Properties

Label 12T46
Order \(72\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $F_9$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $46$
Group :  $F_9$
CHM label :  $[(1/3.3^{3}):2]4_{4}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5)(2,10)(4,8)(7,11), (1,4,2,11,5,8,10,7)(3,9,12,6), (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:  $C_2$
4:  $C_4$
8:  $C_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: None

Low degree siblings

9T15, 18T28, 24T81, 36T49

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 39]
Character table:   
     2  3  3  .  3  3   3   3   3   3
     3  2  .  2  .  .   .   .   .   .

       1a 2a 3a 4a 4b  8a  8b  8c  8d
    2P 1a 1a 3a 2a 2a  4a  4a  4b  4b
    3P 1a 2a 1a 4b 4a  8c  8d  8a  8b
    5P 1a 2a 3a 4a 4b  8b  8a  8d  8c
    7P 1a 2a 3a 4b 4a  8d  8c  8b  8a

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  A -A   B  -B -/B  /B
X.4     1 -1  1  A -A  -B   B  /B -/B
X.5     1 -1  1 -A  A -/B  /B   B  -B
X.6     1 -1  1 -A  A  /B -/B  -B   B
X.7     1  1  1 -1 -1   A   A  -A  -A
X.8     1  1  1 -1 -1  -A  -A   A   A
X.9     8  . -1  .  .   .   .   .   .

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(8)^3