# Properties

 Label 6T15 Order $$360$$ n $$6$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $A_6$

# Related objects

## Group action invariants

 Degree $n$ : $6$ Transitive number $t$ : $15$ Group : $A_6$ CHM label : $A6$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,2,3), (1,2)(3,4,5,6) $|\Aut(F/K)|$: $1$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 3: None

## Low degree siblings

6T15, 10T26, 15T20 x 2, 20T89, 30T88 x 2, 36T555, 40T304, 45T49

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 $1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 1, 1, 1$ $40$ $3$ $(4,5,6)$ $2, 2, 1, 1$ $45$ $2$ $(3,4)(5,6)$ $5, 1$ $72$ $5$ $(2,3,4,5,6)$ $5, 1$ $72$ $5$ $(2,3,4,6,5)$ $4, 2$ $90$ $4$ $(1,2)(3,4,5,6)$ $3, 3$ $40$ $3$ $(1,2,3)(4,5,6)$

## Group invariants

 Order: $360=2^{3} \cdot 3^{2} \cdot 5$ Cyclic: No Abelian: No Solvable: No GAP id: [360, 118]
 Character table:  2 3 . 3 . . 2 . 3 2 2 . . . . 2 5 1 . . 1 1 . . 1a 3a 2a 5a 5b 4a 3b 2P 1a 3a 1a 5b 5a 2a 3b 3P 1a 1a 2a 5b 5a 4a 1a 5P 1a 3a 2a 1a 1a 4a 3b X.1 1 1 1 1 1 1 1 X.2 5 2 1 . . -1 -1 X.3 5 -1 1 . . -1 2 X.4 8 -1 . A *A . -1 X.5 8 -1 . *A A . -1 X.6 9 . 1 -1 -1 1 . X.7 10 1 -2 . . . 1 A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5