Properties

 Label 6T16 Degree $6$ Order $720$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $S_6$

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Show commands: Magma

magma: G := TransitiveGroup(6, 16);

Group action invariants

 Degree $n$: $6$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $16$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_6$ CHM label: $S6$ 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), (1,2,3,4,5,6) 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

6T16, 10T32, 12T183 x 2, 15T28 x 2, 20T145, 20T149 x 2, 30T164 x 2, 30T166 x 2, 30T176 x 2, 36T1252, 40T589, 40T592 x 2, 45T96

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 $1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 1, 1, 1, 1$ $15$ $2$ $(5,6)$ $3, 1, 1, 1$ $40$ $3$ $(4,5,6)$ $2, 2, 1, 1$ $45$ $2$ $(3,4)(5,6)$ $4, 1, 1$ $90$ $4$ $(3,4,5,6)$ $3, 2, 1$ $120$ $6$ $(2,3)(4,5,6)$ $5, 1$ $144$ $5$ $(2,3,4,5,6)$ $2, 2, 2$ $15$ $2$ $(1,2)(3,4)(5,6)$ $4, 2$ $90$ $4$ $(1,2)(3,4,5,6)$ $3, 3$ $40$ $3$ $(1,2,3)(4,5,6)$ $6$ $120$ $6$ $(1,2,3,4,5,6)$

magma: ConjugacyClasses(G);

Group invariants

 Order: $720=2^{4} \cdot 3^{2} \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 720.763 magma: IdentifyGroup(G);
 Character table:  2 4 4 1 4 3 1 . 4 3 1 1 3 2 1 2 . . 1 . 1 . 2 1 5 1 . . . . . 1 . . . . 1a 2a 3a 2b 4a 6a 5a 2c 4b 3b 6b 2P 1a 1a 3a 1a 2b 3a 5a 1a 2b 3b 3b 3P 1a 2a 1a 2b 4a 2a 5a 2c 4b 1a 2c 5P 1a 2a 3a 2b 4a 6a 1a 2c 4b 3b 6b X.1 1 -1 1 1 -1 -1 1 -1 1 1 -1 X.2 5 -3 2 1 -1 . . 1 -1 -1 1 X.3 9 -3 . 1 1 . -1 -3 1 . . X.4 5 -1 -1 1 1 -1 . 3 -1 2 . X.5 10 -2 1 -2 . 1 . 2 . 1 -1 X.6 16 . -2 . . . 1 . . -2 . X.7 5 1 -1 1 -1 1 . -3 -1 2 . X.8 10 2 1 -2 . -1 . -2 . 1 1 X.9 9 3 . 1 -1 . -1 3 1 . . X.10 5 3 2 1 1 . . -1 -1 -1 -1 X.11 1 1 1 1 1 1 1 1 1 1 1 

magma: CharacterTable(G);