Properties

Label 24T18
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_4:S_3$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $18$
Group :  $D_4:S_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,10,2,5,9)(3,7,12,4,8,11)(13,17,22)(14,18,21)(15,19,24)(16,20,23), (1,11,2,12)(3,10,4,9)(5,7,6,8)(13,20,14,19)(15,17,16,18)(21,24,22,23), (1,14,2,13)(3,23,4,24)(5,21,6,22)(7,19,8,20)(9,17,10,18)(11,15,12,16)
$|\Aut(F/K)|$:  $12$

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

24T18, 24T23

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

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 39]
Character table:   
      2  4  3  4  3  4  4  3  2  3  3  3  2   2  3  3
      3  1  1  1  .  .  .  1  1  1  .  .  1   1  1  1

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

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  2  .  .  . -1  1 -1  .  . -1   1  2 -2
X.10     2 -2  2  .  .  . -1  1 -1  .  .  1  -1 -2  2
X.11     2  2  2  .  .  . -1 -1 -1  .  . -1  -1  2  2
X.12     2  2  2  .  .  . -1 -1 -1  .  .  1   1 -2 -2
X.13     2  . -2  .  A -A  2  . -2  .  .  .   .  .  .
X.14     2  . -2  . -A  A  2  . -2  .  .  .   .  .  .
X.15     4  . -4  .  .  . -2  .  2  .  .  .   .  .  .

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