Properties

Label 12T3
Order \(12\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_6$

Related objects

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Group action invariants

Degree $n$:  $12$
Transitive number $t$:  $3$
Group:  $D_6$
CHM label:  $D_{6}(6)[x]2=1/2[3:2]E(4)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,11)(2,8)(3,9)(4,6)(5,7)(10,12), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11)
$|\Aut(F/K)|$:  $12$

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: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $S_3$, $D_{6}$ x 2

Low degree siblings

6T3 x 2

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, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 2)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
$ 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 3)( 2,12)( 4,10)( 5,11)( 6, 8)( 7, 9)$
$ 6, 6 $ $2$ $6$ $( 1, 4, 9,12, 5, 8)( 2, 7,10, 3, 6,11)$
$ 3, 3, 3, 3 $ $2$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$

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