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Properties

Label 12T89
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 $t$ :  $89$
Group :  $C_2\times C_4:D_4:C_3$
CHM label :  $[(1/2.2^{2})^{3}]2A_{4}(6)_{4}{n4}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12)(2,3), (1,12)(4,6)(5,7)(10,11), (1,7,9,3,5,11)(2,4,10,12,6,8)
$|\Aut(F/K)|$:  $2$

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: $A_4\times C_2$

Low degree siblings

12T89, 12T92 x 2, 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, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 4, 4, 1, 1, 1, 1 $ $6$ $4$ $( 4, 6, 5, 7)( 8,10, 9,11)$
$ 4, 4, 1, 1, 1, 1 $ $6$ $4$ $( 4, 6, 5, 7)( 8,11, 9,10)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $12$ $2$ $( 2, 3)( 6, 7)( 8,10)( 9,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $12$ $2$ $( 2, 3)( 6, 7)( 8,11)( 9,10)$
$ 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3,12)( 4, 6)( 5, 7)( 8,10)( 9,11)$
$ 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3,12)( 4, 6)( 5, 7)( 8,11)( 9,10)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 2,12, 3)( 4, 5)( 6, 7)( 8,10, 9,11)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 2,12, 3)( 4, 5)( 6, 7)( 8,11, 9,10)$
$ 6, 6 $ $16$ $6$ $( 1, 4, 8, 2, 6,11)( 3, 7,10,12, 5, 9)$
$ 6, 6 $ $16$ $6$ $( 1, 4, 8, 3, 7,10)( 2, 6,11,12, 5, 9)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 4, 8)( 2, 7,11)( 3, 6,10)( 5, 9,12)$
$ 6, 6 $ $16$ $6$ $( 1, 4, 8,12, 5, 9)( 2, 7,11, 3, 6,10)$
$ 6, 6 $ $16$ $6$ $( 1, 8, 6, 2,10, 4)( 3,11, 5,12, 9, 7)$
$ 6, 6 $ $16$ $6$ $( 1, 8, 7, 3,11, 4)( 2,10, 5,12, 9, 6)$
$ 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, 1002]
Character table:    
      2  6  6  6  5  5  4  4  4  4  5  5   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 4a 4b 2c 2d 2e 2f 4c 4d  6a  6b 3a  6c  6d  6e 3b  6f 2g
     2P 1a 1a 1a 2b 2b 1a 1a 1a 1a 2b 2b  3b  3b 3b  3b  3a  3a 3a  3a 1a
     3P 1a 2a 2b 4a 4b 2c 2d 2e 2f 4c 4d  2f  2e 1a  2g  2e  2f 1a  2g 2g
     5P 1a 2a 2b 4a 4b 2c 2d 2e 2f 4c 4d  6e  6d 3b  6f  6b  6a 3a  6c 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 -1  1   A  -A  A  -A -/A  /A /A -/A -1
X.6      1 -1  1 -1  1 -1  1 -1  1 -1  1  /A -/A /A -/A  -A   A  A  -A -1
X.7      1 -1  1 -1  1  1 -1  1 -1 -1  1 -/A  /A /A -/A   A  -A  A  -A -1
X.8      1 -1  1 -1  1  1 -1  1 -1 -1  1  -A   A  A  -A  /A -/A /A -/A -1
X.9      1  1  1  1  1 -1 -1 -1 -1  1  1 -/A -/A /A  /A  -A  -A  A   A  1
X.10     1  1  1  1  1 -1 -1 -1 -1  1  1  -A  -A  A   A -/A -/A /A  /A  1
X.11     1  1  1  1  1  1  1  1  1  1  1   A   A  A   A  /A  /A /A  /A  1
X.12     1  1  1  1  1  1  1  1  1  1  1  /A  /A /A  /A   A   A  A   A  1
X.13     3 -3  3  1 -1 -1  1  3 -3  1 -1   .   .  .   .   .   .  .   . -3
X.14     3 -3  3  1 -1  1 -1 -3  3  1 -1   .   .  .   .   .   .  .   . -3
X.15     3  3  3 -1 -1 -1 -1  3  3 -1 -1   .   .  .   .   .   .  .   .  3
X.16     3  3  3 -1 -1  1  1 -3 -3 -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