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Properties

Label 12T17
Order \(36\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_3\times C_3):C_4$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $17$
Group :  $(C_3\times C_3):C_4$
CHM label :  $[3^{2}]4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,10,6)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: $C_3^2:C_4$

Low degree siblings

6T10 x 2, 9T9, 12T17, 18T10, 36T14

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

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [36, 9]
Character table:    
     2  2  .  2  2  2  .
     3  2  2  .  .  .  2

       1a 3a 4a 2a 4b 3b
    2P 1a 3a 2a 1a 2a 3b
    3P 1a 1a 4b 2a 4a 1a

X.1     1  1  1  1  1  1
X.2     1  1 -1  1 -1  1
X.3     1  1  A -1 -A  1
X.4     1  1 -A -1  A  1
X.5     4  1  .  .  . -2
X.6     4 -2  .  .  .  1

A = -E(4)
  = -Sqrt(-1) = -i