Properties

Label 8T39
Order \(192\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^3:S_4$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $39$
Group :  $C_2^3:S_4$
CHM label :  $[2^{3}]S(4)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3)(2,8)(4,6)(5,7), (1,6)(2,3,5,4), (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:  $C_2$
6:  $S_3$
24:  $S_4$ x 3
96:  $V_4^2:S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $S_4$

Low degree siblings

8T39 x 5, 16T442 x 3, 24T333 x 6, 24T431 x 2, 32T2213 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 $ $6$ $2$ $(3,4)(7,8)$
$ 2, 2, 1, 1, 1, 1 $ $12$ $2$ $(3,7)(4,8)$
$ 4, 2, 1, 1 $ $24$ $4$ $(2,3,5,4)(7,8)$
$ 3, 3, 1, 1 $ $32$ $3$ $(2,3,7)(4,8,5)$
$ 2, 2, 2, 2 $ $12$ $2$ $(1,2)(3,4)(5,6)(7,8)$
$ 2, 2, 2, 2 $ $6$ $2$ $(1,2)(3,7)(4,8)(5,6)$
$ 2, 2, 2, 2 $ $6$ $2$ $(1,2)(3,8)(4,7)(5,6)$
$ 6, 2 $ $32$ $6$ $(1,2,3,6,5,4)(7,8)$
$ 4, 4 $ $24$ $4$ $(1,2,3,7)(4,8,6,5)$
$ 4, 4 $ $24$ $4$ $(1,2,3,8)(4,7,6,5)$
$ 4, 4 $ $12$ $4$ $(1,2,6,5)(3,7,4,8)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,6)(2,5)(3,4)(7,8)$

Group invariants

Order:  $192=2^{6} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [192, 1493]
Character table:   
      2  6  5  4  3  1  4  5  5  1  3  3  4  6
      3  1  .  .  .  1  .  .  .  1  .  .  .  1

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

X.1      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  1
X.3      2  2  .  . -1  .  2  2 -1  .  .  2  2
X.4      3 -1 -1  1  . -1  3 -1  . -1  1 -1  3
X.5      3 -1  1 -1  .  1  3 -1  .  1 -1 -1  3
X.6      3  3 -1 -1  . -1 -1 -1  .  1  1 -1  3
X.7      3  3  1  1  .  1 -1 -1  . -1 -1 -1  3
X.8      3 -1 -1  1  . -1 -1  3  .  1 -1 -1  3
X.9      3 -1  1 -1  .  1 -1  3  . -1  1 -1  3
X.10     4  .  2  .  1 -2  .  . -1  .  .  . -4
X.11     4  . -2  .  1  2  .  . -1  .  .  . -4
X.12     6 -2  .  .  .  . -2 -2  .  .  .  2  6
X.13     8  .  .  . -1  .  .  .  1  .  .  . -8