Group action invariants
Degree $n$: | $6$ | |
Transitive number $t$: | $16$ | |
Group: | $S_6$ | |
CHM label: | $S6$ | |
Parity: | $-1$ | |
Primitive: | yes | |
Nilpotency class: | $-1$ (not nilpotent) | |
$\card{\Aut(F/K)}$: | $1$ | |
Generators: | (1,2), (1,2,3,4,5,6) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ Resolvents shown for degrees $\leq 47$
Subfields
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, 45T96Siblings 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 $ | $120$ | $6$ | $(1,2,3,4,5,6)$ |
Group invariants
Order: | $720=2^{4} \cdot 3^{2} \cdot 5$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | no | |
Label: | 720.763 |
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 |