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Properties

Label 6T3
Order \(12\)
n \(6\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{6}$

Related objects

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Group action invariants

Degree $n$ :  $6$
Transitive number $t$ :  $3$
Group :  $D_{6}$
CHM label :  $D(6) = S(3)[x]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4)(2,3)(5,6), (1,2,3,4,5,6)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Low degree siblings

6T3, 12T3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 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