Properties

Label 24T14
Order \(24\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3:D_4$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $14$
Group :  $C_3:D_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9)(2,10)(3,8)(4,7)(5,6)(11,23)(12,24)(13,22)(14,21)(15,20)(16,19)(17,18), (1,19,14,7)(2,20,13,8)(3,6,15,18)(4,5,16,17)(9,11,21,24)(10,12,22,23)
$|\Aut(F/K)|$:  $24$

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:  $D_{6}$

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: $S_3$, $D_{6}$ x 2

Degree 8: $D_4$

Degree 12: $D_6$, $(C_6\times C_2):C_2$, $(C_6\times C_2):C_2$

Low degree siblings

12T13, 12T15

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

Group invariants

Order:  $24=2^{3} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [24, 8]
Character table:   
     2  3  2  2  2  2  2  2  2  3
     3  1  .  .  1  1  1  1  1  1

       1a 2a 4a 6a 6b 2b 3a 6c 2c
    2P 1a 1a 2c 3a 3a 1a 3a 3a 1a
    3P 1a 2a 4a 2b 2c 2b 1a 2b 2c
    5P 1a 2a 4a 6c 6b 2b 3a 6a 2c

X.1     1  1  1  1  1  1  1  1  1
X.2     1 -1 -1  1  1  1  1  1  1
X.3     1 -1  1 -1  1 -1  1 -1  1
X.4     1  1 -1 -1  1 -1  1 -1  1
X.5     2  .  .  1 -1 -2 -1  1  2
X.6     2  .  . -1 -1  2 -1 -1  2
X.7     2  .  .  . -2  .  2  . -2
X.8     2  .  .  A  1  . -1 -A -2
X.9     2  .  . -A  1  . -1  A -2

A = -E(3)+E(3)^2
  = -Sqrt(-3) = -i3