Properties

 Label 20T15 Order $$60$$ n $$20$$ Cyclic No Abelian No Solvable No Primitive No $p$-group No Group: $A_5$

Related objects

Group action invariants

 Degree $n$ : $20$ Transitive number $t$ : $15$ Group : $A_5$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,17)(2,18)(3,4)(5,7)(6,8)(9,20)(10,19)(11,14)(12,13)(15,16), (1,6,10,13,17)(2,5,9,14,18)(3,8,12,15,20)(4,7,11,16,19) $|\Aut(F/K)|$: $2$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $A_5$

Degree 10: $A_{5}$

Low degree siblings

5T4, 6T12, 10T7, 12T33, 15T5, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 3, 3, 1, 1$ $20$ $3$ $( 3, 7, 9)( 4, 8,10)( 5,11,18)( 6,12,17)(13,15,19)(14,16,20)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $15$ $2$ $( 1, 2)( 3,13)( 4,14)( 5, 6)( 7,19)( 8,20)( 9,15)(10,16)(11,17)(12,18)$ $5, 5, 5, 5$ $12$ $5$ $( 1, 4, 6,12,15)( 2, 3, 5,11,16)( 7,17,14,10,19)( 8,18,13, 9,20)$ $5, 5, 5, 5$ $12$ $5$ $( 1, 4,18,11,13)( 2, 3,17,12,14)( 5,16, 8,19, 9)( 6,15, 7,20,10)$

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