Properties

Label 24T24
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 $t$ :  $24$
Group :  $D_{12}:C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,20)(2,19)(3,18)(4,17)(5,15)(6,16)(7,14)(8,13)(9,12)(10,11)(21,23)(22,24), (1,16,17,8,9,23)(2,15,18,7,10,24)(3,6,19,21,11,13)(4,5,20,22,12,14), (1,21,18,14,9,6,2,22,17,13,10,5)(3,24,20,16,11,7,4,23,19,15,12,8)
$|\Aut(F/K)|$:  $4$

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: $D_{6}$ x 3

Degree 8: $Q_8:C_2$

Degree 12: $S_3 \times C_2^2$

Low degree siblings

24T19, 24T24

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

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  3  4  4  3  3  3
      3  1  .  1  .   1   1   1  .   1  .  1  1  1  1  1  1  1  1

        1a 2a 2b 2c 12a 12b 12c 4a 12d 4b 2d 3a 6a 4c 4d 6b 6c 4e
     2P 1a 1a 1a 1a  6a  6a  6a 2b  6a 2b 1a 3a 3a 2b 2b 3a 3a 2b
     3P 1a 2a 2b 2c  4e  4e  4c 4a  4d 4b 2d 1a 2b 4d 4c 2d 2d 4e
     5P 1a 2a 2b 2c 12b 12a 12c 4a 12d 4b 2d 3a 6a 4c 4d 6c 6b 4e
     7P 1a 2a 2b 2c 12b 12a 12d 4a 12c 4b 2d 3a 6a 4d 4c 6b 6c 4e
    11P 1a 2a 2b 2c 12a 12b 12d 4a 12c 4b 2d 3a 6a 4d 4c 6c 6b 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  .  -1  -1  -1  .  -1  .  2 -1 -1  2  2 -1 -1  2
X.10     2  .  2  .  -1  -1   1  .   1  . -2 -1 -1 -2 -2  1  1  2
X.11     2  .  2  .   1   1  -1  .  -1  . -2 -1 -1  2  2  1  1 -2
X.12     2  .  2  .   1   1   1  .   1  .  2 -1 -1 -2 -2 -1 -1 -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  -A   C  .  -C  .  . -1  1 -B  B  D -D  .
X.16     2  . -2  .   A  -A  -C  .   C  .  . -1  1  B -B -D  D  .
X.17     2  . -2  .  -A   A   C  .  -C  .  . -1  1 -B  B -D  D  .
X.18     2  . -2  .  -A   A  -C  .   C  .  . -1  1  B -B  D -D  .

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