↑

Properties

Label 8T40
Order \(192\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $Q_8:S_4$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $40$
Group :  $Q_8:S_4$
CHM label :  $1/2[2^{4}]S(4)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5)(4,8), (1,8)(2,3)(4,5)(6,7), (1,2,3)(5,6,7), (2,3)(4,8)(6,7)
$|\Aut(F/K)|$:  $2$

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

8T40, 16T444, 16T445, 24T332 x 2, 24T430 x 2, 32T2215 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 $ $1$ $1$ $()$
$ 4, 1, 1, 1, 1 $ $12$ $4$ $(3,4,7,8)$
$ 2, 2, 1, 1, 1, 1 $ $6$ $2$ $(3,7)(4,8)$
$ 2, 2, 2, 1, 1 $ $24$ $2$ $(2,3)(4,8)(6,7)$
$ 3, 3, 1, 1 $ $32$ $3$ $(2,3,4)(6,7,8)$
$ 2, 2, 2, 2 $ $12$ $2$ $(1,2)(3,4)(5,6)(7,8)$
$ 8 $ $24$ $8$ $(1,2,3,4,5,6,7,8)$
$ 6, 2 $ $32$ $6$ $(1,2,3,5,6,7)(4,8)$
$ 8 $ $24$ $8$ $(1,2,3,8,5,6,7,4)$
$ 4, 4 $ $6$ $4$ $(1,2,5,6)(3,4,7,8)$
$ 4, 2, 2 $ $12$ $4$ $(1,2,5,6)(3,7)(4,8)$
$ 4, 4 $ $6$ $4$ $(1,2,5,6)(3,8,7,4)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$

Group invariants

Order:  $192=2^{6} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [192, 1494]
Character table:    
      2  6  4  5  3  1  4  3  1  3  5  4  5  6
      3  1  .  .  .  1  .  .  1  .  .  .  .  1

        1a 4a 2a 2b 3a 2c 8a 6a 8b 4b 4c 4d 2d
     2P 1a 2a 1a 1a 3a 1a 4d 3a 4b 2d 2a 2d 1a
     3P 1a 4a 2a 2b 1a 2c 8a 2d 8b 4b 4c 4d 2d
     5P 1a 4a 2a 2b 3a 2c 8a 6a 8b 4b 4c 4d 2d
     7P 1a 4a 2a 2b 3a 2c 8a 6a 8b 4b 4c 4d 2d

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