Properties

 Label 6T3 Order $$12$$ n $$6$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $S_3\times C_2$

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Group action invariants

Degree $n$ :  $6$
Transitive number $t$ :  $3$
Group :  $S_3\times C_2$
CHM label :  $D(6) = S(3)[x]2$
Parity:  $-1$
Primitive:  No
Generators:   (1,2,3,4,5,6), (1,4)(2,3)(5,6)
$|\Aut(F/K)|$:  $2$
Low degree resolvents:
 2: 2T1, 2T1, 2T1 4: 4T2 6: 3T2

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Low degree siblings

6T3b, 12T3
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

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

Group invariants

 Order: $12=2^{2} \cdot 3$ Cyclic: No Abelian: No Solvable: Yes GAP id: [12, 4]
 Character table:  2 2 2 2 1 1 2 3 1 . . 1 1 1 1a 2a 2b 6a 3a 2c 2P 1a 1a 1a 3a 3a 1a 3P 1a 2a 2b 2c 1a 2c 5P 1a 2a 2b 6a 3a 2c X.1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 1 X.3 1 -1 1 -1 1 -1 X.4 1 1 -1 -1 1 -1 X.5 2 . . 1 -1 -2 X.6 2 . . -1 -1 2