# Properties

 Label 6T15 Degree $6$ Order $360$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $A_6$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(6, 15);

## Group action invariants

 Degree $n$: $6$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $15$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $A_6$ CHM label: $A6$ 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: (1,2,3), (1,2)(3,4,5,6) magma: Generators(G);

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 3: None

## Low degree siblings

6T15, 10T26, 15T20 x 2, 20T89, 30T88 x 2, 36T555, 40T304, 45T49

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 1A $1^{6}$ $1$ $1$ $()$ 2A $2^{2},1^{2}$ $45$ $2$ $(1,2)(3,4)$ 3A $3,1^{3}$ $40$ $3$ $(1,2,3)$ 3B $3^{2}$ $40$ $3$ $(1,2,3)(4,5,6)$ 4A $4,2$ $90$ $4$ $(1,2,3,4)(5,6)$ 5A1 $5,1$ $72$ $5$ $(1,3,4,5,2)$ 5A2 $5,1$ $72$ $5$ $(1,2,3,4,5)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $360=2^{3} \cdot 3^{2} \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 360.118 magma: IdentifyGroup(G); Character table:

 1A 2A 3A 3B 4A 5A1 5A2 Size 1 45 40 40 90 72 72 2 P 1A 1A 3A 3B 2A 5A2 5A1 3 P 1A 2A 1A 1A 4A 5A2 5A1 5 P 1A 2A 3A 3B 4A 1A 1A Type 360.118.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ 360.118.5a R $5$ $1$ $−1$ $2$ $−1$ $0$ $0$ 360.118.5b R $5$ $1$ $2$ $−1$ $−1$ $0$ $0$ 360.118.8a1 R $8$ $0$ $−1$ $−1$ $0$ $−ζ5−1−ζ5$ $−ζ5−2−ζ52$ 360.118.8a2 R $8$ $0$ $−1$ $−1$ $0$ $−ζ5−2−ζ52$ $−ζ5−1−ζ5$ 360.118.9a R $9$ $1$ $0$ $0$ $1$ $−1$ $−1$ 360.118.10a R $10$ $−2$ $1$ $1$ $0$ $0$ $0$

magma: CharacterTable(G);