Properties

Label 12T25
Order \(48\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2 \times A_4$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $25$
Group :  $C_2^2 \times A_4$
CHM label :  $2A_{4}(6)[x]2=[1/4.2^{6}]3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7)(6,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11)
$|\Aut(F/K)|$:  $4$

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: $C_2$

Degree 3: $C_3$

Degree 4: None

Degree 6: $C_6$, $A_4\times C_2$ x 2

Low degree siblings

12T25 x 2, 12T26 x 2, 16T58, 24T49 x 3, 24T50

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

Group invariants

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

        1a 2a 2b  6a  6b  6c  3a  6d  6e  6f  3b 2c 2d 2e 2f 2g
     2P 1a 1a 1a  3b  3b  3b  3b  3a  3a  3a  3a 1a 1a 1a 1a 1a
     3P 1a 2a 2b  2g  2e  2f  1a  2g  2e  2f  1a 2c 2d 2e 2f 2g
     5P 1a 2a 2b  6d  6e  6f  3b  6a  6b  6c  3a 2c 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   A  -A   A  -A  /A -/A  /A -/A  1 -1  1 -1 -1
X.6      1 -1  1  /A -/A  /A -/A   A  -A   A  -A  1 -1  1 -1 -1
X.7      1 -1  1 -/A  /A  /A -/A  -A   A   A  -A -1  1 -1 -1  1
X.8      1 -1  1  -A   A   A  -A -/A  /A  /A -/A -1  1 -1 -1  1
X.9      1  1  1   A   A  -A  -A  /A  /A -/A -/A -1 -1 -1  1 -1
X.10     1  1  1  /A  /A -/A -/A   A   A  -A  -A -1 -1 -1  1 -1
X.11     1  1  1 -/A -/A -/A -/A  -A  -A  -A  -A  1  1  1  1  1
X.12     1  1  1  -A  -A  -A  -A -/A -/A -/A -/A  1  1  1  1  1
X.13     3 -1 -1   .   .   .   .   .   .   .   . -1 -1  3  3  3
X.14     3 -1 -1   .   .   .   .   .   .   .   .  1  1 -3  3 -3
X.15     3  1 -1   .   .   .   .   .   .   .   . -1  1  3 -3 -3
X.16     3  1 -1   .   .   .   .   .   .   .   .  1 -1 -3 -3  3

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