# Properties

 Label 10T7 Order $$60$$ n $$10$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $A_{5}$

# Related objects

## Group action invariants

 Degree $n$ : $10$ Transitive number $t$ : $7$ Group : $A_{5}$ CHM label : $A_{5}(10)$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,9)(3,4)(5,10)(6,7), (1,3,5,7,9)(2,4,6,8,10) $|\Aut(F/K)|$: $1$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 5: None

## Low degree siblings

5T4, 6T12, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 1$ $20$ $3$ $( 2, 4, 5)( 3, 6, 9)( 7, 8,10)$ $2, 2, 2, 2, 1, 1$ $15$ $2$ $( 2, 7)( 4,10)( 5, 8)( 6, 9)$ $5, 5$ $12$ $5$ $( 1, 2, 3, 6, 8)( 4, 9, 7, 5,10)$ $5, 5$ $12$ $5$ $( 1, 2, 9, 6, 7)( 3, 8, 4,10, 5)$

## 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