# Properties

 Label 20T30 Degree $20$ Order $120$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $S_5$

# Related objects

## Group action invariants

 Degree $n$: $20$ Transitive number $t$: $30$ Group: $S_5$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $2$ Generators: (1,3)(2,4)(5,13)(6,14)(15,17)(16,18)(19,20), (1,6,10,13,17)(2,5,9,14,18)(3,8,12,15,20)(4,7,11,16,19)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 4: None

Degree 5: $S_5$

Degree 10: $S_5$

## Low degree siblings

5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T32, 20T35, 24T202, 30T22, 30T25, 30T27, 40T62

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1$ $10$ $2$ $( 5,17)( 6,18)( 7, 9)( 8,10)(11,12)(13,15)(14,16)$ $3, 3, 3, 3, 3, 3, 1, 1$ $20$ $3$ $( 3, 7, 9)( 4, 8,10)( 5,11,18)( 6,12,17)(13,15,19)(14,16,20)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $15$ $2$ $( 1, 2)( 3,13)( 4,14)( 5, 6)( 7,19)( 8,20)( 9,15)(10,16)(11,17)(12,18)$ $6, 6, 3, 3, 2$ $20$ $6$ $( 1, 2)( 3,13, 9,19, 7,15)( 4,14,10,20, 8,16)( 5,11,18)( 6,12,17)$ $4, 4, 4, 4, 4$ $30$ $4$ $( 1, 3, 7, 9)( 2, 4, 8,10)( 5,11,16,19)( 6,12,15,20)(13,17,14,18)$ $5, 5, 5, 5$ $24$ $5$ $( 1, 4, 6,12,15)( 2, 3, 5,11,16)( 7,17,14,10,19)( 8,18,13, 9,20)$

## Group invariants

 Order: $120=2^{3} \cdot 3 \cdot 5$ Cyclic: no Abelian: no Solvable: no GAP id: [120, 34]
 Character table:  2 3 2 1 3 1 2 . 3 1 1 1 . 1 . . 5 1 . . . . . 1 1a 2a 3a 2b 6a 4a 5a 2P 1a 1a 3a 1a 3a 2b 5a 3P 1a 2a 1a 2b 2a 4a 5a 5P 1a 2a 3a 2b 6a 4a 1a X.1 1 1 1 1 1 1 1 X.2 1 -1 1 1 -1 -1 1 X.3 4 -2 1 . 1 . -1 X.4 4 2 1 . -1 . -1 X.5 5 1 -1 1 1 -1 . X.6 5 -1 -1 1 -1 1 . X.7 6 . . -2 . . 1