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Properties

Label 6T11
Degree $6$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_4\times C_2$

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Show commands: Magma

magma: G := TransitiveGroup(6, 11);
 

Group action invariants

Degree $n$:  $6$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $11$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_4\times C_2$
CHM label:  $2S_{4}(6) = [2^{3}]S(3) = 2 wr S(3)$
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)(2,4), (1,3,5)(2,4,6), (3,6)
magma: Generators(G);
 

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: None

Degree 3: $S_3$

Low degree siblings

6T11, 8T24 x 2, 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 $ $1$ $1$ $()$
$ 2, 1, 1, 1, 1 $ $3$ $2$ $(3,6)$
$ 2, 2, 1, 1 $ $6$ $2$ $(2,3)(5,6)$
$ 4, 1, 1 $ $6$ $4$ $(2,3,5,6)$
$ 2, 2, 1, 1 $ $3$ $2$ $(2,5)(3,6)$
$ 2, 2, 2 $ $6$ $2$ $(1,2)(3,6)(4,5)$
$ 3, 3 $ $8$ $3$ $(1,2,3)(4,5,6)$
$ 6 $ $8$ $6$ $(1,2,3,4,5,6)$
$ 4, 2 $ $6$ $4$ $(1,2,4,5)(3,6)$
$ 2, 2, 2 $ $1$ $2$ $(1,4)(2,5)(3,6)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $48=2^{4} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  48.48
magma: IdentifyGroup(G);
 
Character table:    
      2  4  4  3  3  4  3  1  1  3  4
      3  1  .  .  .  .  .  1  1  .  1

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

magma: CharacterTable(G);