# Properties

 Label 6T12 Degree $6$ Order $60$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $\PSL(2,5)$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(6, 12);

## Group action invariants

 Degree $n$: $6$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $12$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $\PSL(2,5)$ CHM label: $L(6) = PSL(2,5) = A_{5}(6)$ Parity: $1$ magma: IsEven(G); Primitive: yes magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,4)(5,6), (1,2,3,4,6) magma: Generators(G);

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 3: None

## Low degree siblings

5T4, 10T7, 12T33, 15T5, 20T15, 30T9

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Label Cycle Type Size Order Representative $1^{6}$ $1$ $1$ $()$ $2^{2},1^{2}$ $15$ $2$ $(3,6)(4,5)$ $5,1$ $12$ $5$ $(2,3,5,4,6)$ $5,1$ $12$ $5$ $(2,4,3,6,5)$ $3^{2}$ $20$ $3$ $(1,2,3)(4,5,6)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $60=2^{2} \cdot 3 \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 60.5 magma: IdentifyGroup(G); Character table:

 1A 2A 3A 5A1 5A2 Size 1 15 20 12 12 2 P 1A 1A 3A 5A2 5A1 3 P 1A 2A 1A 5A2 5A1 5 P 1A 2A 3A 1A 1A Type 60.5.1a R $1$ $1$ $1$ $1$ $1$ 60.5.3a1 R $3$ $−1$ $0$ $−ζ5−1−ζ5$ $−ζ5−2−ζ52$ 60.5.3a2 R $3$ $−1$ $0$ $−ζ5−2−ζ52$ $−ζ5−1−ζ5$ 60.5.4a R $4$ $0$ $1$ $−1$ $−1$ 60.5.5a R $5$ $1$ $−1$ $0$ $0$

magma: CharacterTable(G);