Properties

Label 12T34
Order \(72\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\wr C_2$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $34$
Group :  $S_3\wr C_2$
CHM label :  $F_{36}:2(12e)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8)(2,3)(4,5)(6,7)(9,12)(10,11), (1,3,5,7,9,11)(2,4,6,8,10,12)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: $C_3^2:D_4$

Low degree siblings

6T13 x 2, 9T16, 12T34, 12T35 x 2, 12T36 x 2, 18T34 x 2, 18T36, 24T72 x 2, 36T53, 36T54 x 2

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 40]
Character table:   
     2  3  3  1  2  1  2  2  1  1
     3  2  .  2  .  1  1  1  1  2

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

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     4  . -2  . -1  2  .  .  1
X.7     4  . -2  .  1 -2  .  .  1
X.8     4  .  1  .  .  . -2  1 -2
X.9     4  .  1  .  .  .  2 -1 -2