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Properties

Label 12T40
Order \(72\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_3:S_3.C_2$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $40$
Group :  $C_2\times C_3:S_3.C_2$
CHM label :  $F_{36}(6)[x]2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5,9)(4,8,12), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11), (2,10)(3,11)(4,8)(5,9), (1,7)(2,8,10,4)(3,9,11,5)(6,12)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $C_4\times C_2$
36:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: $C_3^2:C_4$

Low degree siblings

12T40, 12T41 x 2, 18T27 x 2, 24T76 x 2, 36T35, 36T36

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 45]
Character table:    
      2  3  3  1  3  3  3  3  3  1  1  1  3
      3  2  .  2  .  .  .  .  .  2  2  2  2

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

X.1      1  1  1  1  1  1  1  1  1  1  1  1
X.2      1  1  1 -1 -1 -1 -1  1  1  1  1  1
X.3      1  1  1 -1 -1  1  1 -1 -1 -1  1 -1
X.4      1  1  1  1  1 -1 -1 -1 -1 -1  1 -1
X.5      1 -1  1  A -A  A -A -1  1  1  1  1
X.6      1 -1  1 -A  A -A  A -1  1  1  1  1
X.7      1 -1  1  A -A -A  A  1 -1 -1  1 -1
X.8      1 -1  1 -A  A  A -A  1 -1 -1  1 -1
X.9      4  . -2  .  .  .  .  . -2  1  1  4
X.10     4  . -2  .  .  .  .  .  2 -1  1 -4
X.11     4  .  1  .  .  .  .  . -1  2 -2 -4
X.12     4  .  1  .  .  .  .  .  1 -2 -2  4

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