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Properties

Label 8T40
Degree $8$
Order $192$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $Q_8:S_4$

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Show commands: Magma

magma: G := TransitiveGroup(8, 40);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $40$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $Q_8:S_4$
CHM label:  $1/2[2^{4}]S(4)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
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)
magma: Generators(G);
 

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 $ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $192=2^{6} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  192.1494
magma: IdentifyGroup(G);
 
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  3 -1  . -1  1  .  1 -1 -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 -1  1  . -1  1  . -1 -1 -1  3  3
X.8      3  1 -1 -1  . -1 -1  .  1 -1  1  3  3
X.9      3  1 -1 -1  . -1  1  . -1  3  1 -1  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

magma: CharacterTable(G);