Properties

Label 12T44
Order \(72\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3:S_4$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $44$
Group :  $C_3:S_4$
CHM label :  $1/2[3:2]S(4)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11)(2,10)(3,9)(4,8)(5,7), (1,7,10)(2,5,11)(3,6,9), (2,8,11)(3,6,12)(4,7,10), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$ x 4
18:  $C_3^2:C_2$
24:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: $S_4$

Degree 6: None

Low degree siblings

12T44 x 2, 18T37, 18T40, 24T79 x 3, 36T23, 36T56

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 $ $18$ $2$ $( 2, 3)( 5, 9)( 6,11)( 7,10)( 8,12)$
$ 3, 3, 3, 1, 1, 1 $ $8$ $3$ $( 2, 8,11)( 3, 6,12)( 4, 7,10)$
$ 3, 3, 3, 3 $ $8$ $3$ $( 1, 2, 3)( 4, 8,12)( 5, 6, 7)( 9,10,11)$
$ 4, 4, 4 $ $18$ $4$ $( 1, 2, 4,11)( 3, 9, 6,12)( 5,10, 8, 7)$
$ 6, 6 $ $6$ $6$ $( 1, 2, 9,10, 5, 6)( 3, 4,11,12, 7, 8)$
$ 3, 3, 3, 3 $ $8$ $3$ $( 1, 2,12)( 3, 7,11)( 4, 5, 6)( 8, 9,10)$
$ 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$
$ 3, 3, 3, 3 $ $2$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 43]
Character table:   
     2  3  2  .  .  2  2  .  3  2
     3  2  .  2  2  .  1  2  1  2

       1a 2a 3a 3b 4a 6a 3c 2b 3d
    2P 1a 1a 3a 3b 2b 3d 3c 1a 3d
    3P 1a 2a 1a 1a 4a 2b 1a 2b 1a
    5P 1a 2a 3a 3b 4a 6a 3c 2b 3d

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