Properties

Label 24T47
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times S_4$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $47$
Group :  $C_2\times S_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,10,13,18,22)(2,5,9,14,17,21)(3,7,12,15,19,24)(4,8,11,16,20,23), (1,23,14,12)(2,24,13,11)(3,10)(4,9)(5,8,18,19)(6,7,17,20)(15,22)(16,21)
$|\Aut(F/K)|$:  $8$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
12:  $D_{6}$
24:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $S_3$, $D_{6}$ x 2, $S_4$, $S_4$, $S_4\times C_2$ x 2

Degree 8: None

Degree 12: $D_6$, $S_4$, $C_2\times S_4$, $C_2 \times S_4$ x 2, $C_2 \times S_4$ x 2

Low degree siblings

6T11 x 2, 8T24 x 2, 12T21, 12T22, 12T23 x 2, 12T24 x 2, 16T61, 24T46, 24T48 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, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 3,16)( 4,15)( 5,18)( 6,17)( 9,22)(10,21)(11,24)(12,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 2)( 3, 4)( 5,17)( 6,18)( 7, 8)( 9,10)(11,23)(12,24)(13,14)(15,16)(19,20) (21,22)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 3)( 2, 4)( 5,12)( 6,11)( 7,22)( 8,21)( 9,20)(10,19)(13,15)(14,16)(17,24) (18,23)$
$ 4, 4, 4, 4, 2, 2, 2, 2 $ $6$ $4$ $( 1, 3,14,16)( 2, 4,13,15)( 5,23)( 6,24)( 7,22,20, 9)( 8,21,19,10)(11,17) (12,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 4)( 2, 3)( 5,24)( 6,23)( 7,21)( 8,22)( 9,19)(10,20)(11,18)(12,17)(13,16) (14,15)$
$ 4, 4, 4, 4, 2, 2, 2, 2 $ $6$ $4$ $( 1, 4,14,15)( 2, 3,13,16)( 5,11)( 6,12)( 7,21,20,10)( 8,22,19, 9)(17,23) (18,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $8$ $3$ $( 1, 5,21)( 2, 6,22)( 3, 8,12)( 4, 7,11)( 9,13,17)(10,14,18)(15,20,24) (16,19,23)$
$ 6, 6, 6, 6 $ $8$ $6$ $( 1, 6,10,13,18,22)( 2, 5, 9,14,17,21)( 3, 7,12,15,19,24)( 4, 8,11,16,20,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,13)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7,19)( 8,20)( 9,21)(10,22)(11,23) (12,24)$

Group invariants

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

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

X.1      1  1  1  1  1  1  1  1  1  1
X.2      1  1 -1 -1 -1  1  1  1 -1 -1
X.3      1  1 -1  1  1 -1 -1  1 -1 -1
X.4      1  1  1 -1 -1 -1 -1  1  1  1
X.5      2  2 -2  .  .  .  . -1  1 -2
X.6      2  2  2  .  .  .  . -1 -1  2
X.7      3 -1 -1 -1  1 -1  1  .  .  3
X.8      3 -1 -1  1 -1  1 -1  .  .  3
X.9      3 -1  1 -1  1  1 -1  .  . -3
X.10     3 -1  1  1 -1 -1  1  .  . -3