Properties

Label 24T19
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{12}:C_2$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $19$
Group :  $D_{12}:C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,24,2,23)(3,18,4,17)(5,19,6,20)(7,13,8,14)(9,15,10,16)(11,22,12,21), (1,12,2,11)(3,5,4,6)(7,9,8,10)(13,15,14,16)(17,20,18,19)(21,23,22,24), (1,19)(2,20)(3,13)(4,14)(5,15)(6,16)(7,22)(8,21)(9,24)(10,23)(11,18)(12,17)
$|\Aut(F/K)|$:  $12$

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:  $C_2^3$
12:  $D_{6}$ x 3
16:  $Q_8:C_2$
24:  $S_3 \times C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $S_3$, $D_{6}$ x 2

Degree 8: $Q_8:C_2$

Degree 12: $D_6$

Low degree siblings

24T24 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$
$ 12, 12 $ $2$ $12$ $( 1, 3,10,12, 5, 7, 2, 4, 9,11, 6, 8)(13,19,21,16,18,24,14,20,22,15,17,23)$
$ 12, 12 $ $2$ $12$ $( 1, 3,10,12, 5, 7, 2, 4, 9,11, 6, 8)(13,20,21,15,18,23,14,19,22,16,17,24)$
$ 12, 12 $ $2$ $12$ $( 1, 4,10,11, 5, 8, 2, 3, 9,12, 6, 7)(13,19,21,16,18,24,14,20,22,15,17,23)$
$ 12, 12 $ $2$ $12$ $( 1, 4,10,11, 5, 8, 2, 3, 9,12, 6, 7)(13,20,21,15,18,23,14,19,22,16,17,24)$
$ 6, 6, 3, 3, 3, 3 $ $2$ $6$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)(13,17,22,14,18,21)(15,20,24,16,19,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)(13,18,22)(14,17,21)(15,19,24) (16,20,23)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1, 6, 9, 2, 5,10)( 3, 8,11, 4, 7,12)(13,17,22,14,18,21)(15,20,24,16,19,23)$
$ 6, 6, 3, 3, 3, 3 $ $2$ $6$ $( 1, 6, 9, 2, 5,10)( 3, 8,11, 4, 7,12)(13,18,22)(14,17,21)(15,19,24)(16,20,23)$
$ 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,11, 2,12)( 3, 6, 4, 5)( 7,10, 8, 9)(13,15,14,16)(17,20,18,19)(21,23,22,24)$
$ 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,11, 2,12)( 3, 6, 4, 5)( 7,10, 8, 9)(13,16,14,15)(17,19,18,20)(21,24,22,23)$
$ 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,12, 2,11)( 3, 5, 4, 6)( 7, 9, 8,10)(13,15,14,16)(17,20,18,19)(21,23,22,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1,13)( 2,14)( 3,23)( 4,24)( 5,22)( 6,21)( 7,20)( 8,19)( 9,18)(10,17)(11,16) (12,15)$
$ 4, 4, 4, 4, 4, 4 $ $6$ $4$ $( 1,13, 2,14)( 3,23, 4,24)( 5,22, 6,21)( 7,20, 8,19)( 9,18,10,17)(11,16,12,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1,15)( 2,16)( 3,22)( 4,21)( 5,24)( 6,23)( 7,18)( 8,17)( 9,19)(10,20)(11,13) (12,14)$
$ 4, 4, 4, 4, 4, 4 $ $6$ $4$ $( 1,15, 2,16)( 3,22, 4,21)( 5,24, 6,23)( 7,18, 8,17)( 9,19,10,20)(11,13,12,14)$

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 37]
Character table:   
      2  4  3  4   3   3   3   3  3  3  3  3  3  4  4  3  3  3  3
      3  1  1  1   1   1   1   1  1  1  1  1  1  1  1  .  .  .  .

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

X.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
X.3      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
X.5      1 -1  1   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 -1  1  1
X.7      1  1  1  -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 -1 -1 -1
X.9      2 -2  2  -1   1   1  -1  1 -1 -1  1  2 -2 -2  .  .  .  .
X.10     2 -2  2   1  -1  -1   1  1 -1 -1  1 -2  2  2  .  .  .  .
X.11     2  2  2  -1  -1  -1  -1 -1 -1 -1 -1  2  2  2  .  .  .  .
X.12     2  2  2   1   1   1   1 -1 -1 -1 -1 -2 -2 -2  .  .  .  .
X.13     2  . -2   .   B  -B   .  .  2 -2  .  .  B -B  .  .  .  .
X.14     2  . -2   .  -B   B   .  .  2 -2  .  . -B  B  .  .  .  .
X.15     2  . -2   A   C  -C  -A  D -1  1 -D  . -B  B  .  .  .  .
X.16     2  . -2   A  -C   C  -A -D -1  1  D  .  B -B  .  .  .  .
X.17     2  . -2  -A   C  -C   A -D -1  1  D  . -B  B  .  .  .  .
X.18     2  . -2  -A  -C   C   A  D -1  1 -D  .  B -B  .  .  .  .

A = -E(12)^7+E(12)^11
  = Sqrt(3) = r3
B = -2*E(4)
  = -2*Sqrt(-1) = -2i
C = -E(4)
  = -Sqrt(-1) = -i
D = -E(3)+E(3)^2
  = -Sqrt(-3) = -i3