Properties

Label 12T16
Order \(36\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3^2$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $16$
Group :  $S_3^2$
CHM label :  $[3^{2}]E(4)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (2,10,6)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 2
12:  $D_{6}$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: $S_3^2$

Low degree siblings

6T9, 9T8, 18T9, 18T11 x 2, 36T13

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

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [36, 10]
Character table:   
     2  2  .  2  1  2  1  2  1  1
     3  2  2  1  1  .  1  1  2  2

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

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