↑

Properties

Label 24T48
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times S_4$

Learn more about

Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $48$
Group :  $C_2\times S_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,24,4,18,21,14)(2,23,3,17,22,13)(5,7,11,10,19,16)(6,8,12,9,20,15), (1,6,12,23)(2,5,11,24)(3,8,14,19)(4,7,13,20)(9,15,22,18)(10,16,21,17)
$|\Aut(F/K)|$:  $4$

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: $S_4$

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

Degree 8: $S_4\times C_2$

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, 24T46, 24T47, 24T48

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

Group invariants

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

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