Properties

Label 12T45
Order \(72\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times S_4$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $45$
Group :  $C_3\times S_4$
CHM label :  $S(4)[x]C(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10)(2,5)(6,9), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$
24:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: $S_4$

Degree 6: None

Low degree siblings

18T30, 18T33, 24T80, 24T84, 36T20, 36T52

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 42]
Character table:   
      2  3  2  .   .   2   3   2   .   2   3   2  2  3  3  3
      3  2  1  2   2   1   1   1   2   1   1   1  1  1  2  2

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

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3
C = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3