Properties

Label 24T33
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_6:C_4$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $33$
Group :  $D_6:C_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,12)(4,11)(5,22)(6,21)(7,8)(9,17)(10,18)(15,23)(16,24)(19,20), (1,15,6,19,9,24,13,3,17,7,21,11)(2,16,5,20,10,23,14,4,18,8,22,12)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$
8:  $D_{4}$ x 2, $C_4\times C_2$
12:  $D_{6}$
16:  $C_2^2:C_4$
24:  $S_3 \times C_4$, $D_{12}$, $(C_6\times C_2):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 6: $D_{6}$

Degree 8: $C_2^2:C_4$

Degree 12: $S_3 \times C_4$, $D_{12}$, $(C_6\times C_2):C_2$

Low degree siblings

24T33

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

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 14]
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 12a 4a 12b 6a 2c 6b 4b 4c 3a 6c 2d 2e 12c 12d 4d
     2P 1a 1a 1a  6b 2e  6b 3a 1a 3a 2d 2e 3a 3a 1a 1a  6b  6b 2d
     3P 1a 2a 2b  4b 4c  4b 2e 2c 2d 4d 4a 1a 2b 2d 2e  4d  4d 4b
     5P 1a 2a 2b 12b 4a 12a 6a 2c 6b 4b 4c 3a 6c 2d 2e 12d 12c 4d
     7P 1a 2a 2b 12c 4c 12d 6a 2c 6b 4d 4a 3a 6c 2d 2e 12a 12b 4b
    11P 1a 2a 2b 12d 4c 12c 6a 2c 6b 4d 4a 3a 6c 2d 2e 12b 12a 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   A -A   A -1  1 -1 -A  A  1  1 -1 -1  -A  -A  A
X.6      1 -1  1  -A  A  -A -1  1 -1  A -A  1  1 -1 -1   A   A -A
X.7      1  1  1   A  A   A -1 -1 -1 -A -A  1  1 -1 -1  -A  -A  A
X.8      1  1  1  -A -A  -A -1 -1 -1  A  A  1  1 -1 -1   A   A -A
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   B  .  -B  1  . -1  .  . -1  1  2 -2   B  -B  .
X.14     2  . -2  -B  .   B  1  . -1  .  . -1  1  2 -2  -B   B  .
X.15     2  . -2   C  .  -C -1  .  1  .  . -1  1 -2  2  -C   C  .
X.16     2  . -2  -C  .   C -1  .  1  .  . -1  1 -2  2   C  -C  .
X.17     2  .  2   A  .   A  1  .  1  D  . -1 -1 -2 -2  -A  -A -D
X.18     2  .  2  -A  .  -A  1  .  1 -D  . -1 -1 -2 -2   A   A  D

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