Properties

Label 12T28
Order \(48\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times D_4$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $28$
Group :  $S_3\times D_4$
CHM label :  $D(4)[x]S(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7)(3,9)(5,11), (1,5)(2,10)(4,8)(7,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
6:  $S_3$
8:  $D_{4}$ x 2, $C_2^3$
12:  $D_{6}$ x 3
16:  $D_4\times C_2$
24:  $S_3 \times C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $D_{6}$

Low degree siblings

12T28 x 3, 24T52 x 2, 24T53 x 2, 24T54 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, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 6)( 3,11)( 5, 9)( 8,12)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 2, 8)( 4,10)( 6,12)$
$ 2, 2, 2, 2, 2, 1, 1 $ $6$ $2$ $( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
$ 12 $ $4$ $12$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$
$ 4, 4, 4 $ $6$ $4$ $( 1, 2, 7, 8)( 3,12, 9, 6)( 4, 5,10,11)$
$ 6, 6 $ $4$ $6$ $( 1, 2, 9,10, 5, 6)( 3, 4,11,12, 7, 8)$
$ 6, 6 $ $2$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
$ 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 3)( 2, 8)( 4, 6)( 5,11)( 7, 9)(10,12)$
$ 6, 3, 3 $ $4$ $6$ $( 1, 3, 5, 7, 9,11)( 2,10, 6)( 4,12, 8)$
$ 4, 4, 4 $ $2$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$
$ 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$
$ 3, 3, 3, 3 $ $2$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 38]
Character table:   
      2  4  4  3  3  3   2  3  2  3  4  2  3  3  3  4
      3  1  .  1  .  .   1  .  1  1  .  1  1  1  1  1

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

X.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
X.3      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
X.5      1 -1  1 -1  1  -1  1 -1  1 -1  1 -1 -1  1  1
X.6      1  1 -1 -1 -1   1  1 -1  1  1 -1  1 -1  1  1
X.7      1  1 -1 -1  1  -1 -1  1  1  1 -1 -1  1  1  1
X.8      1  1  1  1 -1  -1 -1 -1  1  1  1 -1 -1  1  1
X.9      2  . -2  .  .  -1  .  1 -1  .  1  2 -2 -1  2
X.10     2  . -2  .  .   1  . -1 -1  .  1 -2  2 -1  2
X.11     2  .  2  .  .  -1  . -1 -1  . -1  2  2 -1  2
X.12     2  .  2  .  .   1  .  1 -1  . -1 -2 -2 -1  2
X.13     2 -2  .  .  .   .  .  . -2  2  .  .  .  2 -2
X.14     2  2  .  .  .   .  .  . -2 -2  .  .  .  2 -2
X.15     4  .  .  .  .   .  .  .  2  .  .  .  . -2 -4