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Properties

Label 12T99
Order \(192\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^5.C_6$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $99$
Group :  $C_2^5.C_6$
CHM label :  $[(1/2.2^{2})^{3}]2A_{4}(6)_{2}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12)(2,3), (4,6)(5,7)(8,10)(9,11), (2,3)(4,5)(8,10,9,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:  $2$

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:  $A_4\times C_2$
48:  12T29
96:  $C_2^4:C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: $A_4\times C_2$

Low degree siblings

12T99, 12T105 x 2, 16T418 x 2, 24T483 x 2, 24T484 x 2, 24T501 x 2, 24T502 x 4, 24T503 x 2, 24T504 x 2, 24T505 x 2, 24T506 x 2, 24T507 x 4

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 $ $3$ $2$ $( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 4, 6)( 5, 7)( 8,10)( 9,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 4, 6)( 5, 7)( 8,11)( 9,10)$
$ 4, 2, 2, 1, 1, 1, 1 $ $12$ $4$ $( 2, 3)( 6, 7)( 8,10, 9,11)$
$ 4, 2, 2, 1, 1, 1, 1 $ $12$ $4$ $( 2, 3)( 6, 7)( 8,11, 9,10)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3,12)( 4, 5)( 6, 7)( 8,10)( 9,11)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3,12)( 4, 5)( 6, 7)( 8,11)( 9,10)$
$ 4, 4, 4 $ $4$ $4$ $( 1, 2,12, 3)( 4, 6, 5, 7)( 8,10, 9,11)$
$ 4, 4, 4 $ $4$ $4$ $( 1, 2,12, 3)( 4, 6, 5, 7)( 8,11, 9,10)$
$ 12 $ $16$ $12$ $( 1, 4, 8, 2, 6,11,12, 5, 9, 3, 7,10)$
$ 12 $ $16$ $12$ $( 1, 4, 8, 3, 7,10,12, 5, 9, 2, 6,11)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 4, 8)( 2, 7,11)( 3, 6,10)( 5, 9,12)$
$ 6, 6 $ $16$ $6$ $( 1, 4, 8,12, 5, 9)( 2, 7,11, 3, 6,10)$
$ 12 $ $16$ $12$ $( 1, 8, 6, 2,10, 5,12, 9, 7, 3,11, 4)$
$ 12 $ $16$ $12$ $( 1, 8, 7, 3,11, 5,12, 9, 6, 2,10, 4)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 8, 4)( 2,11, 7)( 3,10, 6)( 5,12, 9)$
$ 6, 6 $ $16$ $6$ $( 1, 8, 5,12, 9, 4)( 2,11, 6, 3,10, 7)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$

Group invariants

Order:  $192=2^{6} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [192, 191]
Character table:    
      2  6  6  6  5  5  4  4  5  5  4  4   2   2   2   2   2   2   2   2  6
      3  1  .  .  .  .  .  .  .  .  1  1   1   1   1   1   1   1   1   1  1

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

X.1      1  1  1  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  -1   1   1  1
X.3      1 -1  1  1 -1  A -A  1 -1 -A  A  -A   A   1  -1   A  -A   1  -1 -1
X.4      1 -1  1  1 -1 -A  A  1 -1  A -A   A  -A   1  -1  -A   A   1  -1 -1
X.5      1 -1  1  1 -1  A -A  1 -1 -A  A   C  -C -/D  /D  /C -/C  -D   D -1
X.6      1 -1  1  1 -1  A -A  1 -1 -A  A -/C  /C  -D   D  -C   C -/D  /D -1
X.7      1 -1  1  1 -1 -A  A  1 -1  A -A  /C -/C  -D   D   C  -C -/D  /D -1
X.8      1 -1  1  1 -1 -A  A  1 -1  A -A  -C   C -/D  /D -/C  /C  -D   D -1
X.9      1  1  1  1  1 -1 -1  1  1 -1 -1   D   D  -D  -D  /D  /D -/D -/D  1
X.10     1  1  1  1  1 -1 -1  1  1 -1 -1  /D  /D -/D -/D   D   D  -D  -D  1
X.11     1  1  1  1  1  1  1  1  1  1  1 -/D -/D -/D -/D  -D  -D  -D  -D  1
X.12     1  1  1  1  1  1  1  1  1  1  1  -D  -D  -D  -D -/D -/D -/D -/D  1
X.13     3  3  3 -1 -1 -1 -1 -1 -1  3  3   .   .   .   .   .   .   .   .  3
X.14     3  3  3 -1 -1  1  1 -1 -1 -3 -3   .   .   .   .   .   .   .   .  3
X.15     3 -3  3 -1  1  A -A -1  1  B -B   .   .   .   .   .   .   .   . -3
X.16     3 -3  3 -1  1 -A  A -1  1 -B  B   .   .   .   .   .   .   .   . -3
X.17     6 -2 -2 -2  2  .  .  2 -2  .  .   .   .   .   .   .   .   .   .  6
X.18     6 -2 -2  2 -2  .  . -2  2  .  .   .   .   .   .   .   .   .   .  6
X.19     6  2 -2 -2 -2  .  .  2  2  .  .   .   .   .   .   .   .   .   . -6
X.20     6  2 -2  2  2  .  . -2 -2  .  .   .   .   .   .   .   .   .   . -6

A = -E(4)
  = -Sqrt(-1) = -i
B = -3*E(4)
  = -3*Sqrt(-1) = -3i
C = E(12)^11
D = -E(3)
  = (1-Sqrt(-3))/2 = -b3