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Properties

Label 12T95
Order \(192\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_4^2:C_3:C_2$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $95$
Group :  $C_2\times C_4^2:C_3:C_2$
CHM label :  $[1/2.4^{3}]S(3)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,9)(6,12), (1,5)(2,10)(4,8)(7,11), (1,10,7,4)(3,6,9,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
12:  $D_{6}$
24:  $S_4$
48:  $S_4\times C_2$
96:  12T62

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4$

Low degree siblings

12T95, 12T96, 12T97, 24T471, 24T472 x 2, 24T473 x 2, 24T474 x 2, 24T475 x 2, 24T476, 24T477, 24T478, 24T479 x 2, 24T480 x 2, 32T2216 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 $ $3$ $2$ $( 3, 9)( 6,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $12$ $2$ $( 2, 3)( 5, 6)( 8, 9)(11,12)$
$ 4, 4, 1, 1, 1, 1 $ $12$ $4$ $( 2, 3, 8, 9)( 5, 6,11,12)$
$ 4, 4, 1, 1, 1, 1 $ $3$ $4$ $( 2, 5, 8,11)( 3, 6, 9,12)$
$ 4, 4, 1, 1, 1, 1 $ $6$ $4$ $( 2, 5, 8,11)( 3,12, 9, 6)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 8)( 3, 9)( 5,11)( 6,12)$
$ 4, 4, 1, 1, 1, 1 $ $3$ $4$ $( 2,11, 8, 5)( 3,12, 9, 6)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3, 9)( 4, 5)( 6,12)( 7, 8)(10,11)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)$
$ 6, 6 $ $32$ $6$ $( 1, 2, 3, 7, 8, 9)( 4, 5, 6,10,11,12)$
$ 8, 4 $ $12$ $8$ $( 1, 2, 4, 5, 7, 8,10,11)( 3, 6, 9,12)$
$ 8, 4 $ $12$ $8$ $( 1, 2, 4, 5, 7, 8,10,11)( 3,12, 9, 6)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 2, 7, 8)( 3, 9)( 4, 5,10,11)( 6,12)$
$ 8, 4 $ $12$ $8$ $( 1, 2,10,11, 7, 8, 4, 5)( 3, 6, 9,12)$
$ 8, 4 $ $12$ $8$ $( 1, 2,10,11, 7, 8, 4, 5)( 3,12, 9, 6)$
$ 4, 4, 2, 2 $ $3$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 9)( 6,12)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 4, 7,10)( 2, 8)( 3,12, 9, 6)( 5,11)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
$ 4, 4, 2, 2 $ $3$ $4$ $( 1, 7)( 2,11, 8, 5)( 3,12, 9, 6)( 4,10)$

Group invariants

Order:  $192=2^{6} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [192, 944]
Character table:    
      2  6  6  4  4   6  5  6   6  4  1  1  4  4  4  4  4  6  5  6  6
      3  1  .  .  .   .  .  .   .  .  1  1  .  .  .  .  .  .  .  1  .

        1a 2a 2b 4a  4b 4c 2c  4d 2d 3a 6a 8a 8b 4e 8c 8d 4f 4g 2e 4h
     2P 1a 1a 1a 2c  2c 2c 1a  2c 1a 3a 3a 4f 4f 2c 4h 4h 2c 2c 1a 2c
     3P 1a 2a 2b 4a  4d 4c 2c  4b 2d 1a 2e 8d 8c 4e 8b 8a 4h 4g 2e 4f
     5P 1a 2a 2b 4a  4b 4c 2c  4d 2d 3a 6a 8a 8b 4e 8c 8d 4f 4g 2e 4h
     7P 1a 2a 2b 4a  4d 4c 2c  4b 2d 3a 6a 8d 8c 4e 8b 8a 4h 4g 2e 4f

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

A = -1-2*E(4)
  = -1-2*Sqrt(-1) = -1-2i
B = E(4)
  = Sqrt(-1) = i