# Properties

 Label 5T5 Degree $5$ Order $120$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $S_5$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(5, 5);

## Group action invariants

 Degree $n$: $5$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $5$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_5$ CHM label: $S5$ Parity: $-1$ magma: IsEven(G); Primitive: yes magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,2), (1,2,3,4,5) magma: Generators(G);

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Prime degree - none

## Low degree siblings

6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 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 $1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 1, 1, 1$ $10$ $2$ $(4,5)$ $3, 1, 1$ $20$ $3$ $(3,4,5)$ $2, 2, 1$ $15$ $2$ $(2,3)(4,5)$ $4, 1$ $30$ $4$ $(2,3,4,5)$ $3, 2$ $20$ $6$ $(1,2)(3,4,5)$ $5$ $24$ $5$ $(1,2,3,4,5)$

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 2 1 3 2 1 . 3 1 1 1 . . 1 . 5 1 . . . . . 1 1a 2a 3a 2b 4a 6a 5a 2P 1a 1a 3a 1a 2b 3a 5a 3P 1a 2a 1a 2b 4a 2a 5a 5P 1a 2a 3a 2b 4a 6a 1a X.1 1 -1 1 1 -1 -1 1 X.2 4 -2 1 . . 1 -1 X.3 5 -1 -1 1 1 -1 . X.4 6 . . -2 . . 1 X.5 5 1 -1 1 -1 1 . X.6 4 2 1 . . -1 -1 X.7 1 1 1 1 1 1 1 

magma: CharacterTable(G);