Properties

Label 24T34
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{24}$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $34$
Group :  $D_{24}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8)(2,7)(3,6)(4,5)(9,23)(10,24)(11,21)(12,22)(13,19)(14,20)(15,18)(16,17), (1,6)(2,5)(7,23)(8,24)(9,21)(10,22)(11,19)(12,20)(13,17)(14,18)(15,16)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $D_{6}$

Degree 8: $D_{8}$

Degree 12: $D_{12}$

Low degree siblings

24T34

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

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 7]
Character table:   
      2  4  2  4  2   3   3   3   3  3  3  3  3   3   3  3
      3  1  .  1  .   1   1   1   1  1  1  1  1   1   1  1

        1a 2a 2b 2c 24a 24b 12a 12b 8a 8b 3a 6a 24c 24d 4a
     2P 1a 1a 1a 1a 12b 12b  6a  6a 4a 4a 3a 3a 12a 12a 2b
     3P 1a 2a 2b 2c  8b  8a  4a  4a 8b 8a 1a 2b  8b  8a 4a
     5P 1a 2a 2b 2c 24d 24c 12b 12a 8b 8a 3a 6a 24b 24a 4a
     7P 1a 2a 2b 2c 24c 24d 12b 12a 8a 8b 3a 6a 24a 24b 4a
    11P 1a 2a 2b 2c 24b 24a 12a 12b 8b 8a 3a 6a 24d 24c 4a
    13P 1a 2a 2b 2c 24b 24a 12a 12b 8b 8a 3a 6a 24d 24c 4a
    17P 1a 2a 2b 2c 24c 24d 12b 12a 8a 8b 3a 6a 24a 24b 4a
    19P 1a 2a 2b 2c 24d 24c 12b 12a 8b 8a 3a 6a 24b 24a 4a
    23P 1a 2a 2b 2c 24a 24b 12a 12b 8a 8b 3a 6a 24c 24d 4a

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      2  .  2  .   .   .  -2  -2  .  .  2  2   .   . -2
X.6      2  .  2  .   1   1  -1  -1 -2 -2 -1 -1   1   1  2
X.7      2  .  2  .  -1  -1  -1  -1  2  2 -1 -1  -1  -1  2
X.8      2  . -2  .   A  -A   .   .  A -A  2 -2   A  -A  .
X.9      2  . -2  .  -A   A   .   . -A  A  2 -2  -A   A  .
X.10     2  . -2  .   B  -B  -D   D -A  A -1  1   C  -C  .
X.11     2  . -2  .   C  -C   D  -D -A  A -1  1   B  -B  .
X.12     2  . -2  .  -C   C   D  -D  A -A -1  1  -B   B  .
X.13     2  . -2  .  -B   B  -D   D  A -A -1  1  -C   C  .
X.14     2  .  2  .   D   D   1   1  .  . -1 -1  -D  -D -2
X.15     2  .  2  .  -D  -D   1   1  .  . -1 -1   D   D -2

A = -E(8)+E(8)^3
  = -Sqrt(2) = -r2
B = -E(24)+E(24)^11
C = -E(24)^17+E(24)^19
D = -E(12)^7+E(12)^11
  = Sqrt(3) = r3