4)$

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 3a 2a 5a 5b 2P 1a 3a 1a 5b 5a 3P 1a 1a 2a 5b 5a 5P 1a 3a 2a 1a 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