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Properties

Label 12T104
Order \(192\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_2^2.\SL(2,3)):C_2$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $104$
Group :  $(C_2^2.\SL(2,3)):C_2$
CHM label :  $[(1/2.2^{2})^{3}]2A_{4}(6)_{8}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12)(2,3), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (6,7)(8,10)(9,11)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
12:  $A_4$
24:  $A_4\times C_2$
48:  16T60
96:  12T60

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

12T104 x 3, 24T496 x 2, 24T497 x 2, 24T498 x 2, 24T499 x 4, 24T500 x 4

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, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 8, 9)(10,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $12$ $2$ $( 6, 7)( 8,10)( 9,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 2, 1, 1 $ $12$ $2$ $( 2, 3)( 4, 6)( 5, 7)( 8, 9)(10,11)$
$ 4, 2, 2, 2, 1, 1 $ $12$ $4$ $( 2, 3)( 4, 6, 5, 7)( 8,10)( 9,11)$
$ 4, 2, 2, 2, 1, 1 $ $12$ $4$ $( 2, 3)( 4, 6, 5, 7)( 8,11)( 9,10)$
$ 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 $ $16$ $12$ $( 1, 4,10, 3, 7, 9,12, 5,11, 2, 6, 8)$
$ 12 $ $16$ $12$ $( 1, 4,10, 2, 6, 8,12, 5,11, 3, 7, 9)$
$ 6, 6 $ $16$ $6$ $( 1, 4,10,12, 5,11)( 2, 6, 9, 3, 7, 8)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 4,10)( 2, 6, 9)( 3, 7, 8)( 5,11,12)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 8, 6)( 2,10, 4)( 3,11, 5)( 7,12, 9)$
$ 6, 6 $ $16$ $6$ $( 1, 8, 7,12, 9, 6)( 2,10, 5, 3,11, 4)$
$ 12 $ $16$ $12$ $( 1, 8, 4, 2,10, 6,12, 9, 5, 3,11, 7)$
$ 12 $ $16$ $12$ $( 1, 8, 5, 3,11, 6,12, 9, 4, 2,10, 7)$
$ 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, 194]
Character table:    
      2  6  6  4  6  4  4  4  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 4c 4d 12a 12b  6a  3a  3b  6b 12c 12d 2e
     2P 1a 1a 1a 1a 1a 2a 2a 2e 2e  6b  6b  3b  3b  3a  3a  6a  6a 1a
     3P 1a 2a 2b 2c 2d 4a 4b 4d 4c  4c  4d  2e  1a  1a  2e  4c  4d 2e
     5P 1a 2a 2b 2c 2d 4a 4b 4c 4d 12c 12d  6b  3b  3a  6a 12a 12b 2e
     7P 1a 2a 2b 2c 2d 4a 4b 4d 4c 12b 12a  6a  3a  3b  6b 12d 12c 2e
    11P 1a 2a 2b 2c 2d 4a 4b 4d 4c 12d 12c  6b  3b  3a  6a 12b 12a 2e

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

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