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Properties

Label 12T92
Order \(192\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_4:D_4:C_3$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $92$
Group :  $C_2\times C_4:D_4:C_3$
CHM label :  $[2^{4}]A_{4}(6)_{4}{n4}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12)(2,3)(4,5), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,12)(6,7), (2,8)(3,9)(4,10)(5,11)(6,7)
$|\Aut(F/K)|$:  $2$

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
48:  $C_2^2 \times A_4$
96:  12T60

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: $A_4$

Low degree siblings

12T89 x 2, 12T92, 24T453 x 2, 24T454 x 2, 24T455 x 4, 24T456 x 4, 24T463, 24T464, 24T465 x 4, 24T466 x 4, 32T2188 x 2

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, 2, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 4, 5)(10,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 4, 5)( 6, 7)( 8, 9)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 4, 5)( 8, 9)(10,11)$
$ 4, 4, 1, 1, 1, 1 $ $6$ $4$ $( 2, 8, 3, 9)( 4,10, 5,11)$
$ 2, 2, 2, 2, 2, 1, 1 $ $12$ $2$ $( 2, 8)( 3, 9)( 4,10)( 5,11)( 6, 7)$
$ 4, 4, 1, 1, 1, 1 $ $6$ $4$ $( 2, 8, 3, 9)( 4,11, 5,10)$
$ 2, 2, 2, 2, 2, 1, 1 $ $12$ $2$ $( 2, 8)( 3, 9)( 4,11)( 5,10)( 6, 7)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 2, 4)( 3, 5,12)( 6, 8,11)( 7, 9,10)$
$ 6, 3, 3 $ $16$ $6$ $( 1, 2, 4)( 3, 5,12)( 6, 9,11, 7, 8,10)$
$ 6, 6 $ $16$ $6$ $( 1, 2, 4,12, 3, 5)( 6, 8,11, 7, 9,10)$
$ 6, 3, 3 $ $16$ $6$ $( 1, 2, 4,12, 3, 5)( 6, 9,11)( 7, 8,10)$
$ 6, 3, 3 $ $16$ $6$ $( 1, 4, 2)( 3,12, 5)( 6,10, 8, 7,11, 9)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 4, 2)( 3,12, 5)( 6,11, 8)( 7,10, 9)$
$ 6, 3, 3 $ $16$ $6$ $( 1, 4, 3,12, 5, 2)( 6,10, 9)( 7,11, 8)$
$ 6, 6 $ $16$ $6$ $( 1, 4, 3,12, 5, 2)( 6,11, 9, 7,10, 8)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 6,12, 7)( 2, 3)( 4,10, 5,11)( 8, 9)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 6,12, 7)( 2, 3)( 4,11, 5,10)( 8, 9)$
$ 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, 1002]
Character table:    
      2  6  4  6  4  6  5  4  5  4  2   2   2   2   2  2   2   2  5  5  6
      3  1  1  .  1  .  .  .  .  .  1   1   1   1   1  1   1   1  .  .  1

        1a 2a 2b 2c 2d 4a 2e 4b 2f 3a  6a  6b  6c  6d 3b  6e  6f 4c 4d 2g
     2P 1a 1a 1a 1a 1a 2d 1a 2d 1a 3b  3b  3b  3b  3a 3a  3a  3a 2d 2d 1a
     3P 1a 2a 2b 2c 2d 4a 2e 4b 2f 1a  2a  2g  2c  2a 1a  2c  2g 4c 4d 2g
     5P 1a 2a 2b 2c 2d 4a 2e 4b 2f 3b  6d  6f  6e  6a 3a  6c  6b 4c 4d 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  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  -1 -1  1 -1
X.5      1 -1 -1  1  1  1 -1 -1  1  A  -A  -A   A -/A /A  /A -/A -1  1 -1
X.6      1 -1 -1  1  1  1 -1 -1  1 /A -/A -/A  /A  -A  A   A  -A -1  1 -1
X.7      1 -1  1 -1  1  1 -1  1 -1  A  -A   A  -A -/A /A -/A  /A  1  1  1
X.8      1 -1  1 -1  1  1 -1  1 -1 /A -/A  /A -/A  -A  A  -A   A  1  1  1
X.9      1  1 -1 -1  1  1  1 -1 -1  A   A  -A  -A  /A /A -/A -/A -1  1 -1
X.10     1  1 -1 -1  1  1  1 -1 -1 /A  /A -/A -/A   A  A  -A  -A -1  1 -1
X.11     1  1  1  1  1  1  1  1  1  A   A   A   A  /A /A  /A  /A  1  1  1
X.12     1  1  1  1  1  1  1  1  1 /A  /A  /A  /A   A  A   A   A  1  1  1
X.13     3 -3 -3  3  3 -1  1  1 -1  .   .   .   .   .  .   .   .  1 -1 -3
X.14     3 -3  3 -3  3 -1  1 -1  1  .   .   .   .   .  .   .   . -1 -1  3
X.15     3  3 -3 -3  3 -1 -1  1  1  .   .   .   .   .  .   .   .  1 -1 -3
X.16     3  3  3  3  3 -1 -1 -1 -1  .   .   .   .   .  .   .   . -1 -1  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(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3