Properties

Label 8T24
Order \(48\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_4\times C_2$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $24$
Group :  $S_4\times C_2$
CHM label :  $E(8):D_{6}=S(4)[x]2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3)(2,8)(4,6)(5,7), (2,3)(4,5), (1,8)(2,3)(4,5)(6,7), (1,2,3)(4,6,5), (1,5)(2,6)(3,7)(4,8)
$|\Aut(F/K)|$:  $2$

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 4: $S_4$

Low degree siblings

6T11 x 2, 8T24, 12T21, 12T22, 12T23 x 2, 12T24 x 2, 16T61, 24T46, 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 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1 $ $6$ $2$ $(3,8)(4,7)$
$ 3, 3, 1, 1 $ $8$ $3$ $(2,3,8)(4,7,5)$
$ 2, 2, 2, 2 $ $3$ $2$ $(1,2)(3,8)(4,7)(5,6)$
$ 4, 4 $ $6$ $4$ $(1,2,3,8)(4,7,6,5)$
$ 2, 2, 2, 2 $ $6$ $2$ $(1,4)(2,5)(3,6)(7,8)$
$ 6, 2 $ $8$ $6$ $(1,4,8,6,3,7)(2,5)$
$ 4, 4 $ $6$ $4$ $(1,4,8,5)(2,6,3,7)$
$ 2, 2, 2, 2 $ $3$ $2$ $(1,4)(2,7)(3,6)(5,8)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,6)(2,5)(3,4)(7,8)$

Group invariants

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

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