Properties

 Label 6T11 Order $$48$$ n $$6$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $S_4\times C_2$

Related objects

Group action invariants

Degree $n$ :  $6$
Transitive number $t$ :  $11$
Group :  $S_4\times C_2$
CHM label :  $2S_{4}(6) = [2^{3}]S(3) = 2 wr S(3)$
Parity:  $-1$
Primitive:  No
Generators:  (1,3,4,6), (1,4)(2,6,5,3)
$|\Aut(F/K)|$:  $2$
Low degree resolvents:
 2: $C_2$ x 3 4: $V_4$ 6: $S_3$ 12: $S_3\times C_2$ 24: $S_4$

Subfields

Degree 2: None

Degree 3: $S_3$

Low degree siblings

6T11, 8T24 x 2, 12T21, 12T22, 12T23 x 2, 12T24 x 2, 16T61, 24T46, 24T47, 24T48 x 2
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 1, 1, 1, 1$ $3$ $2$ $(3,6)$ $2, 2, 1, 1$ $6$ $2$ $(2,3)(5,6)$ $4, 1, 1$ $6$ $4$ $(2,3,5,6)$ $2, 2, 1, 1$ $3$ $2$ $(2,5)(3,6)$ $2, 2, 2$ $6$ $2$ $(1,2)(3,6)(4,5)$ $3, 3$ $8$ $3$ $(1,2,3)(4,5,6)$ $6$ $8$ $6$ $(1,2,3,4,5,6)$ $4, 2$ $6$ $4$ $(1,2,4,5)(3,6)$ $2, 2, 2$ $1$ $2$ $(1,4)(2,5)(3,6)$

Group invariants

 Order: $48=2^{4} \cdot 3$ Cyclic: No Abelian: No Solvable: Yes GAP id: [48, 48]
 Character table:  2 4 4 3 3 4 3 1 1 3 4 3 1 . . . . . 1 1 . 1 1a 2a 2b 4a 2c 2d 3a 6a 4b 2e 2P 1a 1a 1a 2c 1a 1a 3a 3a 2c 1a 3P 1a 2a 2b 4a 2c 2d 1a 2e 4b 2e 5P 1a 2a 2b 4a 2c 2d 3a 6a 4b 2e X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 1 1 -1 -1 -1 X.3 1 -1 1 -1 1 -1 1 -1 1 -1 X.4 1 1 -1 -1 1 -1 1 1 -1 1 X.5 2 -2 . . 2 . -1 1 . -2 X.6 2 2 . . 2 . -1 -1 . 2 X.7 3 -1 -1 1 -1 -1 . . 1 3 X.8 3 -1 1 -1 -1 1 . . -1 3 X.9 3 1 -1 -1 -1 1 . . 1 -3 X.10 3 1 1 1 -1 -1 . . -1 -3