Properties

Label 12T19
Order \(36\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times (C_3 : C_4)$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $19$
Group :  $C_3\times (C_3 : C_4)$
CHM label :  $[3^{2}]4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,6,10)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $S_3$, $C_6$
12:  $C_{12}$, $C_3 : C_4$
18:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: $S_3\times C_3$

Low degree siblings

36T5

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

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [36, 6]
Character table:   
      2  2   1   1   2  2   2   2   1  1   2  2   2   2  1  2   1   2   2
      3  2   2   2   1  1   1   2   2  2   1  1   1   2  2  2   2   2   2

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

X.1      1   1   1   1  1   1   1   1  1   1  1   1   1  1  1   1   1   1
X.2      1   1   1  -1 -1  -1   1   1  1  -1 -1  -1   1  1  1   1   1   1
X.3      1   1   1   B  B   B  -1  -1 -1  -B -B  -B   1  1 -1  -1   1  -1
X.4      1   1   1  -B -B  -B  -1  -1 -1   B  B   B   1  1 -1  -1   1  -1
X.5      1   A  /A -/A -1  -A   A  /A  1 -/A -1  -A  /A  1  1   A   A  /A
X.6      1  /A   A  -A -1 -/A  /A   A  1  -A -1 -/A   A  1  1  /A  /A   A
X.7      1   A  /A  /A  1   A   A  /A  1  /A  1   A  /A  1  1   A   A  /A
X.8      1  /A   A   A  1  /A  /A   A  1   A  1  /A   A  1  1  /A  /A   A
X.9      1   A  /A   C  B -/C  -A -/A -1  -C -B  /C  /A  1 -1  -A   A -/A
X.10     1   A  /A  -C -B  /C  -A -/A -1   C  B -/C  /A  1 -1  -A   A -/A
X.11     1  /A   A -/C  B   C -/A  -A -1  /C -B  -C   A  1 -1 -/A  /A  -A
X.12     1  /A   A  /C -B  -C -/A  -A -1 -/C  B   C   A  1 -1 -/A  /A  -A
X.13     2  -1  -1   .  .   .  -2   1  1   .  .   .   2 -1 -2   1   2  -2
X.14     2  -1  -1   .  .   .   2  -1 -1   .  .   .   2 -1  2  -1   2   2
X.15     2 -/A  -A   .  .   .   D   A  1   .  .   . -/D -1 -2  /A  -D  /D
X.16     2  -A -/A   .  .   .  /D  /A  1   .  .   .  -D -1 -2   A -/D   D
X.17     2 -/A  -A   .  .   .  -D  -A -1   .  .   . -/D -1  2 -/A  -D -/D
X.18     2  -A -/A   .  .   . -/D -/A -1   .  .   .  -D -1  2  -A -/D  -D

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = -E(4)
  = -Sqrt(-1) = -i
C = -E(12)^7
D = -2*E(3)
  = 1-Sqrt(-3) = 1-i3