# Properties

 Label 20T35 Degree $20$ Order $120$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $S_5$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(20, 35);

## Group action invariants

 Degree $n$: $20$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $35$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_5$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $2$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,3)(2,4)(5,13)(6,14)(7,8)(9,10)(11,12)(15,17)(16,18), (1,6,10,14,18)(2,5,9,13,17)(3,8,11,16,19)(4,7,12,15,20) magma: Generators(G);

## 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: None

Degree 10: $S_5$

## Low degree siblings

5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 3, 3, 1, 1$ $20$ $3$ $( 3, 7,10)( 4, 8, 9)( 5,11,17)( 6,12,18)(13,16,20)(14,15,19)$ $6, 6, 6, 1, 1$ $20$ $6$ $( 3,14,10,19, 7,15)( 4,13, 9,20, 8,16)( 5,12,17, 6,11,18)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1$ $10$ $2$ $( 3,19)( 4,20)( 5, 6)( 7,14)( 8,13)( 9,16)(10,15)(11,12)(17,18)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $15$ $2$ $( 1, 2)( 3,13)( 4,14)( 5, 6)( 7,20)( 8,19)( 9,15)(10,16)(11,18)(12,17)$ $5, 5, 5, 5$ $24$ $5$ $( 1, 3, 6,12,15)( 2, 4, 5,11,16)( 7,17,14, 9,20)( 8,18,13,10,19)$ $4, 4, 4, 4, 4$ $30$ $4$ $( 1, 3, 8,10)( 2, 4, 7, 9)( 5,12,16,20)( 6,11,15,19)(13,18,14,17)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $120=2^{3} \cdot 3 \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Label: 120.34 magma: IdentifyGroup(G);
 Character table:  2 3 1 1 2 3 . 2 3 1 1 1 1 . . . 5 1 . . . . 1 . 1a 3a 6a 2a 2b 5a 4a 2P 1a 3a 3a 1a 1a 5a 2b 3P 1a 1a 2a 2a 2b 5a 4a 5P 1a 3a 6a 2a 2b 1a 4a X.1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 1 -1 X.3 4 1 1 -2 . -1 . X.4 4 1 -1 2 . -1 . X.5 5 -1 1 1 1 . -1 X.6 5 -1 -1 -1 1 . 1 X.7 6 . . . -2 1 . 

magma: CharacterTable(G);