# Properties

 Label 6T14 Degree $6$ Order $120$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $\PGL(2,5)$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(6, 14);

## Group action invariants

 Degree $n$: $6$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $14$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $\PGL(2,5)$ CHM label: $L(6):2 = PGL(2,5) = S_{5}(6)$ 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)(3,4)(5,6), (1,2,3,4,6) magma: Generators(G);

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 3: None

## Low degree siblings

5T5, 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 $1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 1, 1$ $30$ $4$ $(3,4,6,5)$ $2, 2, 1, 1$ $15$ $2$ $(3,6)(4,5)$ $5, 1$ $24$ $5$ $(2,3,5,4,6)$ $2, 2, 2$ $10$ $2$ $(1,2)(3,4)(5,6)$ $3, 3$ $20$ $3$ $(1,2,3)(4,5,6)$ $6$ $20$ $6$ $(1,2,3,6,5,4)$

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