Properties

Label 12T29
Order \(48\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_4\times A_4$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $29$
Group :  $C_4\times A_4$
CHM label :  $[1/2.4^{2}]3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7)(3,9)(4,10)(6,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $C_6$
12:  $A_4$, $C_{12}$
24:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: None

Degree 6: $C_6$

Low degree siblings

16T57, 24T55, 24T56

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

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 31]
Character table:   
      2  4  4  4   2   2   2   2  4  4  4   2   2   2   2  4  4
      3  1  .  .   1   1   1   1  1  .  .   1   1   1   1  1  1

        1a 2a 2b 12a 12b  6a  3a 4a 4b 4c  6b  3b 12c 12d 2c 4d
     2P 1a 1a 1a  6a  6a  3b  3b 2c 2c 2c  3a  3a  6b  6b 1a 2c
     3P 1a 2a 2b  4a  4d  2c  1a 4d 4c 4b  2c  1a  4a  4d 2c 4a
     5P 1a 2a 2b 12c 12d  6b  3b 4a 4b 4c  6a  3a 12a 12b 2c 4d
     7P 1a 2a 2b 12b 12a  6a  3a 4d 4c 4b  6b  3b 12d 12c 2c 4a
    11P 1a 2a 2b 12d 12c  6b  3b 4d 4c 4b  6a  3a 12b 12a 2c 4a

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   A  -A  -1   1 -A  A -A  -1   1   A  -A -1  A
X.4      1 -1  1  -A   A  -1   1  A -A  A  -1   1  -A   A -1 -A
X.5      1 -1  1   B  -B  /C -/C -A  A -A   C  -C -/B  /B -1  A
X.6      1 -1  1 -/B  /B   C  -C -A  A -A  /C -/C   B  -B -1  A
X.7      1 -1  1  /B -/B   C  -C  A -A  A  /C -/C  -B   B -1 -A
X.8      1 -1  1  -B   B  /C -/C  A -A  A   C  -C  /B -/B -1 -A
X.9      1  1  1   C   C -/C -/C -1 -1 -1  -C  -C  /C  /C  1 -1
X.10     1  1  1  /C  /C  -C  -C -1 -1 -1 -/C -/C   C   C  1 -1
X.11     1  1  1 -/C -/C  -C  -C  1  1  1 -/C -/C  -C  -C  1  1
X.12     1  1  1  -C  -C -/C -/C  1  1  1  -C  -C -/C -/C  1  1
X.13     3 -1 -1   .   .   .   . -3  1  1   .   .   .   .  3 -3
X.14     3 -1 -1   .   .   .   .  3 -1 -1   .   .   .   .  3  3
X.15     3  1 -1   .   .   .   .  D  A -A   .   .   .   . -3 -D
X.16     3  1 -1   .   .   .   . -D -A  A   .   .   .   . -3  D

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(12)^7
C = -E(3)
  = (1-Sqrt(-3))/2 = -b3
D = -3*E(4)
  = -3*Sqrt(-1) = -3i