↑

Properties

Label 24T29
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times D_{12}$

Learn more about

Group action invariants

Degree $n$:  $24$
Transitive number $t$:  $29$
Group:  $C_2\times D_{12}$
Parity:  $1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $4$
Generators:  (3,12)(4,11)(5,22)(6,21)(7,8)(9,17)(10,18)(15,23)(16,24)(19,20), (1,24)(2,23)(3,21)(4,22)(5,20)(6,19)(7,17)(8,18)(9,15)(10,16)(11,13)(12,14), (1,20,2,19)(3,9,4,10)(5,24,6,23)(7,13,8,14)(11,17,12,18)(15,21,16,22)

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
$16$:  $D_4\times C_2$
$24$:  $S_3 \times C_2^2$, $D_{12}$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 6: $D_{6}$ x 3

Degree 8: $D_4\times C_2$

Degree 12: $S_3 \times C_2^2$, $D_{12}$ x 2

Low degree siblings

24T29 x 3

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 $ $2$ $12$ $( 1, 4,18,19, 9,12, 2, 3,17,20,10,11)( 5, 7,21,23,14,15, 6, 8,22,24,13,16)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1, 5, 9,14,17,22)( 2, 6,10,13,18,21)( 3, 8,11,16,19,23)( 4, 7,12,15,20,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 5)( 2, 6)( 3,15)( 4,16)( 7,11)( 8,12)( 9,22)(10,21)(13,18)(14,17)(19,24) (20,23)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1, 6, 9,13,17,21)( 2, 5,10,14,18,22)( 3, 7,11,15,19,24)( 4, 8,12,16,20,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,24)(10,23)(11,21)(12,22)(13,19)(14,20)(15,17) (16,18)$
$ 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 7, 2, 8)( 3,22, 4,21)( 5,12, 6,11)( 9,15,10,16)(13,19,14,20)(17,24,18,23)$
$ 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 $ $2$ $6$ $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,13)( 2,14)( 3,15)( 4,16)( 5,18)( 6,17)( 7,19)( 8,20)( 9,21)(10,22)(11,24) (12,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5,17)( 6,18)( 7,20)( 8,19)( 9,22)(10,21)(11,23) (12,24)$
$ 12, 12 $ $2$ $12$ $( 1,15,18, 8, 9,24, 2,16,17, 7,10,23)( 3, 5,20,21,11,14, 4, 6,19,22,12,13)$
$ 12, 12 $ $2$ $12$ $( 1,16,18, 7, 9,23, 2,15,17, 8,10,24)( 3, 6,20,22,11,13, 4, 5,19,21,12,14)$
$ 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, 36]
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 6a 2d 6b 2e 4a 3a 6c 2f 2g 12c 12d 4b
     2P 1a 1a 1a 1a  6c  6c 3a 1a 3a 1a 2b 3a 3a 1a 1a  6c  6c 2b
     3P 1a 2a 2b 2c  4b  4b 2g 2d 2f 2e 4a 1a 2b 2f 2g  4a  4a 4b
     5P 1a 2a 2b 2c 12b 12a 6a 2d 6b 2e 4a 3a 6c 2f 2g 12d 12c 4b
     7P 1a 2a 2b 2c 12b 12a 6a 2d 6b 2e 4a 3a 6c 2f 2g 12d 12c 4b
    11P 1a 2a 2b 2c 12a 12b 6a 2d 6b 2e 4a 3a 6c 2f 2g 12c 12d 4b

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  .  2  .  .  2 -2  2 -2   .   .  .
X.10     2  . -2  .   .   .  2  . -2  .  .  2 -2 -2  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  .   1   1 -1  . -1  . -2 -1 -1  2  2   1   1 -2
X.14     2  .  2  .   1   1  1  .  1  .  2 -1 -1 -2 -2  -1  -1 -2
X.15     2  . -2  .   A  -A -1  .  1  .  . -1  1 -2  2  -A   A  .
X.16     2  . -2  .  -A   A -1  .  1  .  . -1  1 -2  2   A  -A  .
X.17     2  . -2  .   A  -A  1  . -1  .  . -1  1  2 -2   A  -A  .
X.18     2  . -2  .  -A   A  1  . -1  .  . -1  1  2 -2  -A   A  .

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