# Properties

 Label 20T89 Order $$360$$ n $$20$$ Cyclic No Abelian No Solvable No Primitive No $p$-group No Group: $A_6$

# Related objects

## Group action invariants

 Degree $n$ : $20$ Transitive number $t$ : $89$ Group : $A_6$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,5,14)(2,6,13)(3,15,18)(4,16,17)(7,12,20)(8,11,19), (1,19,6)(2,20,5)(3,17,12)(4,18,11)(9,16,14)(10,15,13) $|\Aut(F/K)|$: $2$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 4: None

Degree 5: None

Degree 10: $\PSL(2,9)$

## Low degree siblings

6T15 x 2, 10T26, 15T20 x 2, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $45$ $2$ $( 5,11)( 6,12)( 7, 9)( 8,10)(13,15)(14,16)(17,19)(18,20)$ $3, 3, 3, 3, 3, 3, 1, 1$ $40$ $3$ $( 3, 5,11)( 4, 6,12)( 7,17,13)( 8,18,14)( 9,15,19)(10,16,20)$ $3, 3, 3, 3, 3, 3, 1, 1$ $40$ $3$ $( 3, 8,10)( 4, 7, 9)( 5,18,16)( 6,17,15)(11,14,20)(12,13,19)$ $4, 4, 4, 4, 2, 2$ $90$ $4$ $( 1, 2)( 3, 4)( 5,17,11,19)( 6,18,12,20)( 7,16, 9,14)( 8,15,10,13)$ $5, 5, 5, 5$ $72$ $5$ $( 1, 3, 7,17,16)( 2, 4, 8,18,15)( 5,13,10,11,20)( 6,14, 9,12,19)$ $5, 5, 5, 5$ $72$ $5$ $( 1, 3, 7,12,14)( 2, 4, 8,11,13)( 5,20,15,10,18)( 6,19,16, 9,17)$

## 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 2a 3a 3b 4a 5a 5b 2P 1a 1a 3a 3b 2a 5b 5a 3P 1a 2a 1a 1a 4a 5b 5a 5P 1a 2a 3a 3b 4a 1a 1a X.1 1 1 1 1 1 1 1 X.2 5 1 2 -1 -1 . . X.3 5 1 -1 2 -1 . . X.4 8 . -1 -1 . A *A X.5 8 . -1 -1 . *A A X.6 9 1 . . 1 -1 -1 X.7 10 -2 1 1 . . . A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5