# Properties

 Label 6T12 Order $$60$$ n $$6$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $\PSL(2,5)$

# Related objects

## 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

None

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 3: None

## Low degree siblings

5T4, 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 $1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 1, 1$ $15$ $2$ $(3,6)(4,5)$ $5, 1$ $12$ $5$ $(2,3,5,4,6)$ $5, 1$ $12$ $5$ $(2,4,3,6,5)$ $3, 3$ $20$ $3$ $(1,2,3)(4,5,6)$

## 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