↑

Properties

Label 12T54
Order \(96\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_4:S_4$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $54$
Group :  $C_4:S_4$
CHM label :  $[(1/2.2^{2})^{3}]D(6)_{4}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12)(2,3), (1,2)(3,12)(4,11)(5,10)(6,9)(7,8), (2,3)(4,9)(5,8)(6,10)(7,11), (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:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
8:  $D_{4}$
12:  $D_{6}$
24:  $S_4$, $D_{12}$
48:  $S_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $D_{6}$

Low degree siblings

12T54, 16T191 x 2, 24T128 x 2, 24T170, 24T171 x 2, 24T172 x 2, 32T397

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

Group invariants

Order:  $96=2^{5} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [96, 187]
Character table:    
      2  5  5  5  3  3  3  3  4  4  2  2   2   2  5
      3  1  .  .  .  .  .  .  .  1  1  1   1   1  1

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

X.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
X.3      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
X.5      2  2  2  .  .  .  . -2 -2 -1 -1   1   1  2
X.6      2  2  2  .  .  .  .  2  2 -1 -1  -1  -1  2
X.7      2 -2  2  .  .  .  .  .  .  2 -2   .   . -2
X.8      2 -2  2  .  .  .  .  .  . -1  1   A  -A -2
X.9      2 -2  2  .  .  .  .  .  . -1  1  -A   A -2
X.10     3 -1 -1 -1  1 -1  1 -1  3  .  .   .   .  3
X.11     3 -1 -1 -1  1  1 -1  1 -3  .  .   .   .  3
X.12     3 -1 -1  1 -1 -1  1  1 -3  .  .   .   .  3
X.13     3 -1 -1  1 -1  1 -1 -1  3  .  .   .   .  3
X.14     6  2 -2  .  .  .  .  .  .  .  .   .   . -6

A = -E(12)^7+E(12)^11
  = Sqrt(3) = r3