Properties

Label 12T12
Order \(24\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{12}$

Related objects

Learn more about

Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $D_{6}$

Low degree siblings

12T12, 24T13

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

Group invariants

Order:  $24=2^{3} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [24, 6]
Character table:   
     2  3  2  2   2  2  2  2   2  3
     3  1  .  .   1  1  1  1   1  1

       1a 2a 2b 12a 6a 4a 3a 12b 2c
    2P 1a 1a 1a  6a 3a 2c 3a  6a 1a
    3P 1a 2a 2b  4a 2c 4a 1a  4a 2c
    5P 1a 2a 2b 12b 6a 4a 3a 12a 2c
    7P 1a 2a 2b 12b 6a 4a 3a 12a 2c
   11P 1a 2a 2b 12a 6a 4a 3a 12b 2c

X.1     1  1  1   1  1  1  1   1  1
X.2     1 -1 -1   1  1  1  1   1  1
X.3     1 -1  1  -1  1 -1  1  -1  1
X.4     1  1 -1  -1  1 -1  1  -1  1
X.5     2  .  .   . -2  .  2   . -2
X.6     2  .  .  -1 -1  2 -1  -1  2
X.7     2  .  .   1 -1 -2 -1   1  2
X.8     2  .  .   A  1  . -1  -A -2
X.9     2  .  .  -A  1  . -1   A -2

A = -E(12)^7+E(12)^11
  = Sqrt(3) = r3