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Properties

Label 12T26
Order \(48\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2 \times A_4$

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $26$
Group :  $C_2^2 \times A_4$
CHM label :  $A_{4}(12)x2^{2}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,5)(2,4,3)(6,8,7)(10,12,11), (1,4)(2,5)(3,9)(6,12)(7,10)(8,11), (1,7)(2,11)(3,12)(4,10)(5,8)(6,9), (1,11,6)(2,9,7)(3,10,5)(4,8,12)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $C_6$ x 3
12:  $A_4$, $C_6\times C_2$
24:  $A_4\times C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: $A_4\times C_2$ x 3

Low degree siblings

12T25 x 3, 12T26, 16T58, 24T49 x 3, 24T50

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 49]
Character table:    
      2  4  4  4  4   2   2   2  2   2   2   2  2  4  4  4  4
      3  1  .  .  .   1   1   1  1   1   1   1  1  1  .  1  1

        1a 2a 2b 2c  6a  6b  6c 3a  6d  6e  6f 3b 2d 2e 2f 2g
     2P 1a 1a 1a 1a  3b  3b  3b 3b  3a  3a  3a 3a 1a 1a 1a 1a
     3P 1a 2a 2b 2c  2g  2f  2d 1a  2d  2g  2f 1a 2d 2e 2f 2g
     5P 1a 2a 2b 2c  6e  6f  6d 3b  6c  6a  6b 3a 2d 2e 2f 2g

X.1      1  1  1  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 -1  1 -1  1
X.3      1 -1  1 -1  -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 -1  1  1 -1
X.5      1 -1 -1  1   A  -A  -A  A -/A  /A -/A /A -1  1 -1  1
X.6      1 -1 -1  1  /A -/A -/A /A  -A   A  -A  A -1  1 -1  1
X.7      1 -1  1 -1 -/A -/A  /A /A   A  -A  -A  A  1  1 -1 -1
X.8      1 -1  1 -1  -A  -A   A  A  /A -/A -/A /A  1  1 -1 -1
X.9      1  1 -1 -1 -/A  /A -/A /A  -A  -A   A  A -1  1  1 -1
X.10     1  1 -1 -1  -A   A  -A  A -/A -/A  /A /A -1  1  1 -1
X.11     1  1  1  1   A   A   A  A  /A  /A  /A /A  1  1  1  1
X.12     1  1  1  1  /A  /A  /A /A   A   A   A  A  1  1  1  1
X.13     3 -1 -1 -1   .   .   .  .   .   .   .  .  3 -1  3  3
X.14     3 -1  1  1   .   .   .  .   .   .   .  . -3 -1  3 -3
X.15     3  1 -1  1   .   .   .  .   .   .   .  .  3 -1 -3 -3
X.16     3  1  1 -1   .   .   .  .   .   .   .  . -3 -1 -3  3

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3