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Properties

Label 12T1
Order \(12\)
n \(12\)
Cyclic Yes
Abelian Yes
Solvable Yes
Primitive No
$p$-group No
Group: $C_{12}$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $1$
Group :  $C_{12}$
CHM label :  $C(4)[x]C(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $1$
Generators:  (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12)
$|\Aut(F/K)|$:  $12$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: $C_4$

Degree 6: $C_6$

Low degree siblings

There are no siblings 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 $ $1$ $1$ $()$
$ 12 $ $1$ $12$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$
$ 6, 6 $ $1$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
$ 4, 4, 4 $ $1$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$
$ 3, 3, 3, 3 $ $1$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 12 $ $1$ $12$ $( 1, 6,11, 4, 9, 2, 7,12, 5,10, 3, 8)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
$ 12 $ $1$ $12$ $( 1, 8, 3,10, 5,12, 7, 2, 9, 4,11, 6)$
$ 3, 3, 3, 3 $ $1$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
$ 4, 4, 4 $ $1$ $4$ $( 1,10, 7, 4)( 2,11, 8, 5)( 3,12, 9, 6)$
$ 6, 6 $ $1$ $6$ $( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$
$ 12 $ $1$ $12$ $( 1,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$

Group invariants

Order:  $12=2^{2} \cdot 3$
Cyclic:  Yes
Abelian:  Yes
Solvable:  Yes
GAP id:  [12, 2]
Character table:    
      2  2   2   2  2   2   2  2   2   2  2   2   2
      3  1   1   1  1   1   1  1   1   1  1   1   1

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

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   A -/A -1  -A  /A  1   A -/A -1  -A  /A
X.4      1  /A  -A -1 -/A   A  1  /A  -A -1 -/A   A
X.5      1 -/A  -A  1 -/A  -A  1 -/A  -A  1 -/A  -A
X.6      1  -A -/A  1  -A -/A  1  -A -/A  1  -A -/A
X.7      1   B  -1 -B   1   B -1  -B   1  B  -1  -B
X.8      1  -B  -1  B   1  -B -1   B   1 -B  -1   B
X.9      1   C  /A -B  -A -/C -1  -C -/A  B   A  /C
X.10     1 -/C   A -B -/A   C -1  /C  -A  B  /A  -C
X.11     1  /C   A  B -/A  -C -1 -/C  -A -B  /A   C
X.12     1  -C  /A  B  -A  /C -1   C -/A -B   A -/C

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