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Properties

Label 24T46
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 $t$ :  $46$
Group :  $C_2\times S_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (5,17)(6,18)(7,8)(9,10)(11,23)(12,24)(19,20)(21,22), (1,3)(2,4)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,15)(14,16), (1,9,18)(2,10,17)(3,11,20)(4,12,19)(5,14,22)(6,13,21)(7,16,24)(8,15,23)
$|\Aut(F/K)|$:  $8$

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: $C_2$

Degree 3: $S_3$

Degree 4: None

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

Degree 8: None

Degree 12: $S_4$, $C_2\times S_4$, $C_2 \times S_4$

Low degree siblings

6T11 x 2, 8T24 x 2, 12T21, 12T22, 12T23 x 2, 12T24 x 2, 16T61, 24T47, 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 $ $3$ $2$ $( 5,17)( 6,18)( 7, 8)( 9,10)(11,23)(12,24)(19,20)(21,22)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 2)( 3,15)( 4,16)( 5,17)( 6,18)( 7, 8)( 9,22)(10,21)(11,24)(12,23)(13,14) (19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 3)( 2, 4)( 5,12)( 6,11)( 7,21)( 8,22)( 9,19)(10,20)(13,15)(14,16)(17,24) (18,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 3)( 2, 4)( 5,24)( 6,23)( 7,22)( 8,21)( 9,20)(10,19)(11,18)(12,17)(13,15) (14,16)$
$ 4, 4, 4, 4, 4, 4 $ $6$ $4$ $( 1, 4,14,15)( 2, 3,13,16)( 5,12, 6,11)( 7,21,19, 9)( 8,22,20,10)(17,24,18,23)$
$ 4, 4, 4, 4, 4, 4 $ $6$ $4$ $( 1, 4,14,15)( 2, 3,13,16)( 5,24, 6,23)( 7,22,19,10)( 8,21,20, 9)(11,17,12,18)$
$ 6, 6, 6, 6 $ $8$ $6$ $( 1, 5, 9,14,18,22)( 2, 6,10,13,17,21)( 3, 7,11,16,20,24)( 4, 8,12,15,19,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $8$ $3$ $( 1, 5,21)( 2, 6,22)( 3, 7,12)( 4, 8,11)( 9,13,17)(10,14,18)(15,19,24) (16,20,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5,18)( 6,17)( 7,20)( 8,19)( 9,22)(10,21)(11,24) (12,23)$

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