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Properties

Label 24T12
Order \(24\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_4\times S_3$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $12$
Group :  $C_4\times S_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6)(2,5)(3,15)(4,16)(7,12)(8,11)(9,21)(10,22)(13,17)(14,18)(19,23)(20,24), (1,16,5,20,9,23,13,3,18,7,22,11)(2,15,6,19,10,24,14,4,17,8,21,12)
$|\Aut(F/K)|$:  $24$

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:  $C_4\times C_2$
12:  $D_{6}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_4$ x 2, $C_2^2$

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

Degree 8: $C_4\times C_2$

Degree 12: $D_6$, $S_3 \times C_4$ x 2

Low degree siblings

12T11 x 2

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

Group invariants

Order:  $24=2^{3} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [24, 5]
Character table:    
      2  3  3   2  3  2  3  3  3  2  3   2  3
      3  1  .   1  .  1  .  1  .  1  1   1  1

        1a 2a 12a 4a 6a 2b 4b 4c 3a 2c 12b 4d
     2P 1a 1a  6a 2c 3a 1a 2c 2c 3a 1a  6a 2c
     3P 1a 2a  4b 4c 2c 2b 4d 4a 1a 2c  4d 4b
     5P 1a 2a 12a 4a 6a 2b 4b 4c 3a 2c 12b 4d
     7P 1a 2a 12b 4c 6a 2b 4d 4a 3a 2c 12a 4b
    11P 1a 2a 12b 4c 6a 2b 4d 4a 3a 2c 12a 4b

X.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
X.3      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
X.5      1 -1   A -A -1  1 -A  A  1 -1  -A  A
X.6      1 -1  -A  A -1  1  A -A  1 -1   A -A
X.7      1  1   A  A -1 -1 -A -A  1 -1  -A  A
X.8      1  1  -A -A -1 -1  A  A  1 -1   A -A
X.9      2  .  -1  . -1  .  2  . -1  2  -1  2
X.10     2  .   1  . -1  . -2  . -1  2   1 -2
X.11     2  .   A  .  1  .  B  . -1 -2  -A -B
X.12     2  .  -A  .  1  . -B  . -1 -2   A  B

A = -E(4)
  = -Sqrt(-1) = -i
B = -2*E(4)
  = -2*Sqrt(-1) = -2i